Computer simulation technique the definitive introduction - harry perros

Computer Simulation Techniques:
The definitive introduction!
by

Harry Perros
Computer Science Department
NC State University
Raleigh, NC

All rights reserved 2007

TO THE READER…

This book is available for free download from my web site:
http://www.csc.ncsu.edu/faculty/perros//simulation.pdf
Self-study: You can use this book to learn the basic simulation techniques by yourself!
At the end of Chapter 1, you will find three examples. Select one of them and do the
corresponding computer assignment given at the end of the Chapter. Then, after you read
each new Chapter, do all the problems at the end of the Chapter and also do the
computer assignment that corresponds to your problem, By the time you reach the end of
the book, you will have developed a very sophisticated simulation model !
You can use any high-level programming language you like.
Errors: I am not responsible for any errors in the book, and if you do find any, I would
appreciate it if you could let me know (hp@csc.ncsu.edu).
Acknowledgment: Please acknowledge this book, if you use it in a course, or in a project,
or in a publication.
Copyright: Please remember that it is illegal to reproduce parts of this book or all of the
book in other publications without my consent.
Enjoy!
Harry Perros, August 2007

TABLE OF CONTENTS

Chapter 1.

Introduction, 1
1.1 The OR approach, 1
1.2 Building a simulation model, 2
1.3 Basic simulation methodology: Examples, 5
1.3.1
The machine interference model, 5
1.3.2
A token-based access scheme, 11
1.3.3
A two-stage manufacturing system, 18
Problems, 20
Computer assignments, 21

Chapter 2.

Generating pseudo-random numbers, 23
2.1 Introduction, 23
2.2 Pseudo-random numbers, 24
2.3 Congruential methods for generating pseudo-random numbers, 25
2.4 General congruential methods, 27
2.5 Composite generators, 28
2.6 Tausworthe generators, 28
2.7 The Mercenne Twister, 29
2.8 Statistical tests of pseudo-random number generators, 29
2.8.1
Runs test, 30
2.8.2
Chi-square test for goodness of fit, 31
Problems, 32

Chapter 3.

Generating stochastic variates, 33
3.2 The inverse transformation method, 33
3.3 Sampling from continuous probability distribution, 35
3.3.1
Sampling from a uniform distribution, 35
3.3.2
Sampling from an exponential distribution, 36
3.3.3
Sampling from an Erlang distribution, 37
3.3.4
Sampling from a normal distribution, 39
3.4 Sampling from a discrete probability distribution, 41
3.4.1
Sampling from a geometric distribution, 42
3.4.2
Sampling from a binomial distribution, 43

ii

Computer Simulation Techniques
3.4.3
Sampling from a Poisson distribution, 44
3.5 Sampling from an empirical probability distribution, 45
3.5.1
Sampling from a discrete probability distribution, 45
3.5.2
Sampling from a continuous probability distribution, 47
3.6 The Rejection Method, 49
3.7 Monte-Carlo methods, 51
Problems, 52
Solutions to the above problems, 55

Chapter 4.

Simulation designs, 59
4.2 Event-advance design, 59
4.3 Future event list, 61
4.3.1
Sequential arrays, 62
4.3.2
Linked lists, 63
4.3.3
Implementation of linked lists, 67
4.3.4
Future event list, 71
4.3.5
Doubly linked lists, 71
4.4 Unit-time advance design, 72
4.4.1
Selecting a unit-time, 75
4.4.2
Implementation, 76
4.4.3
Event-advance vs. unit-time advance, 76
4.5 Activity based simulation design, 76
4.6 Examples, 79
4.6.1
An inventory system, 79
4.6.2
round-robin queue, 82
Problems, 86

Chapter 5.

Estimation techniques for analyzing endogenously created data, 89
5.2 Collecting endogenously created data, 89
5.3 Transient state vs. steady-state simulation, 92
5.3.1
Transient-state simulation, 92
5.3.2
Steady-state simulation, 93
5.4 Estimation techniques for steady-state simulation, 94
5.4.1
Estimation of the confidence interval of the mean of a
random variable, 94
a. Estimation of the autocorrelation coefficients, 97
b. Batch means, 102
c. Replications, 103
d. Regenerative method, 105
The central limit theorem, 108
The jacknife method, 111

Table of Contents

iii
5.4.2

Estimation of other statistics of a random variable, 112
a. Probability that a random variable lies within a fixed
interval, 113
b. Percentile of a probability distribution, 114
c. Variance of the probability distribution, 115
5.5 Estimation techniques for transient state simulation, 117
5.6 Pilot experiments and sequential procedures for achieving a
required accuracy, 118
5.6.1
Independent replications, 118
5.6.2
Batch means, 119
Chapter 6.

Validation of a simulation model, 121

Chapter 7.

Variance reduction techniques, 125
7.2 The antithetic variates technique, 126
7.3 The control variates technique, 132

CHAPTER 1:
INTRODUCTION

1.1

The OR approach

The OR approach to solving problems is characterized by the following steps:
1.

Problem formulation

2.

Construction of the model

3.

Model validation

4.

Using the model, evaluate various available alternatives (solution)

5.

Implementation and maintenance of the solution

The above steps are not carried out just once. Rather, during the course of an OR
exercise, one frequently goes back to earlier steps. For instance, during model validation
one frequently has to examine the basic assumptions made in steps 1 and 2.
The basic feature of the OR approach is that of model building. Operations
Researchers like to call themselves model builders! A model is a representation of the
structure of a real-life system. In general, models can be classified as follows: iconic,
analogue, and symbolic.
An iconic model is an exact replica of the properties of the real-life system, but in
smaller scale. Examples are: model airplanes, maps, etc. An analogue model uses a set of
properties to represent the properties of a real-life system. For instance, a hydraulic
system can be used as an analogue of electrical, traffic and economic systems. Symbolic
models represent the properties of the real-life system through the means of symbols,

2


such as mathematical equations and computer programs. Obviously, simulation models
are symbolic models.
Operations Research models are in general symbolic models and they can be
classified into two groups, namely deterministic models and stochastic models.
Deterministic models are models which do not contain the element of probability.
Examples are: linear programming, non-linear programming and dynamic programming.
Stochastic models are models which contain the element of probability. Examples are:
queueing theory, stochastic processes, reliability, and simulation techniques.
Simulation techniques rely heavily on the element of randomness. However,
deterministic simulation techniques in which there is a no randomness, are not
uncommon. Simulation techniques are easy to learn and are applicable to a wide range of
problems. Simulation is probably the handiest technique to use amongst the array of OR
models. The argument goes that "when everything else fails, then simulate". That is,
when other models are not applicable, then use simulation techniques.
1.2

Building a simulation model

Any real-life system studied by simulation techniques (or for that matter by any other OR
model) is viewed as a system. A system, in general, is a collection of entities which are
logically related and which are of interest to a particular application. The following
features of a system are of interest:
•

Environment: Each system can be seen as a subsystem of a broader system.

•

Interdependency: No activity takes place in total isolation.

•

Sub-systems: Each system can be broken down to sub-systems.

•

Organization: Virtually all systems consist of highly organized elements or
components, which interact in order to carry out the function of the system.

•

Change: The present condition or state of the system usually varies over a long
period of time.

Introduction

3

When building a simulation model of a real-life system under investigation, one
does not simulate the whole system. Rather, one simulates those sub-systems which are
related to the problems at hand. This involves modelling parts of the system at various
levels of detail. This can be graphically depicted using Beard's managerial pyramid as
shown in Figure 1.1. The collection of blackened areas form those parts of the system
that are incorporated in the model.

Levels of
Detail

Figure 1.1: Beard's managerial pyramid

A simulation model is, in general, used in order to study real-life systems which
do not currently exist. In particular, one is interested in quantifying the performance of a
system under study for various values of its input parameters. Such quantified measures
of performance can be very useful in the managerial decision process. The basic steps
involved in carrying out a simulation exercise are depicted in Figure 1.2.
All the relevant variables of a system under study are organized into two groups.
Those which are considered as given and are not to be manipulated (uncontrollable
variable) and those which are to be manipulated so that to come to a solution
(controllable variables). The distinction between controllable and uncontrollable
variables mainly depends upon the scope of the study.

4


a

Define the Problem

Design Simulation
Experiments

Analyze Data

(alternatives)

Run the Simulator

Formulate Sub-Models

Analyze the Results

Combine Sub-Models

Implement Results

Collect Data

Write the Simulation
Program

Debug

Validate Model

Earlier
Steps

a

Figure 1.2: Basic steps involved in carrying out a simulation study.

Another characterization of the relevant variables is whether they are affected or
not during a simulation run. A variable whose value is not affected is called exogenous. A
variable having a value determined by other variables during the course of the simulation
is called endogenous. For instance, when simulating a single server queue, the following
variables may be identified and characterized accordingly.
Exogenous variables
1. The time interval between two successive arrivals.
2. The service time of a customer.
3. Number of servers.
4. Priority discipline.

Introduction

5

Endogenous variables
1. Mean waiting time in the queue.
2. Mean number of customers in the queue.
The above variables may be controllable or uncontrollable depending upon the
experiments we want to carry out. For instance, if we wish to find the impact of the
number of servers on the mean waiting time in the queue, then the number of servers
becomes an controllable variable. The remaining variables-the time interval between two
arrivals and the service time, will remain fixed. (uncontrollable variables)
Some of the variables of the system that are of paramount importance are those
used to define the status of the system. Such variables are known as status variables.
These variables form the backbone of any simulation model. At any instance, during a
simulation run, one should be able to determine how things stand in the system using
these variables. Obviously, the selection of these variables is affected by what kind of
information regarding the system one wants to maintain.
We now proceed to identify the basic simulation methodology through the means
of a few simulation examples.
1.3

Basic simulation methodology: Examples

1.3.1

The machine interference problem
Let us consider a single server queue with a finite population known as the

machine interference problem. This problem arose originally out of a need to model the
behavior of machines. Later on, it was used extensively in computer modelling. Let us
consider M machines. Each machine is operational for a period of time and then it breaks
down. We assume that there is one repairman. A machine remains broken down until it is
fixed by the repairman. Broken down machines are served in a FIFO manner, and the
service is non-preemptive. Obviously, the total down time of a machine is made up of the
time it has to "queue" for the repairman and the time it takes for the repairman to fix it. A

6


machine becomes immediately operational after it has been fixed. Thus, each machine
follows a basic cycle as shown in figure 1.3, which is repeated continuously.

Operational

Broken Down

Queueing
Time

Repair
Time

Figure 1.3: The basic cycle of a machine.

In general, one has information regarding the operational time and the repair time
of a machine. However, in order to determine the down time of a machine, one should be
able to calculate the queueing time for the repairman. If this quantity is known, then one
can calculate the utilization of a machine. Other quantities of interest could be the
utilization of the repairman.
Repairman
Finite
Population
of Machines

Figure 1.4: The machine interference problem.

Let us now look at the repairman's queue. This can be visualized as a single server
queue fed by a finite population of customers, as shown in figure 1.4
For simplicity, we will assume that the operational time of each machine is equal
to 10 units of time. Also, the repair time of each machine is equal to 5 units of time. In
other words, we assume that all the machines have identical constant operational times.

Introduction

7

They also have identical and constant repair times. (This can be easily changed to more
complicated cases where each machine has its own random operational and repair times.)
The first and most important step in building a simulation model of the above
system, is to identify the basic events whose occurrence will alter the status of the
system. This brings up the problem of having to define the status variables of the above
problem. The selection of the status variables depends mainly upon the type of
performance measures we want to obtain about the system under study.

A machine
breaks down

Repairman
busy
?

yes

Join the
queue

Repairman
becomes busy

Repair
starts

Figure 1.5: An arrival event.

In this problem, the most important status variable is n, the number of broken
down machines, i.e., those waiting in the queue plus the one being repaired. If n=0, then
we know that the queue is empty and the repairman is idle. If n=1, then the queue is
empty and the repairman is busy. If n>1, then the repairman is busy and there are n-1
broken down machines in the queue. Now, there are two events whose occurrence will
cause n to change. These are:
a) A machine breaks down, i.e., an arrival occurs at the queue.

8

b) A machine is fixed, i.e., a departure occurs from the queue.
The flow-charts given in figures 1.5, and 1.6 show what happens when each of

these events occur.

A machine
is repaired

Other
machines to be
repaired?

no

Repairman
becomes
idle

A new repair
starts

Figure 1.6: A departure event.

In order to incorporate the above two basic events in the simulation model, we
need a set of variables, known as clocks, which will keep track of the time instants at
which an arrival or departure event will occur. In particular, for this specific model, we
need to associate a clock for each machine. The clock will simply show the time instant
at which the machine will break down, i. e., it will arrive at the repairman's queue.
Obviously, at any instance, only the clocks of the operational machines are of interest. In
addition to these clocks, we require to have another clock which shows the time instant at
which a machine currently being repaired will become operational, i.e., it will cause a
departure event to occur. Thus, in total, if we have m machines, we need m+1 clocks.
Each of these clocks is associated with the occurrence of an event. In particular, m clocks
are associated with m arrival events and one clock is associated with the departure event.

Introduction

9

In addition to these clocks, it is useful to maintain a master clock, which simply keeps
tracks of the simulated time.
The heart of the simulation model centers around the manipulation of these
events. In particular, using the above clocks, the model decides which of all the possible
events will occur next. Then the master clock is advanced to this time instant, and the
model takes action as indicated in the flow-charts given in figures 1.5 and 1.6. This event
manipulation approach is depicted in figure 1.7.

a

Choose
next event

A machine
Breaks down

Queue
empty?

n
o

A machine
is repaired

Create
operation time

a

ye
s

Queue
empty?

Generate a
new service

ye
s

a

n
o
Generate a

a

new service

a

Figure 1.7: Event manipulation.

We are now ready to carry out the hand simulation shown below in table 1. Let us
assume that we have 3 machines. Let CL1, CL2, and CL3 be the clocks associated with
machine 1, 2, and 3 respectively (arrival event clocks). Let CL4 be the clock associated
with the departure event. Finally, let MC be the master clock and let R indicate whether

10


the repairman is busy or idle. We assume that at time zero all three machines are
operational and that CL1=1, CL2=4, CL3=9. (These are known as initial conditions.)

MC

CL1

CL2

CL3

CL4

n

0

1

4

9

-

0

idle

1

-

4

9

6

1

busy

4

-

-

9

6

2

busy

6

16

-

9

11

1

busy

9

16

-

-

11

2

busy

11

16

21

-

16

1

busy

16

-

21

26

21

1

busy

Table 1: Hand simulation for the machine interference problem

We note that in order to schedule a new arrival time we simply have to set the
associated clock to MC+10. Similarly, each time a new repair service begins we set
CL4=MC+5. A very important aspect of this simulation model is that we only check the
system at time instants at which an event takes place. We observe in the above hand
simulation that the master clock in the above example gives us the time instants at which
something happened (i.e., an event occurred). These times are: 0, 1, 4, 6, 9, 11, 16, ... We
note that in-between these instants no event occurs and, therefore, the system's status
remains unchanged. In view of this, it suffices to check the system at time instants at
which an event occurs. Furthermore, having taken care of an event, we simply advance
the Master clock to the next event which has the smallest clock time. For instance, in the
above hand simulation, after we have taken care of a departure of time 11, the simulation
will advance to time 16. This is because following the time instant 11, there are three
events that are scheduled to take place in the future. These are: a) arrival of machine 1 at
time 16; b) arrival of machine 2 at time 21; and c) a departure of machine 3 at time 16.
Obviously, the next event to occur is the latter event at time 16.

Introduction

11

The above simulation can be easily done using a computer program. An outline of
the flow-chart of the simulation program is given in figure 1.8. The actual
implementation of this simulation model is left as an exercise.

Initiate
simulation

MC=0, CL1=1, CL2=4
CL3=9, n=0, R=0 (idle)

A

Next
event

Arrival
(ith machine)
MC=CLi

n=0 ?

Departure
(ith machine)
MC=CLi

No

n=n-1
n=n+1

yes

n=0 ?

Yes

R=0

R=1
No
CL4=MC+5

CL4=MC+5

A

A

CLj=MC+10

A

Figure 1.8: A flowchart of the simulation program.

1.3.2

A token-based access scheme

We consider a computer network consisting of a number of nodes interconnected via a
shared wired or wireless transport medium, as shown in figure 1.9. Access to the shared
medium is controlled by a token. That is, a node cannot transmit on the network unless it
has the token. In this example, we simulate a simplified version of such an access
scheme. Below, we describe how such a token-based access scheme operates.

12


Figure 1.9: Nodes interconnected by a shared medium

There is a single token that visits the nodes in a certain logical sequence. The nodes are
logically connected so that they form a logical ring. In the case of a bus-based or ringbased wireline medium, the order in which the nodes are logically linked may not be the
same with the order in which they are attached to the network. We will assume that the
token never gets lost. A node cannot transmit unless it has the token. When a node
receives the token, from its previous logical upstream node, it may keep it for a period of
time up to T. During this time, the node transmits packets. A packet is assumed to consist
of data and a header. The header consists of the address of the sender, the address of the
destination, and various control fields. The node surrenders the token when: a) time T
has run out, or b) it has transmitted out all the packets in its queue before T runs out, or
c) it receives the token when it has no packets in its queue to transmit. If time T runs out
and the node is in the process of transmitting, it will complete the transmission and then it
will surrender the token. Surrendering the token means, that the node will transmit it to
its next downstream logical neighbour.

token

...
Figure 1.10. The conceptual queueing system.

Introduction

13

Conceptually, this network can be seen as comprising of a number of queues, one
per node. Only the queue that has the token can transmit packets. The token can be seen
as a server, who cyclically switches between the queues, as shown in figure 1.10. Once
the token is switched to a queue, packets waiting in this queue can be transmitted on the
network. The maximum time that a queue can keep the token is T units of time, as
explained above. The time it takes for the token to switch from one queue to the next is
known as switch-over time.
It is much simpler to use the queueing model given in figure 1.10 when
constructing the simulation model of this access scheme. The following events have to be
taken into account in this simulation. For each queue, there is an arrival event and service
completion event. For the token, there is a time of arrival at the next queue event and a
time-out event. For each queue, we keep track of the time of arrival of the next packet,
the number of customers in the queue, and the time a packet is scheduled to depart, if it is
being transmitted. For the token, we keep track of the time of arrival at the next queue,
the number of the queue that may hold the token, and the time when the token has to be
surrendered to the next node, known as time-out.
In the hand simulation given below we assume that the network consists of three
nodes. That is, the queueing system in figure 1.10 consists of three queues. The interarrival times to queues 1, 2, and 3 are constant and they are equal to 10, 15, and 20 unit
times respectively. T is assumed to be equal to 15 unit times. The time it takes to transmit
a packet is assumed to be constant equal to 6 unit times. The switch over time is equal to
1 unit time. For initial conditions we assume that the system is empty at time zero. The
first arrival to queues 1, 2, and 3 will occur at time 2, 4 and 6 respectively. Also, at time
zero, the token is in queue 1. In case when an arrival and a departure occur
simultaneously at the same queue, we will assume that the arrival occurs first.
The logic for each of the events in this simulation is summarized in figures 1.11 to
1.14. The set of all possible events that can occur at a given time in a simulation is known
as the event list. This event-list has to be searched each time in order to locate the next
event. This event manipulation forms the basis of a simulation, and it is summarized in a
flow-chart given in figure 1.15.

14


Arrival
occurs

MC=ACi

Join the
queue

qi = qi + 1

Schedule
Next Arrival
Arrivals

ATi = MC + new
inter-arrival time

Return

Figure 1.11: Arrival event at queue i.

Service is
completed

Departs from
queue

MC = DT i

q = q-1

yes

Queue
empty?

Pass
token

no
Schedule arrival
time

Token
time out
?

yes

H = (H+1) mod3

ANH =
MC + switch over time

Return

no
Schedule new
service

DTi = MC + new service time

Return
Figure 1.12: Service completion at queue i.

Introduction

15

Time-out
occurs

This means that
node is still
transmitting.

Raise a
flag

Return

Figure 1.13 : Time-out of token

Arrival of token
at next queue

Queue
empty
?

MC = ANH

yes

no
TOUT = MC + T

Schedule
time-out

Pass
token
Schedule
next arrival

yes

DT4 i= MC ++ Schedule next
DT = MC
service
(new service time)
(new service time)
completion

Return

Figure 1.14 : Arrival of token at next queue.

H = (H + 1) mod3

ANH = MC +
(switch over time)

16


Initialize
simulation

Locate
next event

Take
appropriate
action

i. e. search event list for
smallest number

i. e. branch to the appropriate part of the
program (or procedure)

If new events
are scheduled
update next
event

no

Is
simulation
over
?
yes
End

Figure 1.15 : Event manipulation.

The following variables represent the clocks used in the flow-charts:

•

MC: Master clock

•

ATi: Arrival time clock at queue i, i=1,2,3

•

DTi: Departure time clock from queue i, i=1,2,3

•

TOUT: Time out clock for token

•

ANH: Arrival time clock of token to next queue

The hand simulation is given in table 2.

Introduction

Queue

17

1

Queue

Depart Queue
Clock
size

Arr
Clock

2

Queue

Depart Queue
Clock
size

Toke

Depart Queue
Clock size

n
Node Time Arriva
No
Out l Next
Clock Node

MC

Arr
Clock

0

2

0

4

0

6

0

1

1

1

2

0

4

0

6

0

2

2

2

12

1

4

0

6

0

3

3

3

12

9

1

4

0

6

0

1

18

4

12

9

1

19

1

6

0

1

18

6

12

9

1

19

1

26

1

1

18

9

12

0

19

1

26

1

1

10

12

0

19

16

1

26

1

2

25

12

22

1

19

16

1

26

1

2

25

16

22

1

19

0

26

1

2

17

22

1

19

0

26

23

1

3

32

19

22

1

34

1

26

23

1

3

32

22

32

2

34

1

26

23

1

3

32

23

32

2

34

1

26

0

3

24

32

30

2

34

1

26

0

1

39

26

32

30

2

34

1

46

1

1

39

30

32

36

1

34

1

46

1

1

39

32

42

36

2

34

1

46

1

1

39

34

42

36

2

49

2

46

1

1

39

36

42

42

1

49

2

46

1

1

39

39

42

42

1

49

2

46

1

1

*

42

52

42

2

49

2

46

1

1

*

42

52

1

49

2

46

1

1

43

52

1

49

2

46

1

2

49

Arr
Clock

3

Table 2: Hand simulation for the token-based access scheme

10

17

24

43
58

18

1.3.3


A two-stage manufacturing system

Let us consider a two-stage manufacturing system as depicted by the queueing network
shown in figure 1.16. The first queue has an infinite capacity, and the second queue has a
finite capacity. When the second queue becomes full, the server at the first queue stops.
In particular, upon service completion at server 1, the server gets blocked if the second
queue is

Figure 1.16: A two-stage queueing network.

full. Server 1 will remain blocked until a customer departs from the second queue. Each
server may break down. For simplicity, we will assume that a server may break down
whether it is busy or idle. A broken down server cannot provide service until it is
repaired. If a customer was in service when the breakdown occurred, the customer may
resume its service after the server is repaired without any loss. That is, it will continue
from where it stopped when the breakdown occurred.
In the hand-simulation given in table 3, it is assumed that the buffer capacity of
the second queue is 3 (this includes the customer in service). All service times, interarrival times, operational and repair times are constant with the following values: interarrival time = 40, service time at node 1 = 20, service time at node 2 = 30, operational for
server 1 = 200, operational time for server 2 = 300, repair time for server 1 = 50, and
repair time for server 2 = 150. Initially the system is assumed to be empty. The first
arrival occurs at time 10, server 1 will break down for the first time at time 80, and server
2 at time 90.

Introduction

19

MC

AC

#
Cust

10

50

1

30

50

0

50

90

1

60

90

1

70

90

0

80

90
90

90

30

Stage 1
Oper
Rep
Clock Clock

0

90

Server
Clock

Server
Status

# Cust

80

busy

80

idle

1

70

80

busy

1

70

80

Server
Clock

Rep
Clock

Server
Status

90

idle

60

90

busy

60

90

busy

90

idle

0

80

Stage 2
Oper
Clock

idle

1

90

90

busy

130

down

1

90

90

busy

0

130

down

0

90

idle

130

1

130

down

0

90

idle

90

130

1

130

down

0

240

down

130

170

2

130

down

0

240

down

130

170

2

150

330

busy

0

240

down

150

170

1

170

330

busy

1

240

down

170

210

2

170

330

busy

1

240

down

170

210

1

190

330

busy

2

240

down

190

210

0

330

idle

3

240

down

210

250

1

330

busy

3

240

down

230

250

1

330

blocked

3

240

down

240

250

1

330

blocked

3

270

540

busy

250

290

2

330

blocked

3

270

540

busy

270

290

1

290

330

busy

3

300

540

busy

290

330

2

290

330

busy

3

300

540

busy

290

330

2

330

blocked

3

300

540

busy

300

330

1

busy

3

330

540

busy

320

330

1

blocked

3

330

540

busy

330

370

2

330

blocked

3

330

540

busy

330

370

1

330

busy

3

360

540

busy

330

370

1

380

down

360

370

1

380

down

2

390

540

busy

370

410

2

380

down

2

390

540

busy

380

410

2

400

580

busy

2

390

540

busy

390

410

2

400

580

busy

1

420

540

busy

230

320

350

busy

20


Table 3: hand simulation for the two-stage manufacturing system

Since we are dealing with integer numbers, it is possible that more than one clock may
have the same value. That is, more than one event may occur at the same time. In this
particular simulation, simultaneous events can be taken care in any arbitrary order. In
general, however, the order with which such events are dealt with may matter, and it has
to be accounted for in the simulation. In a simulation, typically, clocks are represented by
real numbers. Therefore, it is not possible to have events occurring at the same time.
Problems
1.

Do the hand simulation of the machine interference problem, discussed in section
1.3.1, for the following cases:
a. Vary the repair and operational times.
b. Vary the number of repairmen.
c. Assume that the machines are not repaired in a FIFO manner, but according to
which machine has the shortest repair time.

2.

Do the hand simulation of the token-based access scheme, described in section
1.3.2, for the following cases:
a. Vary the inter-arrival times.
b. Vary the number of queues.
c. Assume that packets have priority 1 or 2 (1 being the highest). The packets in a
queue are served according to their priority. Packets with the same priority are
served in a FIFO manner.

3.

Do the hand simulation of the two-stage manufacturing system, described in
section 1.3.3, for the following cases:
a. Assume no breakdowns.

Introduction

21

b. Assume a three-stage manufacturing system. (The third stage is similar to the
second stage.)
Computer assignments
1.

Write a computer program to simulate the machine interference problem as
described in section 1.3.1. Each time an event occurs, print out a line of output to
show the current values of the clocks and of the other status parameters (as in the
hand simulation). Run your simulation until the master clock is equal to 20.
Check by hand whether the simulation advances from event to event properly, and
whether it updates the clocks and the other status parameters correctly.

2.

Write a computer program to simulate the token bus as described in section 1.3.2.
Each time an event occurs, print out a line of output to show the current values of
the clocks and of the other status parameters (as in the hand simulation). Run your
simulation until the master clock is equal to 100. Check by hand whether the
simulation advances from event to event properly, and whether it updates the
clocks and the other status parameters correctly.

3.

Write a computer program to simulate the two-stage manufacturing system as
described in section 1.3.3. Each time an event occurs, print out a line of output to
show the current values of the clocks and of the other status parameters (as in the
hand simulation). Run your simulation until the master clock is equal to 500.
Check by hand whether the simulation advances from event to event properly, and
whether it updates the clocks and the other status parameters correctly.

CHAPTER 2:
GENERATING PSEUDO-RANDOM NUMBERS

2.1

Introduction

We shall consider methods for generating random number uniformly distributed. Based
on these methods, we shall then proceed in the next Chapter to consider methods for
generating random numbers which have a certain distribution, i.e., exponential, normal,
etc.
Numbers which are chosen at random are useful in a variety of applications. For
instance, in numerical analysis, random numbers are used for solving complicated
integrals. In computer programming, random values make a good source of data for
testing the effectiveness of computer algorithms. In simulation, random numbers are used
in order to represent real-life situations.
Let us consider for a moment the machine interference simulation model
discussed in the previous Chapter. In this model it was assumed that the operational time
of a machine was constant. Also, it was assumed that the repair time of a machine was
constant. It is possible that one may identify real-life situations where these two
assumptions are valid. However, in most of the cases one will observe that the time a
machine remains operational varies. Also, the repair time may vary from machine to
machine. If we are able to observe the behavior of a machine over a reasonably long
period, we will find that the operational times can be characterized by a theoretical or an
empirical probability distribution. Similarly, the repair times can be also characterized by
a theoretical or empirical distribution. Therefore, in order to make the simulation model
more realistic, one would require a built-in mechanism which will produce random
numbers which follow the given distributions. Now, in order to generate such random

24


numbers one needs to be able to generate uniformly distributed random numbers,
otherwise known as pseudo-random numbers. The generation of pseudo-random numbers
is the subject matter of this Chapter.
2.2

Pseudo-random numbers

In a sense, there is no such a thing as a single random number. Rather, we speak of a
sequence of random numbers which follow a specified distribution. Each number in the
sequence has been obtained merely by chance, having nothing to do with other numbers
of the sequence, and each number has a specified probability of falling in any given
range.
Uniformly distributed random numbers in the space [0,1] are usually referred to
as random numbers, whereas random numbers following any other distribution are
referred to as random variates or stochastic variates.
Historically, the first method for creating random numbers by computer was Von
Neuman's mid-square method. His idea was to take the square of the previous random
number and to extract the middle digits. For example, let us assume that we are
generating 10-digit numbers and that the previous value was 5772156649. The square of
this value is 33317792380594909291 and the next number is 7923805949. The question
here that arises is how such a method can give a sequence of random numbers. Well, it
does not, but it appears to be!
The mid-square method was relatively slow and statistically unsatisfactory. It was
later abandoned in favour of congruential methods. At the present time, nearly all
computer codes for generating random numbers use some variation of the congruential
method. As will be seen, this method generates random numbers in a deterministic
manner. However, a sequence of such random numbers appear to be statistically random
Because of that, they are often referred to as pseudo-random numbers.
In general, an acceptable method for generating pseudo-random numbers must
yield sequences which are:
1.

uniformly distributed

Generating Pseudo-random numbers
2.

reproducible, and

4.
2.3

statistically independent

3.

25

non-repeating for any desired length.

Congruential methods for generating pseudo-random numbers

These methods are completely deterministic. Formulae are available for calculating in
advance the exact value of the ith number in a sequence of numbers, before the sequence
is actually generated. Numbers generated in this manner are not random. However, we
treat them as if they were random as long as they pass a certain number of statistical tests
designed to test various properties of random numbers. In particular, if it can be shown
that a sequence of numbers generated using the congruential method are uniformly
distributed and statistically independent, then these numbers can be assumed to be
random (even if they were created in a deterministic fashion).
Congruential methods use the following recursive relationship:
xi+1 = axi + c (mod m)
where xi, a, c and m are all non-negative numbers. Given that the previous random
number was xi, the next random number xi+1 can be generated as follows. Multiply xi by
a and then add c. Then, compute the modulus m of the result. That is, divide the result by
m and set xi+1 equal to the remainder of this division. For example, if x0 = a = c = 7, m =
10 then we can obtain the following sequence of numbers: 7,6,9,0,7,6,9,0,...
The method using the above expression is known as the mixed congruential
method. A simpler variation of this method is the multiplicative congruential method.
This method utilizes the relation xi+1 = axi(mod m). Historically, multiplicative
congruential generators came before the mixed congruential generators. Below we limit
our discussion to mixed congruential generators.
The numbers generated by a congruential method are between 0 and m-1. Quite
frequently in simulation, one needs uniformly distributed random numbers between 0 and

26


1. Such random numbers can be easily obtained from the congruential method by simply
dividing each xi by m.
The number of successively generated pseudo-random numbers after which the
sequence starts repeating itself is called the period. If the period is equal to m, then the
generator is said to have a full period. Theorems from number theory show that the
period depends on m. The larger the value of m, the larger the period. In particular, the
following conditions on a, c, and m guarantee a full period:
1.

m and c have no common divisor.

2.

a = 1 (mod r) if r is a prime factor of m. That is, if r is a prime number
(divisible only by itself and 1) that divides m, then it divides a-1.

3.

a = 1 (mod 4) if m is a multiple of 4.

It is important to note that one should not use any arbitrary values for a, c and m.
Systematic testing of various values for these parameters have led to generators which
have a full period and which are statistically satisfactory. A set of such values is: a =
314, 159, 269, c = 453, 806, 245, and m = 231 (for a 32 bit machine).
In order to get a generator started, we further need an initial seed value for x. It
will become obvious later on that the seed value does not affect the sequence of
generated pseudo-random numbers in the long run.
The implementation of a pseudo-random number generator involves a
multiplication, an addition and a division. A division, being comparatively a slow
operation, can be avoided if m is set equal to the size of the computer word. For, if the
total numerical value of the expression axi+c is less than the word size, then it is in itself
the result of the operation axi+c (mod m), where m is set equal to the word size. Now, let
us assume that a, xi and c have values such that the expression axi+c will give a number
greater than the word size. In this case, when the calculation is performed, an overflow
will occur. If the overflow does not cause the execution of the program to be aborted, but
it simply causes the significant digits to be lost, then the remaining digits left in the
register is the remainder of the division (axi+c)/m. This is because the lost significant


27

digits will represent multiples of the value of m, which is the quotient of the above
division.
In order to demonstrate the above idea, let us consider a fictitious decimal
calculator whose register can accommodate a maximum of 2 digits. Obviously, the
largest number that can be held in the register is 99. Now, we set m equal to 100. For
a=8, x=2, and c=10, we have that axi + c = 26, and 26 (mod 100) = 26. However, if x=20,
then we have that axi + c = 170. If this computation is performed in the above fictitious
calculator, the product axi (which is equal to 8x20) will cause an overflow to occur. The
first significant digit will be lost and thus the register will contain the number 60. If we
now add c (which is equal to 10) to the above result we will obtain 70, which is, the
remaining of the division 170/100.
2.4

General congruential methods

The mixed congruential method described above can be thought of as a special case of a
following generator:
xi+1 = f(xi, xi-1, ...) (mod m),
where f(.) is a function of previously generated pseudo-random numbers. A special case
of the above general congruential method is the quadratic congruential generator. This
has the form:
2

xi+1=a1xi + a2xi-1+ c.
The special case of a1=a2=1, c=0 and m being a power of 2 has been found to be related
to the midsquare method. Another special case that has been considered is the additive
congruential method, which is based on the relation
f(xi, xi-1, ..., xi-k) = a1xi + a2xi-1 + ... akxi-k .

28


The case f(xi, xi-1)=xi+xi-1 has received attention.
2.5

Composite generators

These methods were developed by combining two separate generators (usually
congruential generators). By combining separate generators, one hopes to achieve better
statistical behaviour than either separate generator.
The best known of the composite generators uses the second congruential
generator to shuffle the output of the first congruential generator. In particular, the first
generator is used to fill a vector of size n with its first k generated random numbers. The
second generator is then used to generate a random integer r uniformly distributed over
the numbers 1, 2, …, k. The random number stored in the rth position of the vector is
returned as the first random number of the composite generator. The first generator then
replaces the random number in the rth position with a new random number. The next
random number that will be returned by the composite generator, is the one selected by
the second generator from the updated vector of random numbers. The procedure repeats
itself in this fashion. It has been demonstrated that such a combined generator has good
statistical properties, even if two separate generators used are bad.
2.6

Tausworthe generators

Tausworthe generators are additive congruential generators obtained when the modulus
m is equal to 2. In particular,
xi = (a1xi-1 + a2xi-2 + ...+ anxi-n) (mod 2)
where xi can be either 0 or 1. This type of generator produces a stream of bits {bi}. In
view of this, it suffices to assume that the coefficients ai are also binary. Thus, xi is
obtained from the above expression by adding some of preceding bits and then carrying


29

out a modulo 2 operation. This is equivalent to the exclusive OR operation, notated as ⊕
and defined by the following truth table.
A
0
0
1
1

B
0
1
0
1

A⊕B
0
1
1
0

A⊕B is true (i.e. equal to 1), when either A is true and B false, or A is false and B true.
The generated bits can be put together sequentially to form an -bit binary integer
between 0 and 2 -1. Several bit selection techniques have been suggested in the
literature.
In the composite generator scheme discussed earlier on, one of the generators (but
not both) could be a Tausworthe generator.
Tausworthe generators are independent of the computer used and its word size
and have very large cycles. However, they are too slow since they only produce bits. A
fast variation of this generator is the trinomial-based Tausworthe generator. Two or more
such generators have to be combined in order to obtain statistically good output.
2.7

The Mercenne Twister

This is a newer algorithm that has created a great deal of excitement. Its output has
excellent statistical properties and its period is very long, i.e., 219937-1. (219937-1 is a prime
number and it is also a Mersenne prime, since it is one less than a power of two.) The
algorithm uses a seed value which is 19,937 bits long, and is essentially a large linearfeedback shift register.
2.8

Statistical tests of pseudo-random number generators

Pseudo-random number generation techniques are completely deterministic. Numbers
generated in this manner are not random. However, we can treat them as random as long

30


as they pass a certain number of statistical tests. A sequence of pseudo-random numbers
are treated as random if statistically they are uniformly distributed and independent of
each other.
2.8.1

Runs test

This statistical test can be used to test the assumption that the pseudo-random numbers
are independent of each other. We start with a sequence of pseudo-random numbers
between 0 and 1. We then look for unbroken subsequences of numbers, where the
numbers within each subsequence are monotonically increasing. Such a subsequence is
called a run up, and it may be as long as one number.
For example, let us consider the sequence: 0.8, 0.7, 0.75, 0.55, 0.6, 0.7, 0.3,
0.4, 0.5. Starting from the beginning of this sequence (in this case, starting from the left),
we find a run up of length 1, i.e. 0.8, then a run up of length 2, i.e. 0.7, 0.75, followed by
two successive run ups of length 3, i.e. 0.55, 0.6, 0.7, and 0.3, 0.4, 0.5.
In general, let ri be the number of run ups of length i. (In the above example we
have r1=1, r2=1, r3=2.) All run-ups with a length i> 6, are grouped together into a single
_
run-up. The ri values calculated for a particular sequence are then used to calculate the
following statistic:
1
R = n ∑ (ri-nbi)(rj-nbj)aij , 1≤i≤6, 1≤j≤6
1≤i,j≤6

where n is the sample size and bi, i=1,...6, and aij are known coefficients. The aij
coefficient is obtained as the (i,j)th element of the matrix






4529.4 9044.9 13568 18091 22615 27892
9044.9 18097 27139 36187 45234 55789
13568 27139 40721 54281 67852 83685
18091 36187 54281 72414 90470 111580 ,
22615 45234 67852 90470 113262 139476
27892 55789 83685 111580 139476 172860


31

and the bi coefficient is obtained as the ith element of the vector

1 5 11 19
29
1
(b1,...,b6) = ( 6 , 24 , 120 , 720 , 5040 , 840 ) .

For n > 4000, R has a chi-square distribution (see below) with 6 degrees of freedom
_
(d.f). under the null hypothesis that the random numbers are independent and identically
distributed (i.i.d).
2.8.2

Chi-square test for goodness of fit

Having tested the independence assumption using the above test, we now proceed to
check the uniform distribution assumption. The chi-square test, in general, can be used to
check whether an empirical distribution follows a specific theoretical distribution. In our
case, we are concerned about testing whether the numbers produced by a generator are
uniformly distributed.
Let us consider a sequence of pseudo-random numbers between 0 and 1. We
divide the interval [0,1] into k subintervals of equal length, where k > 100. Let fi be the
number of pseudo-random numbers that fall within the ith subinterval. (Make sure that
enough random numbers are generated so that fi > 5.) The fi values are called the
observed values. Now, if these generated random numbers are truly uniformly
distributed, then the mean number of random numbers that fall within each subinterval is
n/k, where n is the sample site. This value is called the theoretical value. The chi-square
test measures whether the difference between the observed and the theoretical values is
due to random fluctuations or due to the fact that the empirical distribution does not
follow the specific theoretical distribution. For the case where the theoretical distribution
is the uniform distribution, the chi-square statistic is given by the expression

χ2 =

!

k k
n
# ( fi " )2 ,
n i=1
k

32


and it has k-1 degrees of freedom. The null hypothesis is that the generated random
numbers are i.i.d. uniformly distributed in [0,1]. This hypothesis is rejected if the
computed value of χ2 is greater than the one obtained from the chi-square tables for k-1
degrees of freedom and 1-a level of significance. (The chi-square tables can be found in
any introductory Statistics book, and of course they are also accessible through the
Internet!)
Problems
1.

Consider the multiplicative congruential method for generating random digits.
Assuming that m=10, determine the length of the cycle for each set of values of a
and x0 given below.
(a) a = 2, x0 =1, 3, 5.
(b) a = 3, x0 = 1, 2, 5.

Computer Assignments
1.

Use the two statistical tests described in section 2.7, to test a random number
generator available at your computer.

CHAPTER 3:
GENERATING STOCHASTIC VARIATES

3.1

Introduction

In the previous Chapter we examined techniques for generating random numbers. In this
Chapter, we discuss techniques for generating random numbers with a specific
distribution. Random numbers following a specific distribution are called random
variates or stochastic variates. Pseudo-random numbers which are uniformly distributed
are normally referred to as random numbers. Random numbers play a major role in the
generation of stochastic variates.
There are many techniques for generating random variates. The inverse
transformation method is one of the most commonly used techniques. This is discussed
below. Sections 3.3 and 3.4 give methods for generating stochastic variates from known
continuous and discrete theoretical distributions. Section 3.5 discusses methods for
obtaining stochastic variates from empirical distributions. Section 3.6 describes an
alternative method for stochastic variates generation known as the rejection method.
3.2

The inverse transformation method

This method is applicable only to cases where the cumulative density function can be
inversed analytically. Assume that we wish to generate stochastic variates from a
probability density function (pdf) f(x). Let F(x) be its cumulative density function. We
note that F(x) is defined in the region [0,1]. We explore this property of the cumulative
density function to obtain the following simple stochastic variates generator.

34


We first generate a random number r which we set equal to F(x). That is, F(x) =
r. The quantity x is then obtained by inverting F. That is, x = F-1(r), where F-1(r)
indicates the inverse transformation of F.
As an example, let us assume that we want to generate random variates with
probability density function
f(x) = 2x, 0 < x < 1.
_ _
A graphical representation of this probability density function is given in figure 3.1a. We
first calculate the cumulative density function F(x). We have
x
⌠ 2tdt
F(x) = ⌡
0
= x2,

0 < x < 1.
_ _

Let r be a random number. We have
r = x2,
or
x= r.

2
1
f(x)

F(x) r

0
x

1

Figure 3.1a: pdf f(x).

0
x

1

Figure 3.1b: Inversion of F(x).

Generating stochastic variates

35

This inversion is shown graphically in figure 3.1b.
In sections 3 and 4 we employ the inverse transformation method to generate
random variates from various well-known continuous and discrete probability
distributions.
3.3

Sampling from continuous probability distributions

In this section, we use the inverse transformation method to generate variates from a
uniform distribution, an exponential distribution, and an Erlang distribution. We also
describe two techniques for generating variates from the normal distribution.
3.3.1

Sampling from a uniform distribution

The probability density function of the uniform distribution is defined as follows:

 1

b-a
f(x) = 

 0

a<x<b

,

otherwise

and it is shown graphically in figure 3.2.

f(x)
1
b-a

a

b

Figure 3.2: The uniform distribution.

The cumulative density function is:

36

x
1 x
x-a
⌠1
F(x) = ⌡b-a dt = b-a ⌠ dt = b-a .
⌡
a
a

The expectation and variance are given by the following expressions:
b

E(X) =

" f(x)xdx =
a

1
b"a

b

# xdx=

b+a
2

a

(b-a)2
! - E(X)) 2 F(x)dx = 12 .
(x
"
b

!
Var(X) =

a

The inverse transformation method for generating random variates is as follows.
!
x-a
r = F(x) = b-a
or
x = a + (b - a)r.
3.3.2

Sampling from an exponential distribution

The probability density function of the exponential distribution is defined as follows:
f(x) = ae-ax , a > 0, x > 0.
_
The cumulative density function is:
x

F(x) =

x

" f(t)dt = # ae
0

"at

tdt = 1 - e-ax.

0

The expectation and variance are given by the following expressions:
!
!


37

#

E(X) =

$ aet

"at

1
dt dt = a

0

#

Var(X) =
!

2 "at

$ (t " E(X)) e
0

1
tdt dt = a2 .

The inverse transformation method for generating random variates is as follows:
!
r = F(x) = 1 - e-ax
or
1 - r = e-ax
or
1
x = - a log(1-r) = - E(x)log(l-r).

Since 1-F(x) is uniformly distributed in [0,1], we can use the following short-cut
procedure
r = e-ax,
and therefore,
1
x = - a log r.

3.3.3

Sampling from an Erlang distribution

In many occasions an exponential distribution may not represent a real life situation. For
example, the execution time of a computer program, or the time it takes to manufacture
an item, may not be exponentially distributed. It can be seen, however, as a number of
exponentially distributed services which take place successively. If the mean of each of

38


these individual services is the same, then the total service time follows an Erlang
distribution, as shown in figure 3.3.

1/a

1/a

•!•!•

1/a

Figure 3.3 : The Erlang distribution.

The Erlang distribution is the convolution of k exponential distributions having
the same mean 1/a. An Erlang distribution consisting of k exponential distributions is
referred to as Ek. The expected value and the variance of a random variable X that
follows the Erlang distribution are:

k
E(X) = a
and
k
Var(X) = 2 .
a

Erlang variates may be generated by simply reproducing the random process on
which the Erlang distribution is based. This can be accomplished by taking the sum of k
exponential variates, x1, x2, ..., xk with identical mean 1/a. We have

k
x = ∑ xi
i=1
k 
1 k
-1 
= - a ∑ log ri = a log ∑ ri .



i=1
i=1 

3.3.4

39

Sampling from a normal distribution

A random variable X with probability density function

f(x) =

1
e
σ 2π

1 (x-µ)2
σ2

-2

,

-∞ < x < +∞,

where σ is positive, is said to have a normal distribution with parameters µ and σ. The
expectation and variance of X are µ and σ2 respectively. If µ=0 and σ=1, then the normal
distribution is known as the standard normal distribution and its probability density
function is
1

- 2 x2
1
f(x) =
e
, -∞ < x < +∞ .
2π

If a random variable X follows a normal distribution with mean µ and variance σ2, then
the random variable Z defined as follows

Z=

X-µ
σ

follows the standard normal distribution.
In order to generate variates from a normal distribution with parameters µ and σ,
we employ the central limit theorem. (This approach is named after this particular
theorem.) The central limit theorem briefly states that if x1, x2, ... ,xn are n independent
random variates, each having the same probability distribution with E(Xi)=µ and
Var(Xi)=σ2, then the sum ΣXi = X1+X2+...+Xn approaches a normal distribution as n
becomes large. The mean and variance of this normal distribution are:
E(ΣXi) = nµ

40


Var (ΣXi) = nσ2.
The procedure for generating normal variates requires k random numbers r1, r2,...
,rk. Since each ri is a uniformly distributed random number over the interval [0 ,1], we
have that

a+b
1
E(ri) = 2 = 2

(b-a)2
1
Var(ri) = 12 = 12 .

Using the Central Limit theorem, we have that the sum Σri of these k random numbers
approaches the normal distribution. That is
k k 
Σri ~ N2 ,
,
12

or
Σri - k/2
~ N(0, 1).
k/ 12

(3.1)

Now, let us consider the normal distribution with parameters µ and σ from which we
want to generate normal variates. Let x be such a normal variate. Then

x-µ
σ ~ N(0,1).

Equating (3.1) and (3.2) gives

(3.2)


41
Σri - k/2
x-µ
=
,
σ
k/ 12

or

x=σ

12 
k
k Σri - 2 + µ .

This equation provides us with a simple formula for generating normal variates
with a mean µ and standard deviation σ. The value of k has to be very large, since the
larger it is the better the accuracy. Usually, one has to balance accuracy against
efficiency. The smallest value recommended is k=10. (In fact, one can observe that k=12
has computational advantages).
An alternative approach to generating normal variates (known as the direct
approach) is the following. Let r1 and r2 be two uniformly distributed independent
random numbers. Then
1
2

x1 = (-2 loge r1) cos 2πr2
1
2

x2 = (-2 loge r1) sin 2πr2
are two random variates from the standard normal distribution. This method produces
exact results and the speed of calculations compares well with the Central Limit approach
subject to the efficiency of the special function subroutines.
3.4

Sampling from discrete probability distributions

In this section, we use the inverse transformation method to generate variates from a
geometric distribution. Also, we describe a technique for sampling from a binomial
distribution, and a technique for sampling from a Poisson distribution.

42
3.4.1

Sampling from a geometric distribution

Consider a sequence of independent trials, where the outcome of each trial is either a
failure or a success. Let p and q be the probability of a success and failure respectively.
We have that p+q=1. The random variable that gives the number of successive failures
that occur before a success occurs follows the geometric distribution. The probability
density function of the geometric distribution is
p(n) = pqn, n = 0,1,2, . . .,
and its cumulative probability density function is

n

F(n) =

∑pqs , n = 0, 1, 2, . . .

s=0

The expectation and the variance of a random variable following the geometric
distribution are:

p
E(X) = q

p
Var(X) = q2 .

The generation of geometric variates using the inverse transformation method can
be accomplished as follows.

n

F(n)

=

∑pqs

s=0


43
n
= p ∑qs
s=0

=p

1 - qn+1
1-q .

Since p = 1 - q, we have that F(n) = 1-qn+1. From this expression we obtain that 1-F(n)
= qn+1. We observe that 1-F(n) varies between 0 and 1. Therefore, let r be a random
number, then we have
r = qn+1
or
log r = (n+1) log q
or
log r
n = log q - 1.

Alternatively, since (1-F(n))/q=qn, and (1-F(n))/q varies between 0 and 1, we have
r = qn
or
log r
n = log q .

3.4.2

Sampling from a binomial distribution

Consider a sequence of independent trials (Bernoulli trials). Let p be the probability of
success and q=1-p the probability of a failure. Let X be a random variable indicating the
number of successes in n trials. Then, this random variable follows the Binomial
distribution. The probability density function of X is

44


p (k) = n pkqn-k, k = 0, 1, 2, . . .
k
The expectation and variance of the binomial distribution are:
E(X) = np
Var(X) = npq.
We can generate variates from a binomial distribution with a given p and n as
follows. We generate n random numbers, after setting a variable k0 equal to zero. For
each random number ri, i=1, 2, ..., n, a check is made, and the variable ki is incremented
as follows:

ki-1 + 1
ki = 
ki-1

if ri < p
if ri > p

The final quantity kn is the binomial variate. This method for generating variates is
known as the rejection method. This method is discussed in detail below in section 6.
3.4.3

Sampling from a Poisson distribution

The Poisson distribution models the occurrence of a particular event over a time period.
Let λ be the average number of occurrences during a unit time period. Then, the number
of occurrence x during a unit period has the following probability density function
p(n) = e-λ(λn/n!), n = 0, 1, 2, . . .
It can be demonstrated that the time elapsing between two successive occurrences
of the event is exponentially distributed with mean 1/λ, i.e., f(t) = λe-λt . One method for


45

generating Poisson variates involves the generation of exponentially distributed time
intervals t1, t2, t3,... with an expected value equal to 1/λ. These intervals are accumulated
until they exceed 1, the unit time period. That is,
n

n +1

"t <
i

i=1

!

1<

"t .
i

i=1

!

The stochastic variate n is simply the number of events occurred during a unit time
1

period. Now, since ti = -λ logri, n can be obtained by simply summing up random
numbers until the sum for n+1 exceeds the quantity e-λ. That is, n is given by the
following expression:

n

n+1

i=0

i=0

∑ ri > e-λ > ∑ ri .

3.5

Sampling from an empirical probability distribution

Quite often an empirical probability distribution may not be approximated satisfactorily
by one of the well-known theoretical distributions. In such a case, one is obliged to
generate variates which follow this particular empirical probability distribution. In this
section, we show how one can sample from a discrete or a continuous empirical
distribution.
3.5.1

Sampling from a discrete probability distribution

Let X be a discrete random variable, and let p(X = i) = pi, where pi is calculated from
empirical data. Let p(X≤i) = Pi be the cumulative probability. Random variates from this
probability distribution can be easily generated as follows. Now let r be a random
number. Let us assume that r falls between P2 and P3 (see figure 3.4). Then, the random
variate x is equal to 3. In general, if Pi-1<r<Pi then x=i. This method is based on the fact

46


that pi=Pi-Pi-1 and that since r is a random number, it will fall in the interval (Pi, Pi-1)
pi% of the time.

1

• •!•
P3
r
P2
P1

0

1

2

3

• •!•

n

x

Figure 3.4: Sampling from an empirical discrete probability distribution.

As an example, let us consider the well-known newsboy problem. Let X be the
number of newspapers sold by a newsboy per day. From historical data we have the
following distribution for X.

X

1

2

3

4

5

f(x)

0.20

0.20

0.30

0.15

0.15

The cumulative probability distribution is:
X

1

2

3

4

5

f(x)

0.20

0.40

0.70

0.85

1

The random variate generator can be summarized as follows:
1.

Sample a random number r.

2.

47

Locate the interval within which r falls in order to determine the random
variate x.
•

then

x=5

•

If 0.70 < r ≤ 0.85

then

x=4

•

If 0.40 < r ≤ 0.70

then

x=3

•

If 0.20 < r ≤ 0.40

then

x=2

•
3.5.2

If 0.85 < r ≤ 1.00

Otherwise

then

x=1

Sampling from a continuous probability distribution

Let us assume that the empirical observations of a random variable X can be summarized
into the histogram shown in figure 3.5. From this histogram, a set of values (xi, f(xi)) can
be obtained,

f(x3 )
f(x2 )

f(x 4)

f(x1 )

f(x 5)
f(x6 )

x1

x2

x3

x4

x5

x6

f(x 7)
x7

Figure 3.5: Histogram of a random variable X.

where xi is the midpoint of the ith interval, and f(xi) is the length of the ith rectangle.
Using this set of values we can approximately construct the cumulative probability
distribution shown in figure 3..6, where F(xi) = Σ1≤k≤1f(xk). The cumulative distribution
is assumed to be monotonically increasing within each interval [F(xi-1), F(xi)].

48

F(x7 )
F(x6 )
F(x5 )
F(x4 )
F(x3 )
F(x2 )

F(x1 )

x1

x2

x3

x4

x5

x6

x7

Figure 3.6: The cumulative distribution.

Now, let r be a random number and let us assume that F(xi-1)<r<F(xi). Then, using linear
interpolation, the random variate x can be obtained as follows:

r - F(xi-1)
x = xi-1 + (xi - xi-1) F(xi) - F(xi-1) ,

where xi is the extreme right point of the ith interval.

f(x 2)
f(x)

f(x3)

f(x1 )

f(x4 )
f(x n)

x1

x2

x3

x4

•!•!•

xn

Figure 3.7: "Discretizing" the probability density function.


49

This approach can be also used to generate stochastic variates from a known
continuous probability distribution f(x). We first obtain a set of values (xi, f(xi)) as shown
in figure 3.7. This set of values is then used in place of the exact probability density
function. (This is known as "discretizing" the probability density function.) Using this set
of values we can proceed to construct the cumulative probability distribution and then
obtain random variates as described above. The accuracy of this approach depends on
how close the successive xi points are.

1
c f(x)

a

b

Figure 3.8: Normalized f(x)

3.6

The Rejection method

The rejection technique can be used to generate random variates, if f(x) is bounded and x
has a finite range, say a ≤ x ≤ b. The following steps are involved:

•

Normalize the range of f(x) by a scale factor c so that cf(x) ≤ 1, a ≤ x ≤ b. (See
figure 3.8)

•

Define x as a linear function of r, i.e. x = a + (b-a) r, where r is a random number.

•

Generate pairs of random numbers (r1, r2).

•

Accept the pair and use x = a + (b-a)r1 as a random variate whenever the pair
satisfies the relationship r2 ≤ cf(a + (b-a)r1), i.e. the pair (x,r2) falls under the
curve in figure 3.8.

The idea behind this approach is that the probability of r2 being less than or equal to
cf(x) is p[r2 ≤ cf(x)] = cf(x). Consequently, if x is chosen at random from the range (a,b)

50


and then rejected if r2>cf(x), the probability density function of the accepted x's will be
exact.
We demonstrate the rejection method by giving two different examples. The first
example deals with random variate generation, and the second example deals with a
numerical integration problem.
Example 1:
Use the rejection method to generate random variates with probability density function
f(x)=2x, 0≤x≤1.
This can be accomplished using the following procedure:
1.

Select c such that df(x)≤1, i.e. c = 1/2.

2.

Generate r1, and set x = r1.

3.

Generate r2. If r2 < cf(r1) = (1/2)2r1 = r1 then accept r2, otherwise, go back
to step 2.

Example 2:
Use the rejection method to compute the area of the first quadrant of a unit circle.
We first note that any pair of uniform numbers (r1, r2) defined over the unit
interval corresponds to a point within the unit square. A pair (r1, r2) lies on the
circumference if
2

2

r1 +r2 = 1.

The numerical integration can be accomplished by carrying out the following two steps
for a large number of times:
1.

Generate a pair of random numbers (r1, r2).

2.

If r2 < f(r1), , where f(r1) =

1 " r12 ,

then r2 is under (or on) the curve and

hence the pair (r1, r2) is accepted. Otherwise, it is rejected.
!


51

The area under the curve can be obtained as the ratio

total number of acceptable pairs
area = total number of generated pairs .

The rejection method is not very efficient when c(b-a) becomes very large. The
method of mixtures can be used, whereby the distribution is broken into pieces and the
pieces are then sampled in proportion to the amount of distribution area each contains.
This process is identical to the rejection method for each piece of the distribution, plus a
straightforward sampling of data.
3.7

Monte Carlo methods

Monte Carlo methods comprise that branch of experimental mathematics which is
concerned with experiments on random numbers. Monte Carlo methods are usually
associated with problems of theoretical interest, as opposed to the simulation methods
described in this book, otherwise known as direct simulation. Unlike direct simulation
techniques, Monte Carlo methods are not concerned with the passage of time. Every
Monte Carlo computation that leads to quantitative results may be regarded as estimating
the value of a multiple integral.
The previously demonstrated rejection method for calculating the integral of a
function is an example of Monte Carlo, known as hit-or-miss Monte Carlo. An alternative
method for calculating an integral of a function is the crude Monte Carlo method. Let us
assume that we wish to calculate the one-dimensional integral
1
⌠
θ = ⌡f(x)dx .
0
Let ζ1,ζ2,...ζn, be random numbers between 0 and 1. Then, the quantities fi = f(ζi) are
independent random variates with expectation θ. Therefore,

52

1 n
f = n ∑fi
i=1

_

is an unbiased estimator of θ. Its variance can be estimated using the expression

_
1 n
s2 = n-1 ∑ (fi - f) 2 .
i=1

_

Thus, the standard error of f is s/ n .
The above technique of evaluating θ is based on the following idea. In general, if
X is a random variable that takes values in [0,1] and f(x) is a function of this random
variable, then
1
⌠
E(f(x) = ⌡f(x)g(x)dx
0
where g(x) is the density function of X. Assuming that X is uniformly distributed in (0,1),
i.e. g(X) = 1, we obtain
1
⌠f(x)dx .
E(f(x)) = ⌡
0
_

Thus, f is an unbiased estimate of E(f(X)).
This technique is more efficient than the technique mentioned in the previous
section.
Problems
1.

Use the inverse transformation method to generate random variates with
probability density function


53

3x2
f(x) = 
0
2.

, 0≤x≤1
, otherwise

Apply the inverse transformation method and devise specific formulae that yield
the value of variate x given a random number r. (Note that f(x) below needs to be
normalized.)
$5x
0" x"4
f(x) = %
& x # 2 4 < x " 10

3.

Set up a procedure to generate stochastic variates from
!
$x
0 " x " 1/2
f(x) = %
&1# x 1/2 < x " 1

4.

A customer in a bank may receive service which has an Erlang distribution E3
!
(each phase with mean 10) or an Erlang distribution E4 (each phase with mean 5)
with probability 0.4 and 0.6 respectively. Set-up a procedure to generate random
variates of a customer's service.

5.

Use the rejection method to generate stochastic variates from
f(x) = (x-3)4, 0 ≤ x ≤ 10

6.

Modify the procedure for generating stochastic variates from an Erlang
distribution, in order to generate variates from a Coxian distribution. A Coxian
distribution consists of a series of exponential distributions, each with a different
mean, and it has the following structure:

54


µ1

a1

µ2
b1

a2

...

µn

b2

After receiving an exponential service

time with parameter µ1, there

is a

probability b1(=1-a1) of departing, or a probability a1 of receiving another
exponential service time with parameter µ2, and so on until the kth exponential
service is received.
1.

Test statistically one of the stochastic variates generator discussed in this Chapter.

2.

Consider the machine interference problem. Change your simulation program so
that the operational time and the repair time of a machine are exponentially
distributed with the same means as before. Make sure that your clocks are defined
as real variables. Run your simulation model as before. Each time an event
occurs, print out a line of output to show the new value of the clocks and the other
relevant parameters.

3.

Consider the token-based access scheme. Change your simulation program so that
the inter-arrival times are exponentially distributed with the same means as
before. The switch over time and the time period T remain constant as before. The
packet transmission time is calculated as follows. We assume that 80% of the
transmitted packets are due to interactive traffic (i.e. computer/terminal type of
traffic), and 20% of the transmitted packets are due to file transfer. Packets due to
interactive traffic tend to be short, whereas packets due to file transfers tend to be
large. Consequently, we will assume that the time to transmit an interactive packet
is exponentially distributed with mean 2.5, and the time to transmit a file transfer
packet is exponentially distributed with mean 20.


55

Make sure that your clocks are defined as real variables. Run your simulation
model as before. Each time an event occurs, print out a line of output to show the
new value of the clocks and the other relevant parameters
4.

Consider the two-stage manufacturing system. Change your simulation program
so that the inter-arrival, service, operational, and repair times are all exponentially
distributed with the same means as before. Make sure that your clocks are defined
as real variables. Run your simulation model as before. Each time an event
occurs, print out a line of output to show the new value of the clocks and the other
relevant parameters.

Solutions to the above problems

1.

1
1
x3 1
1
⌠f(x) dx = ⌠3x dx = 3 3  = 33 = 1
⌡
⌡
0
0
0
x
t3 x 3x3
⌡
F(x) = ⌠3t2 dt = 3 3  = 3 = x3
0
0
Hence, r = x3 or x =

2.

F(x)

F(x)

3

r.

x
⌠stdt
=⌡
,x < 4
0
t2 x 5
= 5 2  = 2 x2
0
4
x
⌠5tdt = ⌠(t - 2)dt ,x > 4
=⌡
⌡
0
4
2 4 t2
x
t
= 5 2  + 2 - 2t 
0
4
x2
16 
= 40 + 2 - 2x -  2 - 8

56


 5 x2
2
F(x) = 
 40 + x2 - 2x

2

0≤x≤4
4 < x ≤ 10

In order to normalize f(x), we have
4
10
⌠5xdx + = ⌠(x - 2)dx
⌡
⌡
0
4
or
4 x2
10
5
= 2 x2  + 2 - 2x 
0
4
5
100
42
= 2 16 + 2 = 2 x 10 - 2 + 2 - 4
Thus,
1
 70 5 x2
 2
F(x) = 
 1 40 + x2 - 2x
 70 

2

0≤x≤4
4 < x ≤ 10

Procedure:
Draw a random number r.
40
5
If r < 70 then r = 140 x2 or x =

140
5 xr .


1 
x2
Otherwise, r = 70 40 + 2 - 2x , from which one can solve for x.
5.

Step 1. Use calculus to establish the bounds of f(x)
74

34

1

2

3

10

Thus c = 1/74
Step 2:
2.1
2.2

generate r1. Then 10r1.
generate r2.
1
if r2 < 4 f(10r1) then accept 10r1 as a stochastic variate.
7
Otherwise go back to 2.1.

57

CHAPTER 4:
SIMULATION DESIGNS

4.1

Introduction

In this Chapter, we examine three different designs for building simulation models. The
first two designs are: a) event-advance and b)unit-time advance. Both these designs are
event-based but utilize different ways of advancing the time. The third design is activitybased. The event-advance design is the most popular simulation design.
4.2

Event-advance design

This is the design employed in the three examples described in Chapter 1. The basic idea
behind this design is that the status of the system changes each time an event occurs.
During the time that elapses between two successive events, the system's status remains
unchanged. In view of this, it suffices to monitor the system's status. In order to
implement this idea, each event is associated with a clock. The value of this clock gives
the time instance in the future that this event will occur. The simulation model, upon
completion of processing an event, say at time t1, regroups all the possible events that
will occur in the future and finds the one with the smallest clock value. It then advances
the time, i.e., the master clock, to this particular time when the next event will occur, say
time t2. It takes appropriate action as dictated by the occurrence of this event, and then
repeats the process of finding the next event (say at time t3). The simulation model,
therefore, moves through time by simply visiting the time instances at which events
occur,. In view of this it is known as event-advance design.

60

In the machine interference problem, described in section 3.1 of Chapter 1, there

are two types of events. That is, the event of an arrival at the repairman's queue, and the
event of a departure from the repairman's queue. These events are known as primary
events. Quite often the occurrence of a primary event may trigger off the creation of a
new event. For instance, the occurrence of an arrival at the repairman's queue may trigger
off the creation of a departure event (if this arrival occurs at a time when the repairman is
idle). Such triggered events are known as conditional events. The basic approach of this
design is shown in the flow chart in figure 4.1.

A
Find next
event

Future
event
list

Advance
time

Take appropriate
action depending
on the type
of event

A

no

Any
conditional
events
?
yes
Create a new
event(s)

Future
event
list

A

Figure 4.1:

The event-advance simulation design.

Simulation designs
4.3

61

Future event list

Let us assume that a simulation model is currently at time t. The collection of all events
scheduled to occur in the future (i.e., events with clock greater than t) is known as the
future event list. For each event scheduled to occur in the future, the list contains the
following information:

•

Time of occurrence (i.e., value of the event's clock)

•

Type of event

The event type is used in order to determine what action should be taken when the event
occurs. For instance, using the event type the program can determine which procedure to
call or to which statement in the program to go to.
In each of the simulation examples described in section 3 of Chapter 1, there were
only a few events. For instance, in the case of the machine interference problem there
were only two: an arrivals at the repairman's queue and a service-ending (departure)
event. However, when simulating complex systems, the number of events may be very
large. In such cases, finding the next event might require more than a few comparisons.
Naturally, it is important to have an efficient algorithm for finding the next event since
this operation may well account for a large percentage of the total computations involved
in a simulation program. The efficiency of this algorithm depends upon the amount of
information kept in the future event list, and the way this information is stored in the
computer. An event list should be stored in such a way so as to lend itself to an efficient
execution of the following operations.

•

Locating the next future event time and the associated event type.

•

Deleting an event from the list after it has occurred.

•

Inserting newly scheduled events in the event list.

Below we examine two different schemes for storing an event list. In the first
scheme, the event list is stored in a sequential array. In the second scheme, it is stored as
a linked list.

62

4.3.1


Sequential arrays

In this scheme, all future event times are stored sequentially in an array. The simplest
way to implement this, is to associate each event type with an integer number i. The clock
associated with this event is always stored in the ith location of the array . For instance, in
figure 4.2, the clock CL1 for event type 1 is kept in the first location of the array, the
clock CL2 for the event type 2 is kept in the second position of the array, and so on.

CL 1

CL 2

CL 3

• !• !•

CL n

Figure 4.2: Future event list stored sequentially in an array.

Finding the next event is reduced to the problem of locating the smallest value in
an array. The following simple algorithm can be used to find the smallest value in an
array A.
num ← 1
help ← A(1)
for i ← 1,n
if help ≤ A(i) then continue else
help ← A(i)
num ← i
Variable num will eventually contain the location of the array with the smallest value. If
num=i, then the next event is of type i and it will occur at time A(i).
An event is not deleted from the array after it has occurred. However, its clock
should not be considered when locating the next event. This can be avoided by simply
setting its clock to a very large value so that the above algorithm will never select this
event as the next event to occur. Finally, a newly-scheduled event j is inserted in the list
by simply updating its clock given by A(j).

Simulation designs

63

The advantage of storing an event list in a sequential array is that insertions of
new events and deletions of caused events can be done very easily (i.e., in constant time).
The time it takes to find the smallest number in the array depends, in general, on the
length of the array n (i.e., its complexity is linear in time). Locating the smallest number
in array does not take much time if the array is small. However, if the array is large, it
becomes very time consuming. For such cases, one should store the future event list in a
linked list.

F

a

1

a

2

a

3

. . .

a

n 0

Figure 4.3: A linked list.

4.3.2

Linked lists

A linked list representation of data allows us to store each data element of the list in a
different part of the memory. In order to access the data elements in the list in their
correct order, we store along with a data element the address of the next data element.
This is a pointer pointing to the location of the next data element. This pointer is often
referred to as a link. The data element and the link (or links) is generally referred to as a
node. In general, a node may consist of a number of data elements and links. Linked lists
are drawn graphically as shown in figure 4.3. Each node is represented by a box
consisting of as many compartments as the number of data elements and links stored in
the node. In the example given in figure 4.3, each node consists of two compartments,
one for storing a data element and the other for storing the pointer to the next node.
Pointer F points to the first node in the list. If the linked list is empty, i.e., it contains no
nodes, then F is set to zero. The pointer of the last node is always set to zero indicating
that this is the last node in the linked list structure. Due to the fact that two successive
nodes are connected by a single pointer, this data structure is known as a singly linked
list.

64


F

CLi

i

. . .

CL j j

CL n

0

Figure 4.4: Future event list stored sequentially in a linked list.

A single linked list can be used to store a future event list as shown in figure 4.4.
Each node will consist of two data elements, namely a clock CLi showing the future time
of an event, and a value i indicating the type of event. The nodes are arranged in an
ascending order so that CLi ≤ CLj ≤ . . . ≤ CLn.

T
5

7

4

1

12

8

1

3

10

4

13

F

P

0

Figure 4.5: An implementation of a singly linked list

In order to get a better insight into linked list structures, let us consider the
following example. Integer numbers are stored arbitrarily in various locations of an array
T. It is required that these numbers are maintained in an ascending order. This can be
achieved by setting up an equally dimensioned array, P, as shown in figure 4.5. Each
location i of P is associated with the corresponding location i of T. The content of P(i) is
a pointer to a location in array T which contains the next larger number after T(i). For
instance, P(1)=7. That means that the next larger number after T(1) is in the 7th location
of T, i.e. it is T(7). Thus, array P permits the occupied locations of T to be linked an

Simulation designs

65

ascending order. Pointer F (=5) points to the lowest integer number. This is a simple
implementation of a singly linked list. A node consists of a location in T and its
corresponding location in P. For instance, the second node is given by the pair (T(3),
P(3)) and it contains (4, 1).
Linked list structures provide the means to do insertions and deletions very easily.
For instance, let us assume that we want to delete number 10. Using pointer F, we first
check the value stored in the first node (T(5), P(5)). Using pointer P(5) we can locate the
second node (T(3), P(3)). Pointer P(3) gives the address of the third node (T(1), P(1)),
whose pointer P(1) contains the address of the sought node. Given, therefore, that we
know the address of the first node of the linked list, we can easily visit all the other nodes
of the list. Note that we can only move forward and not backwards. This is due to the fact
that the nodes are singly linked. At each node, we only know the location of the next
node, and we do not know the location of the previous node.

T
5

4

4

1

12

8

1

3

10

4

13

F

P

0

Figure 4.6: Deletion of node containing the value 10.

Now, deletion of a node simply means that we change the value of the pointer of
the previous node, so that this particular node is no longer part of the linked list. In
particular, the previous node is (T(1), P(1)) = (5, 4). Now, the node containing 10 has a
pointer P(7) = 4. In order to delete the node, it suffices to set P(1) = P(7) = 4, as shown in

66


figure 4.6. The information pertaining to this node will be still in the arrays T and P.
However, it will no longer be accessible from the linked list. In figure 4.6, the above
deletion is shown using the standard graphical representation of linked lists.

F

1

4

5

10

12

13 0

Figure 4.6: Deletion of node containing the value 10 using the linked list representation.

Insertions can be carried out just as easily. Let us assume that we want to insert
number 6 in the linked list. The first step is to locate the two successive nodes in between
of which 6 will be inserted. Starting from the first node, we sequentially search the nodes
until we find that the new node has to be inserted between (5, 4) and (12, 8). In order to
do this, we first have to acquire an unused location in T. Let this be T(2). We set T(2)=6
and then we link (T(2), P(2)) to the list by appropriately rearranging the pointers P(2) and
P(1). That is P(1)=2 and P(2)=4, as shown in figure 4.7.

T
5

2

6

4

4

1

12

8

1

3

10

4

13

F

P

0

Figure 4.7: Insertion of node containing the value 6.

Simulation designs

67

Using the standard graphical representation of linked lists, the above insertion is shown
figure 4.8.

F

1

4

5

10

12

13 0

6

Figure 4.8: Insertion of node containing the value 6 using the linked list representation.

4.3.3

Implementation of linked lists

In order to process linked list structures, one has to be able to carry out the following
basic operations.

•

Organize information such as data elements and pointers into a node.

•

Access a node through the means of a pointer.

•

Create a new node(s) or delete an existing unused node(s).

Programming languages provide commands that permit the user to carry out the above
operations. However, if such commands are not available, the user has to set-up an
appropriate storage scheme where these basic operations can be implemented. Such a
storage scheme can be easily set up by using sequential arrays as shown in the previous
section. If each node consists of K data elements and a pointer, then this storage scheme
requires K+1 arrays. The arrays should be long enough in order to accommodate the
maximum number of nodes that might be generated during the execution of the
simulation program. All the unused nodes have to be linked together in order to form a
pool of free nodes. Creating a new node would simply require getting a node from the
pool of free nodes. Similarly, discarding a node from the linked list would simply require
that this node be transferred back to the pool.

68

In the following, we assume that each node consists of two fields, namely a

DATA field containing a data element and a LINK field containing a pointer.
Furthermore, we will assume a storage scheme whereby either fields of node i can be
accessed using DATA(i) and LINK(i). The pool of free nodes is assumed to be managed
by the following two functions:
1.

GETNODE (X): This provides a pointer X to a free node. This command can
be used when creating a new node.

2.

RET (X): This returns a node with address X back to the pool of free nodes.

Create a linked list
The following procedure creates a linked list consisting of two nodes containing the
integer numbers 1 and 4. This procedure can be easily expanded to create the linked list
given in section 4.3.2.
Procedure
CREATE (F)
// F points to the first node of the linked list. //
call GETNODE (F)
DATA (F) ← 1
call GETNODE (I)
LINK (F) ← I
DATA (I) ← 4
LINK (I) ← 0
end CREATE
Deletion of a node
The following procedure deletes the node containing the data element 10 from the linked
list given in the example in section 4.3.2. It is assumed that data element 10 occurs only
once. Otherwise, the first node encountered that contains 10 will be deleted.
Procedure
DELETE (F, C)
// F points to the first node of the linked list and C=10, Initial condition LINK (F)
= F //

Simulation designs

69

If F = 0 then error: list empty
I←J←F
do
If DATA (I) = C then
[LINK (J) ← LINK (I)
RET(I)
exit]
else [J ← I
I ← LINK (I)]
forever
end DELETE
Insertion of a new node
The following procedure inserts a node containing the data element G in the linked list
given in example 4.3.2.
Procedure
INSERT (F, G)
// F points to the first node of the linked list and C is the data element to be
inserted, i. e., C = 6 //
If F = 0 then
[GETNODE (X)
DATA (X) ← C
LINK (X) ← 0
F←X
exit]
If DATA (F) > C then
[GETNODE (X)
DATA (X) ← C
LINK (X) ← F
F←X
exit]
J←F
I ← LINK (J)
do
if DATA (I) > C then
[GETNODE (X)
DATA (X) ← C
LINK (X) ← LINK (J)
LINK (J) ← X
exit]
else [J ← I
I ← LINK (I)]

70

forever
end INSERT

Managing the storage pool
The storage pool contains all the nodes that are not being used. Function GETNODE
removes a node from the pool, and function RET returns a node to the pool. Most highlevel languages provide commands similar to GETNODE and RET. The following set of
procedures can be used to manage the storage pool in the absence of such commands. In
the following, it is assumed as above that the linked list is stored using sequential arrays.
Before using the linked list for the first time, all available nodes are linked
together as follows.
Procedure
INIT (n)
//The number of nodes to be linked is n. AV is the pointer to the first node.//
For i ← 1 to n - 1 do
LINK (I) ← i + 1
end
LINK (n) ← 0
AV ← 1
end INIT
Procedures GETNODE (X) and RET (X) can be implemented as follows.
Procedure GETNODE (X)
// X points to a node to be used by the linked list //
if AV = 0 then error: no more nodes
else [X ← AV
AV ← LINK (AV)]
end GETNODE
Procedure RET (X)
// X points to a node which is to be returned back to the pool //
LINK (X) ← AV
AV ← X
end RET

Simulation designs
4.3.4

71

Future event list

The future event list can be implemented as a singly linked list as described in 4.3.2 and
shown in figure 4.4. Data element CLi and i describe each event scheduled to occur in the
future. The nodes are organized so that CLi ≤ CLj ≤ . . . ≤ CLn. Thus, in order to obtain
the next event, it suffices to look up the node pointed to by F. When this event has
occurred, it can be deleted from the list by simply executing RET(F) and then setting F
← LINK (F). A newly generated event can be appropriately inserted using the insertion
procedure given in 4.3.3.
It is obvious that locating the next event and deleting a caused event can be done
in constant time. However, in order to insert a new event with clock CLm, it is necessary
to search sequentially each node, starting from the first node, until we find the first node
with clock CLk, such that CLm > CLk. The time to carry out this operation, in general,
depends on the number of events n in the list, i. e., it is linear on n. The worst case, in
fact, is to have to search all nodes in the list. Searching a linked list might be time
consuming if n is very large. In this case, one can employ better searching procedures.
For example, a simple solution is to maintain a pointer B to a node which is in the middle
of the linked list. This node logically separates the list into two sublists. By comparing
CLm with the clock stored in this node, we can easily establish in which sublist the
insertion is to take place. The actual insertion can then be located by sequentially
searching the nodes of the sublist.
4.3.5

Doubly linked lists

So far we have examined singly linked lists. The main disadvantage of these lists is that
they can be only traversed in one direction, namely from the first node to the last one.
Doubly linked lists allow traversing a linked list in both directions. This is enabled by
linking two successive nodes with two pointers as shown in figure 4.9. Depending upon
the application, a doubly-linked list may be more advantageous than a singly-linked list.

72


A doubly-linked list can be processed using procedures similar to those described in
section 4.3.3.

F

a1

a3

a2

an

• •!•

Figure 4.9: A doubly linked list.

4.4

Unit-time advance design

In the event-advance simulation, the master clock is advanced from event to event.
Alternatively, the master clock can be advanced in fixed increments of time, each
increment being equal to one unit of time. In view of this particular mode of advancing
the master clock, this simulation design is known as the unit-time advance design. Each
time the master clock is advanced by a unit time, all future event clocks are compared
with the current value of the master clock. If any of these clocks is equal to the current
value of the master clock, then the associated event has just occurred and appropriate
action has to take place. If no clock is equal to the current value of the master clock, then
no event has occurred and no action has to take place. In either case, the master clock is
again increased by unit-time and the cycle is repeated. This mode of advancing the
simulation through time is depicted in figure 4.10. The basic approach of the unit-time
design is summarized in the flow-chart in figure 4.11.

E

E

1

2

time

Figure 4.10: The unit-time advance design.

E

n

Simulation designs

73

A

Master clock
is increased
by a unit-time
Is
any future
event clock
= MC
?
yes

no

A

An event has
occurred
Take appropriate
action

Any
conditional
event(s)
?

no

A

yes
Schedule new
event (j)
A
Figure 4.11: The unit-time advance design.

In the flow-chart of the unit-time simulation design, given in figure 4.11, it was
implicitly assumed that a future event clock is a variable which, as in the case of the
event-advance design, contains a future time with respect to the origin. That is, it contains
the time at which the associated event will occur. Alternatively, a future clock can simply
reflect the duration of a particular activity. For instances, in the machine interference
problem, the departure clock will simply contain the duration of a service, rather than the
future time at which the service will be completed. In this case, the unit-time design can

74


be modified as follows. Each time the master clock is advanced by a unit of time, the
value of each future clock is decreased by a unit-time. If any of these clocks becomes
equal to zero, then the associated event has occurred and appropriate action has to take
place. Obviously, the way one defines the future event clock does not affect the unit-time
simulation design.

A
MCL = MCL + 1
>1

ST

=1

ST = ST - 1

ST = ST - 1
End of service
Take appropriate
action

>1

AT

AT = AT - 1

=1
AT = AT - 1
An arrival has
occurred. Take
appropriate action

A

Figure 4.12: A unit-time advance design of a single server queue.

In order to demonstrate the unit-time advance design, we simulate a single queue
served by one server. The population of customers is assumed to be infinite. Figure 4.12
gives the flow-chart of the unit-time design. Variable AT contains the inter-arrival time

Simulation designs

75

between two successive arrivals. Variable ST contains the service time of the customer in
service. Finally, variable MCL contains the master clock of the simulation model.
4.4.1

Selecting a unit-time

The unit-time is readily obtained in the case where all future event clocks are represented
by integer variables. For, each event clock is simply a multiple of the unit-time. However,
quite frequently future event clocks are represented by real variables. In this case, it is
quite likely that an event may occur in between two successive time instants of the master
clock as shown in figure 4.13. Obviously, the exact time of occurrence of an event E is
not known to the master clock. In fact, as far as the simulation is concerned, event E
occurs at time t+1 (or t+2 depending upon how the program is set-up to monitor the
occurrence of events). This introduces inaccuracies when estimating time parameters
related to the occurrence of events. Another complication that might arise is due to the
possibility of having multiple events occurring during the same unit of time.

t

t + 1 Ej

t+2

t+3

Ei t + 4

time

Figure 4.13: Events occurring in between two successive values of the master clock.

In general, a unit-time should be small enough so that at most one event occurs
during the period of a unit of time. However, if it is too small, the simulation program
will spend most of its time in non-productive mode, i.e. advancing the master clock and
checking whether an event has occurred or not. Several heuristic and analytic methods
have been proposed for choosing a unit-time. One can use a simple heuristic rule such as
setting the unit-time equal to one-half of the smallest variate generated. Alternatively,
one can carry out several simulation runs, each with a different unit-time, in order to
observe its impact on the computed results. For instance, one can start with a small unit

76


time. Then, it can be slightly increased. If it is found to have no effect on the computed
results, then it can be further increased, and so on.
4.4.2

Implementation

The main operation related to the processing of the future event list is to compare all the
future event clocks against the master clock each time the master clock is increased by a
unit time. An implementation using a sequential array as described in section 4.3.1 would
suffice in this case.
4.4.3

Event-advance vs. unit-time advance

The unit-time advance method is advantageous in cases where there are many events
which occur at times close to each other. In this case, the next event can be selected fairly
rapidly provided that an appropriate value for the unit-time has been selected. The best
case, in fact, would occur when the events are about a unit-time from each other.
The worst case for the unit-time advance method is when there are few events and
they are far apart from each other. In this case, the unit-time advance design will spend a
lot of non-productive time simply advancing the time and checking if an event has
occurred. In such cases, the event-advance design is obviously preferable.
4.5

Activity-based simulation design

This particular type of simulation design is activity based rather than event based. In an
event oriented simulation, the system is viewed as a world in which events occur that
trigger changes in the system. For instance, in the simulation model of the single server
queue considered in section 4, an arrival or a departure will change the status of the
system. In an activity oriented simulation, the system modelled is viewed as a collection
of activities or processes. For instance, a single server queueing system can be seen as a
collection of the following activities: a) inter arriving, b) being served, and c) waiting for
service. In an activity based design, one mainly concentrates on the set of conditions that

Simulation designs

77

determine when activities start or stop. This design is useful when simulating systems
with complex interactive processing patterns, sometimes referred to as machine-oriented
models.
We demonstrate this design by setting up an activity-based simulation model of
the single server queue problem studied in section 4. Let STi and WTi be the service time
respectively the waiting time of the ith arrival. Also, let ATi+1 be the interarrival time
between the ith and (i+1)st arrival. Finally, let us assume that the ith arrival occurs at time
ai, starts its service at time si and ends its service at time si + STi, as shown in figure
4.14.

ST i
WTi
AT

a

i+1

si

i

ith
arrival

ith arrival
begins its
service

a i+1

(i+1)st
arrival

s i + STi

ith arrival
ends its
service

Figure 4.14: Time components related to the ith arrival.

Let us assume now that we know the waiting time WTi and the service time STi
of the ith customer. Let ATi+1 be the inter-arrival time of the (i+1)st customer. Then,
one the following three situations may occur.
a.

The (i+1)st arrival occurs during the time that the ith arrival is waiting.

b.

The (i+1)st arrival occurs when the ith arrival is in service

c.

The (i+1)st arrival occurs after the ith arrival has departed from the queue

78


Initial
conditions

Empty system
WT = ST = 0, TW = 0

A

Generate
AT

Next arrival occurs
when current arrival
is either waiting or
is in service

TW > AT

Compare
TW with
AT

TW ! AT

Generate
ST

Next arrival
occurs when
server is idle
Generate
ST

TW = ST
TW = ST + (TW - AT)
A
A

Figure 4.15: An activity-based simulation design for a single server queue.

For each of these three cases, the waiting time WTi+1 of the (i+1)st customer can be
easily determined as follows:
a)

WTi+1

= (WTi -ATi+1) + STi
= (WTi + STi) - ATi+1
= TWi - ATi+1,

where TWi is the total waiting time in the system of customer i.
b)

WTi+1

= (WTi + STi) - ATi+1

Simulation designs

79

= TWi - ATi+1
c)

WTi+1= 0

Having calculated WTi+1, we generate a service time STi+1 for the (i+1)st arrival, and
an inter-arrival time ATi+2 for the (i+2)nd arrival. The waiting time of the (i+2)nd arival
can be obtained using the above expressions. The basic mechanism of this activity-based
simulation model is depicted in figure 4.15.
4.6

Examples

In this section, we further highlight the simulation designs discussed in this Chapter by
presenting two different simulation models.
4.6.1

An inventory system

In an inventory system, one is mainly concerned with making decisions in order to
minimize the total cost of the operation. These decisions are mainly related to the
quantity of inventory to be acquired (or produced) and the frequency of acquisitions. The
total cost of running an inventory system is made up of different types of costs. Here, we
will consider the following three costs: a) holding cost, b) setup cost, and c) shortage
cost. The holding cost is related to the cost of keeping one item of inventory over a period
of time. One of the most important components of this cost is that of the invested capital.
The setup cost is related to the cost in placing a new order or changes to production.
Finally, the shortage cost is associated with the cost of not having available a unit of
inventory when demanded. This cost may be in the form of transportation charges (i.e.,
expediting deliveries), increased overtime, and loss of future business.
Let It be the inventory at time t. Let S be the quantity added in the system
between time t and t´. Also, let D be the demand between these time instances. Then, the
inventory at time t´ is

80

It´ = It + S - D.

If It´ is below a certain value, then an order is placed. The time it takes for the ordered
stock to arrive is known as the lead time. We assume that the daily demand and the lead
time follow known arbitrary distributions. The inventory level is checked at the end of
each day. If it is less than or equal to the re-ordering level, an order is placed. The lead
time for the order begins to count from the following day. Orders arrive in the morning
and they can be disposed of during the same day. During stockout days, orders are
backlogged. They are satisfied on the day the order arrives. The fluctuation in the
inventory level is shown in figure 4.16.

Stock
level
Reorder
point
t

1

t

2
time

t

3

t

4

Figure 4.16: An inventory system.

The simulation model is described in the flow chart given in figures 4.17 and 4.18. The
model estimates the total cost of the inventory system for specific values of the
reordering point and the quantity ordered. The model keeps track of It on a daily basis. In
view of this, the model was developed using the unit-time advance design. We note that
this design arises naturally in this case. A unit of time is simply equal to one day. The
lead time is expressed in the same unit-time. The basic input parameters to the simulation
model are the following. a) ROP, reordering point, b) Q, quantity ordered, c) BI, the
beginning inventory, d) probability distributions for variables D and LT representing the
daily demand and lead time, respectively, e) T, the total simulation time, and f) C1, C2,
C3, representing the holding cost per unit per unit time, the setup cost per order, and the

Simulation designs

81

shortage cost per unit per unit time, respectively. The output parameters are TC1, TC2,
TC3 representing the total holding, setup and shortage costs respectively.

Initialization

a

Generate
AT

t>T

yes

Print
results

no
End
>1

=1

LT
=0
LT = LT - 1

order has
arrived
LT = 0
I=I+Q

no order
outstanding

Generate
daily demand D

I=I-D

b

Figure 4.17: A unit-time simulation design of an inventory system.

82


b

no

I < ROP
?

TC1 = TC1 + I*C1

yes
no

a
LT = 0
yes
Generate
LT

TC2 = TC2+ C2

no

I<0

TC1 = TC1 + I*C1
yes

TC3 = TC3 + I*C3

a

a

Figure 4.18: A unit-time simulation design of an inventory system.

4.6.2

A round-robin queue

Let us consider a computer system consisting of N terminals accessing a single CPU. A
user at a terminal spends sometime thinking (i.e., typing a line or thinking what to do
next), upon completion of which it creates a job that is executed at the CPU. (Every time
the user hits the return key a process runs on the CPU). We assume that a user at a
terminal continuously cycles through a think time and a CPU time, as shown in figure
4.19. During the time that a user’s job is being executed, the user cannot submit another
job.

Simulation designs

think

CPU

83

think

CPU

...

think

CPU

Figure 4.19: Cycling through a think time and a CPU time

Jobs accessing the CPU are served in a round robin manner. That is, each job is
allowed to use the CPU for a small quantum of time. If the job is done at the end of this
quantum (or during the quantum), then it departs from the CPU. Otherwise, it is simply
placed at the end of the CPU queue. In this manner, each job in the CPU queue gets a
chance to use the CPU. Furthermore, short jobs get done faster than long ones. A job that
leaves the CPU simply goes back to the originating terminal. At that instance the user at
the terminal goes into a think state. The think time is typically significantly longer than
the duration of a quantum. For instance, the mean think time could be 30 seconds,
whereas a

Terminals

•
•
•

CPU

Figure 4.20: A round-robin queue

quantum could be less than 1 msec. A job, in general, would require many CPU quanta. If
a job requires 5 seconds of CPU time, and a quantum is 1 msec, then it would cycle
through the CPU queue 5000 times.

84

The queueing system reflecting this round-robin queue is shown in figure 4.20.

This system is similar to the machine interference problem. The terminals are the
machines and the CPU is the repairman. A terminal in the think state is like a machine
being operational. The main difference from the machine interference problem is that the
CPU queue is served in a round robin fashion rather in a FIF0 manner.
The basic events associated with this system are: a) arrival of a job at the CPU
queue, and b) service completion at the CPU. A job arriving at the CPU queue may be
either a new arrival (i.e., a user at a terminal hit the return key, thus completing the think
state), or it may be a job that has just received a quantum of CPU time and it requires
further processing. A departing job from the CPU may either go back to the originating
terminal, or it may simply join the end of the CPU queue for further execution. We
observe that during the time that the CPU is busy, departure events occur every quantum
of time. However, new arrivals of jobs at the CPU queue occur at time instances which
may be a few hundreds of quanta apart.
Following the same approach as in the machine interference problem, described in
section 1.3.1 of Chapter 1, we can easily develop an event-advance simulation model. We
observe that in this case, the event-advance design is quite efficient. The future event list
will contain one event associated with a departure from the CPU and the remaining
events will be associated with future new arrivals of jobs. When a departure occurs, most
likely a new departure event will be scheduled to occur in the next quantum of time. This
new event will more likely be the next event to occur. In view of this, most of the time, a
newly created event will be simply inserted at the top of the future event list. Such
insertions can be done in 0(1) time.

F

t

i1

i

i

t

i2

i

2

• • •

t

in

i

n

0

Figure 4.21: Future event list of all new arrivals to the CPU queue.

We now give an alternative simulation model of the round-robin queue which
utilizes both the event-advance and unit-time advance designs! Specifically, all the events

Simulation designs

85

related to new arrivals at the CPU queue are kept in a linked list as shown in figure 4.21.
Each node contains a future event time and a terminal identification number. The event
time simply shows the time at which the user will stop thinking and will access the CPU.
Nodes are ordered in an ascending order of the data element that contains the future event
time. Thus, the next new arrival event is given by the first node of the linked list.

B

t

j1

j

t

1

j2

j

• • •

2

t jk j
k

E

Figure 4.22: Future event list of all new arrivals to the CPU queue.

b

MCL = tarr

yes

Is
CPU
queue
empty
?

no

MCL = MCL + 1

A new arrival
event occurs

A departure
event occurs

b

Is
MCL=tarr
?

no

yes
A new arrival
event occurs
b
Figure 4.23: Hybrid simulation design of the round-robin queue.

b

86

All the information regarding the jobs in the CPU queue, including the one in

service, is maintained in the separate linked shown in figure 4.22. This linked list is
known as a circular singly linked list, since it is singly linked and the last node is linked
to the first node. Each node contains the number of quanta required by a job and its
terminal identification number. Pointers B and E point to the beginning and end of the list
respectively. The nodes are ordered in the same way that the jobs are kept in the CPU
queue. Thus, the first node corresponds to the job currently in service. When a new job
arrives at the CPU queue, a new node is created which is attached after node E. If a job
requires further service upon completion of its quantum, then its node is simply placed at
the end of the list. This is achieved by simply setting E←B and B←LINK(B).
The simulation model operates under the unit-time advance design during the
period of time that the CPU is busy. During the time that CPU is idle, the simulation
model switches to an event-advance design. This hybrid design is summarized in figure
4.23. Note that tarr gives the time of the next new arrival at the CPU queue. The
remaining details of this program. are left up to the reader as an exercise!
Problems
Consider the following systems:
1.

Checkout stands at a supermarket

2.

Teller's window at a bank

3.

Elevators serving an office building

4.

Traffic lights in a configuration of 8 city blocks

5.

Outpatient clinic

6.

Pumps at a gasoline station

7.

Parking lot

8.

Runways at an airport

9.

A telecommunication system with 5 nodes (virtual HDLC three linked lists)

10.

Solar heating of a house

Simulation designs

87

Choose any of the above systems. First, describe how the system operates. (Make
your own assumptions whenever necessary. Make sure that these assumptions do not
render the system trivial!) Then, set up a simulation model to represent the operations of
the system. If you choose an event-based simulation design, then state clearly which are
the state variables and what are the events. For each caused event, state clearly what
action the simulation model will take. If you choose an activity-based design, state
clearly which are the activities and under what conditions they are initiated and
terminated.
1.

Implement the hybrid simulation model of the round-robin queue discussed in
section 4.6.2.

2.

Consider the machine interference problem. Modify your simulation model so
that the event list is maintained in the form of a linked list. Assume that the queue
of broken-down machines can be repaired by more than one server (i.e., there are
more than one repairman repairing machines off the same queue). Parametrize
your program so that it can run for any number of machines and repairmen
(maximum 20 and 10, respectively). Run your simulation model until 20 repairs
have been completed. As before, each time an event occurs, print out the usual
line of output and also additional information pertaining to the event list. Check
by hand that your linked list implementation is correct.

3.

Consider the token-based access scheme problem. Assume that transmissions are
not error-free. That is, when a packet arrives at the destination node, it may
contain errors. In this case, the packet will have to be re-transmitted. The
procedure is as follows:
Upon completion of the transmission of a packet, the host will wait to hear from
the receiving host whether the packet has been received correctly or not. The time

88

for the receiver to notify the sender may be assumed to be constant. If the packet
has been correctly received, the host will proceed with the next transmission. If
the packet has been erroneously transmitted, the sender will re-transmit the
packet. There is 0.01 probability that the packet has been transmitted erroneously.
The sender will re-transmit the packet immediately. This procedure will be
repeated until the packet is correctly transmitted. No more than 5 re-transmissions
will be attempted. After the 5th re-transmission, the packet will be discarded, and
the sender will proceed to transmit another packet. All these re-transmissions take
place while the host has the token. When the token's time-out occurs, the host will
carry on re-transmitting until either the packet is transmitted correctly, or the
packet is discarded.
Describe how you will modify your simulation model in order to
accommodate the above acknowledgement scheme. Can this additional structure
be accommodated without introducing more events? If yes, how? If no, what
additional events need to be introduced? Describe what action will be taken each
time one of these additional events takes place. Also, describe how these new
events will interact with the existing events (i.e., triggering-off each other).

4.

Consider the token-based access scheme problem. Modify the simulation design
in order to take advantage of the structure of the system. Specifically, do not
generate arrivals to each of the nodes. Store the residual inter-arrival time when
the node surrenders the token. Then, when the token comes back to the station,
continue to generate arrivals until the token times-out or it is surrendered by the
node. This change leads to a considerably simpler simulation model.

CHAPTER 5:
ESTIMATION TECHNIQUES FOR ANALYZING
ENDOGENOUSLY CREATED DATA

5.1

Introduction

So far we have examined techniques for building a simulation model. These techniques
were centered around the topics of random number generation and simulation design. The
reason why one develops a simulation model is because one needs to estimate various
performance measures. These measures are obtained by collecting and analyzing
endogenously created data. In this Chapter, we will examine various estimation
techniques that are commonly used in simulation. Before we proceed to discuss these
techniques, we will first discuss briefly how one can collect data generated by a
simulation program.
5.2

Collecting endogenously created data

A simulation model can be seen as a reconstruction of the system under investigation.
Within this reconstructed environment, one can collect data pertaining to parameters of
interest. This is similar to collecting actual data pertaining to parameters of interest in a
real-life system. Using the techniques outlined in the previous Chapters, one can
construct, say, an event simulation model of a system. This model simply keeps track of
the state of the system as it changes through time. Now, one can incorporate additional
logic to the simulation program in order to collect various statistics of interest such as the
frequency of occurrence of a particular activity, and the duration of an activity. These

90


statistics are obtained using data generated from within the simulation programs, known
as endogenous data.
Endogenous data can sometimes be collected by simply introducing in the
simulation program single variables, acting as counters. However, quite frequently one is
interested in the duration of an activity. For instance, in the machine interference problem
one may be interested in the down time of a machine. This is equal to the time the
machine spends queueing up for the repairman plus the duration of its repair. In such
cases, one needs to introduce a storage scheme where the relevant endogenously created
data can be stored. The form of the storage scheme depends, of course, upon the nature of
the problem.
In the machine interference problem, the down time of a machine can be obtained
by keeping the following information: a) time of arrival at the repairman's queue, and b)
time at which the repair was completed. This information can be kept in an array. At the
end of the simulation run, the array will simply contain arrival and departure times for all
the simulated breakdowns. The down time for each breakdown can be easily calculated.
Using this information one can then obtain various statistics such as the mean, the
standard deviation, and percentiles of the down time.

F

ti

i

tj

j

•! • !•

ta k

B

Figure 5.1: The linked list for the repairman's queue.

A more efficient method would be to maintain a linked list representing those
machines which are in the repairman's queue. This linked list is shown in figure 5.1. Each
node contains the following two data elements: a) time of arrival at the repairman's
queue, and b) index number of the machine. The nodes are linked so that to represent the
FIFO manner in which the machines are served. Thus, the first node, pointed by F,
represents the machine currently in service. If a machine arrives at the repairman's queue,
a new node will be appended after the last node, pointed to by B. The total down time of
a machine is calculated at the instance when the machine departs from the repairman.

Estimation Techniques

91

This is equal to the master clock's value at that instance minus its arrival time. In this
fashion we can obtain a sample of observations. Each observation is the duration of a
down time. Such a sample of observations can be analyzed statistically.
Another statistic of interest in is the probability distribution of the number of
broken down machines. In this case, the maximum number of broken down machines
will not exceed m, the total number of machines. In view of this, it suffices to maintain an
array with m + l locations. Location i will contain the total time during which there were i
broken down machines. Each time an arrival or a departure occurs, the appropriate
location of the array is updated. At the end of the simulation run, the probability p(n) that
there are n machines down is obtained by dividing the contents of the nth location by T,
the total simulation time.
As another example on how to collect endogenously created data, let us consider
the simulation model of the token-based access scheme. The simulation program can be
enhanced so that each node is associated with a two-dimensional array, as shown in
figure 5.2. In each array, the first column contains the arrival times of packets, and the
second column contains their departure time. When a packet arrives at the node, its
arrival time is stored in the next available location in the first column. When the packet
departs from the node, its departure time is stored in the corresponding location of the
second column. Thus, each array contains the arrival and departure times of all packets
that have been through the node. Also, it contains the arrival times of all the packets
currently waiting in the queue. Instead of keeping two columns per node, one can keep
one column. When a packet arrives, its arrival time is stored in the next available
location. Upon departure of the packet, its arrival time is substituted by its total time in
the system. These arrays can be processed later in order to obtain various statistics per
node.
An alternative approach is to maintain a linked list per node, as described above
for the machine interference problem. Each time a packet is transmitted, its total time is
calculated by subtracting the current master clock value from it arrival time stored in the
linked list. This value is then stored into a single array containing all durations of the
packets in the order in which they departed.

92

Node 1
Arr Dep

Node 2
Arr Dep

Node 3
Arr Dep

x

x

x

x

x

x

x

x

x

x

Bottom
of queue

x

x
Top of
queue
Current
queue

x
x

x

x

x

x

x

x

x

x

x
x
x

Figure 5.2: Data structure for the token-based access scheme simulation model.

5.3

Transient state vs. steady-state simulation

In general, a simulation model can be used to estimate a parameter of interest during the
transient state or the steady state.
Let us consider the machine interference problem. Let us assume that one is
interested in obtaining statistics pertaining to the number of broken down machines. The
simulation starts by assuming that the system at time zero is at a given state. This is
known as the initial condition. Evidently, the behaviour of the system will be affected by
the particular initial condition. However, if we let the simulation run for a long period, its
statistical behaviour will eventually become independent of the particular initial
condition. In general, the initial condition will affect the behavior of the system for an
initial period of time, say T. Thereafter, the simulation will behave statistically in the
same way whatever the initial condition. During this initial period T, the simulated
system is said to be in a transient state. After period T is over, the simulated system is
said to be in a steady state.
5.3.1 Transient-state simulation
One may be interested in studying the behavior of a system during its transient state. In


93

this case, one is mostly interested in analyzing problems associated with a specific initial
starting condition. This arises, for example, if we want to study the initial operation of a
new plant. Also, one may be forced to study the transient state of a system, if this system
does not have a steady state. Such a case may arise when the system under study is
constantly changing.
5.3.2 Steady-state simulation
Typically, a simulation model is used to study the steady-state behaviour of a system. In
this case, the simulation model has to run long enough so that to get away from the
transient state. There are two basic strategies for choosing the initial conditions. The first
strategy is to begin with an empty system. That is, we assume that there are no activities
going on in the system at the beginning of the simulation. The second strategy is to make
the initial condition to be as representative as possible of the typical states the system
might find itself in. This reduces the duration of the transient period. However, in order to
set the initial conditions properly, an a priori knowledge of the system is required.
One should be careful about the effects of the transient period when collecting
endogenously created data. For, the data created during the transient period are dependent
on the initial condition. Two methods are commonly used to remove the effects of the
transient period. The first one requires a very long simulation run, so that the amount of
data collected during the transient period is insignificant relative to the amount of data
collected during the steady state period. The second method simply requires that no data
collection is carried out during the transient period. This can be easily implemented as
follows. Run the simulation model until it reaches its steady state, and then clear all
statistical accumulations (while leaving the state of the simulated system intact!).
Continue to simulate until a sufficient number of observations have been obtained. These
observations have all been collected during the steady-state. The second method is easy
to implement and it is quite popular.
The problem of determining when the simulation system has reached its steady
state is a difficult one. A simple method involves trying out different transient periods
T1,T2,T3,...,Tk, where T1<T2<T3<...<Tk. Compile steady-state statistics for each

94


simulation run. Choose Ti so that for all the other intervals greater than Ti, the steadystate statistics do not change significantly. Another similar method requires to compute a
moving average of the output and to assume steady-state when the average no longer
changes significantly over time.
5.4

Estimation techniques for steady-state simulation

Most of the performance measures that one would like to estimate through simulation are
related to the probability distribution of an endogenously created random variable. The
most commonly sought measures are the mean and the standard deviation of a random
variable. Also, of importance is the estimation of percentiles of the probability
distribution of an endogenously created random variable.
For instance, in the machine interference problem one may be interested in the
distribution of the down time. In particular, one may settle for the mean and standard
deviation of the down time. However, percentiles can be very useful too. From the
management point of view, one may be interested in the 95% percentile of the down
time. This is the down time such that only 5% of down times are greater than it.
Percentiles often are more meaningful to the management than the mean down time.
5.4.1

Estimation of the confidence interval of the mean of a random variable

Let x1, x2,..., xn be n consecutive endogenously obtained observations of a random
variable. Then

_
1 n
x = n ∑xi
i=1

(5.1)

is an unbiased estimate of the true population mean, i.e., the expectation of the random
variable. In order to obtain the confidence interval for the sample mean we have to first


95

estimate the standard deviation. If the observations x1, x2,... ,xn are independent of each
other, then

1
s2 = n - 1

n

_
∑ (xi -x)2

(5.2)

i=1

or, using the short-cut formula

1
s2 = n - 1

 n 2 (Σxi)2
∑xi - n 
 i=1


and, therefore, we obtain the confidence interval
_
_
s
s
(x - 1.96
, x + 1.96
)
n
n

at 95% confidence. The confidence interval provides an indication of the error associated
with the sample mean . It is a very useful statistical tool and it should be always
computed. Unfortunately, quite frequently it is ignored. The confidence interval tells us
that the true population mean lies within the interval 95% of the time. That is, if we
repeat the above experiment 100 times, 95% of these times, on the average, the true
population mean will be within the interval.
The theory behind the confidence interval is very simple indeed. Observations x1,
x2, …xn are assumed to come from a population known as the parent population whose
mean µ we are trying to estimate. Let σ2 be the variance of the parent population. The
distribution that x follows is known as the sampling distribution. Using the Central Limit
Theorem we have that x follows the normal distribution N(µ, σ/ n ), as shown in figure
!
5.3. Now, let us fix points a and b in this distribution so that 95% of the observations (an
!
!
observation here is a sample mean x ) fall in-between the two points. Points a and b are

!

96


symmetrical around µ. The areas (-∞, a) and (b, +∞) account for 5% of the total
distribution. Using the table of the standard normal distribution, we have that a is 1.96
standard deviation below µ, i.e., a=µ-1.96σ/ n . Likewise, b=µ+1.96σ/ n . Now, if we
consider an arbitrary observation x , this observation will lie in the interval [a,b] 95% of
!
!
the time. That is, its distance from µ will be less than 1.96σ/ n 95% of the time.

!
Therefore, µ will lie in the interval
!

( x - 1.96 σ n , x + 1.96 σ n )
!
! !
95% of the time. If σ is not known, one can use!in its place the sample standard deviation

s.

a

µ

b

Figure 5.3: The normal distribution.

The value 95% is known as the confidence. In general, a confidence interval can
be calculated for any value of confidence. Most typical confidence values are 99%, 95%
and 90%. For each value, points a and b are calculated from the table of the standard
normal distribution. If the sample size is small (less than 30), then we can construct
similar confidence intervals, but points a and b will be obtained using the t distribution,
i.e.,
( x - t.95 s , x + t.95 s )
n
n
!

!


97

with (n-1) degrees of freedom.
In general, the observations x1, x2,..., xn that one obtains endogenously from a
simulation model are correlated. For instance, the down time of a machine depends on the
down time of another machine that was ahead of it in the repairman's queue. In the
presence of correlated observations, the above expression (5.2) for the variance does not
hold. Expression (5.1) for the mean holds for correlated or uncorrelated observations. The
correct procedure, therefore, for obtaining the confidence interval of the sample mean is
to first check if the observations x1, x2,..., xn are correlated. If they are not, one can
proceed as described above. If the observations are correlated, then one has to use a
special procedure to get around this problem. Below, we discuss the following four
procedures for estimating the variance of correlated observations:
a. Estimation of the autocorrelation function.
b. Batch means.
c. Replications.
d. Regenerative method.
a)

Estimation of the autocorrelation coefficients

Let X and Y be two random variables. Let µX and µY be the expectation of X and Y
2
2
respectively. Also, let " X and " Y be the variance of X and Y respectively. We define the

covariance between X and Y to be
!

!

Cov (X,Y) = E[(X-µX) (Y-µY)]
= E(XY) - µX µY
This statistic reflects the dependency between X and Y. If X and Y are uncorrelated, then
Cov(X,Y) = 0. If Cov (X,Y) > 0, then X and Y are positively correlated, and if Cov (X,Y)
2
<0, then they are negatively correlated. If Y is identical to X, then Cov(X,X)= " X .

!

98


(i) uncorrelated, i.e. Cov(X,Y) = 0.

(ii) positive correlation, i.e. Cov (X,Y) > 0.

(iii) negative correlation, i.e. Cov (X,Y) <0.

Figure 5.4: The three cases of correlation.


99

Let us assume that we have obtained actual observations of the random variables
X and Y in the form of (x,y). Then, the scatter diagram can be plotted out. The scatter
diagrams given in figure 5.4 shows the three cases of correlation between X and Y
mentioned above.
The Cov (X,Y) may take values in the region (-∞, +∞). Also, it is not
dimensionless, which makes its interpretation troublesome. In view of this, the
correlation ρxy defined by

ρ XY =

Cov(X,Y)
σx σy

is typically used as the measure of dependency between X and Y. It can be shown that 1≤ρXY≤1. If ρXY is close to 1, then X and Y are highly positively correlated. If ρXY is
close to -1, then they are highly negatively correlated. Finally, if ρXY=0, then they are
independent from each other.
Now, let us assume we have n observations x1, x2, ..., xn. We form the following
n-1 pairs of observations: (x1,x2), (x2,x3), (x3,x4), ..., (xi,xi+1), ..., (xn-1,xn). Now, let us
regard the first observation in each pair as coming from a variable X and the second
observation as coming from a variable Y. Then, in this case ρXY is called the
autocorrelation or the serial correlation coefficient. It can be estimated as follows:
n"1

# (x i "X )(x i+1 "Y )

r1 =

i=1
n"1

2

n"1

# (x i "X) # (x i+1 "Y )

i=1

,
2

i=1

_
_
1 n-1
1 n
where X = n-1 ∑xi !
and Y = n-1 ∑xi . For n reasonably large, ρXY can be
i=1
i=2
approximated by

100

n"1

# (x i "X )(x i+1 "X )

r1 =

i=1

n"1

# (x i "X) 2

i=1

_ 1n
!
where X = n ∑xi is the overall mean.
i=1

We refer to the above estimate of ρXY as r1 in order to remind ourselves that this
is the correlation between observations which are a distance of 1 apart. This autocorrelation is often referred to as lag 1 autocorrelation. In a similar fashion, we can
obtain the lag k autocorrelation, that is the correlation between observations which are a
distance k apart. This can be calculated using the expression:
n"k

# (x i "X )(x i+k "X )

rk =

i=1

n"1

# (x i "X )

)
2

i=1

In practice, the autocorrelation coefficients are usually calculated by computing the series
!
of autocovariances R0, R1, ..., where Rk is given by the formula

Rk =

1 n"k
# (x i "X )(x i+k "X )
n i=1

(5.3)

We then compute rk as the ratio
!
Rk
rk = R0 ,

(5.4)

where R0 = σ2. Typically, rk is not calculated for values of k greater than about n/4.


101

+1

rk

0
lag k
Figure 5.5: A correlogram with short-term correlation.

A useful aid in interpreting the autocorrelation coefficients is the correlogram.
This is a graph in which rk is plotted against lag k. If the time series exhibits a short-term
correlation, then its correlogram will be similar to the one shown in figure 5.5. This is
characterized by a fairly large value of r1 followed by 2 or 3 more coefficients which,
while significantly greater than zero, tend to get successively smaller. Values of rk for
longer lags tend to be approximately zero. If the time series has a tendency to alternate,
with successive observations on different sides of the overall mean, then the correlogram
also tends to alternate as shown in figure 5.6.
+1

rk

0
lag k
Figure 5.6: An alternating correlogram.

102

Let us now return to our estimation problem. Having obtained a sample of n

observations x1,x2,...,xn, we calculate the autocorrelation coefficients using the
expressions (5.3) and (5.4). Then, the variance can be estimated using the expression
n"1

2
s2 = sX [1 + 2 # (1"
k=1

k
rk ) ]
n

2
!
where sX is the standard deviation given by
!

2
sX

!

_
1 n
and X = " x i .
n

b)
!

=

1 n
# (x i "X )
n "1 i=1

!

!

i=1

Batch Means

This is a fairly popular technique. It involves dividing successive observations into
batches as shown in figure 5.7. Each batch contains the same number of observations. Let
the batch size be equal to b. Then batch 1 contains observations x1, x2, ..., xb, batch 2
contains xb+1, xb+2, ...,

| x1, x2, ..., xb | xb+1, xb+2, ..., x2b |
batch 1

...|

xkb+1, xkb+2, ..., x(k+1)b

batch 2

|

batch k

Figure 5.7: The batch means method.

x2b, and so on. The observations close to batch 2 are likely to be correlated with the
observations in batch 2 which are close to batch 1. Also, the observations in batch 2
which are close to batch 3 are likely to be correlated with those in batch 3 which are close

to batch 2, and so on. Let

103

X i be the sample mean of the observations in batch i. If we

choose b to be large enough, then the sequence

X 1 , X 2 , ..., X k can be shown that it is

approximately uncorrelated. Therefore, we can treat these means as a sample of

!
independent observations and calculate their mean and standard deviation. We have
! !

!

_ 1 k _
X =k ∑ Xi
i=1

s2 =

2
1 k
# (X i "X ) .
k "1 i=1

Therefore, for large k (i.e. k ≥30), we obtain the confidence interval
!
_
s _
s
(X - 1.96
, X + 1.96 ).
k
k

For small k, we can construct our confidence interval using the t distribution.
In general, the batch size b has to be large enough so that the successive batch
means are not correlated. If b is not large, then the successive batch means will be
correlated, and the above estimation procedure will yield severely biased estimates. An
estimate of b can be obtained by plotting out the correlogram of the observations
x1,x2,...,xn, which can be obtained from a preliminary simulation run. We can fix b so
that it is 5 times the smallest value b' for which rb' is approximately zero.
c)

Replications

Another approach to constructing a confidence interval for a mean is to replicate the
simulation run several times. Suppose we make n replications, each resulting to n
observations as follows:
replication 1:

x11, x12, ..., x1m

104

replication 2:

x21, x22, ..., x2m

.
.
.
replication n:

xnm, xn2, ..., xnm

For each sample, we can construct the sample mean

_
1 n
X i = m ∑xij .
j=1
_ _
_
We can then treat the sample means X 1, X 2, ..., X n as a sample of independent
observations, thus obtaining
_ 1 n _
X =n ∑ Xi
i=1

s2 =

1 n
# (X i " X ) 2.
n i=1

Using the above statistics we can construct our confidence interval.
!
The problems that arise with this approach are: a) decide on the length of each
simulation run, i.e., the value m, and b) decide on the length of the transient period. One
of the following two approaches can be employed:
Start each simulation run with different values for the seeds of the random
number generators. Allow the simulation to reach its steady state and then collect the
sample observations. Repeat this procedure n times. In order to obtain n samples,
therefore, we have to run n independent simulations, each time having to allow the
simulation to reach its steady state.


105

Alternatively, we can run one very long simulation. Allow first the simulation to
reach its steady state, and then collect the first sample of observations. Subsequently,
instead of terminating the simulation and starting all over again, we extend the simulation
run in order to collect the second sample of observations, then the third sample and so on.
The advantage of this method is that it does not require the simulation to go through a
transient period for each sampling period. However, some of the observations that will be
collected at the beginning of a sampling period will be correlated with observations that
will be collected towards the end of the previous sampling period.
The replication method appears to be similar to the batch means approach.
However, in the batch means method, the batch size is relatively small and, in general,
one collects a large number of batches. In the above case, each sampling period is very
large and one collects only a few samples.
d)

Regenerative method
The last two methods described above can be used to obtain independent or

approximately independent sequences of observations. The method of independent
replications generates independent sequences through independent runs. The batch means
method generates approximately independent sequences by breaking up the output
generated in one run into successive subsequences which are approximately independent.
The regenerative method produces independent subsequences from a single run. Its
applicability, however, is limited to cases which exhibit a particular probabilistic
behaviour.
Let us consider a single server queue. Let t0, t1, t2,... be points at which the
simulation model enters the state where the system is empty. Such time instances occur
when a customer departs and leaves an empty system behind. Let t0 be the instance when
the simulation run starts assuming an empty system. The first customer that will arrive
will see an empty system. During its service, other customers may arive thus forming a
queue. Let t1 be the point at which the last customer departs and leaves an empty system.
That is, t1 is the time instance where the server becomes idle. This will repeat itself as
shown in figure 5.8. It is important to note that the activity of the queue in the interval (ti,

106


ti+1) is independent of its activity during the previous interval (ti-1,ti). That is to say, the
probability of finding customer n in the interval (ti, ti+1) does not depend on the number
of customers in the system during the previous interval. These time instances are known
as regeneration points. The time between two such points is known as the regeneration
cycle or tour. The queue-length distribution observed during a cycle is independent of the
distribution observed during the previous cycles. The same applies for the density
probability distribution of the total time a customer spends in the system.

Number
in system

t0

t1

t2

t3

Time

Figure 5.8: Regeneration points.

The regeneration points identified above in relation with the queue-length (or
waiting time) probability distribution are the time points when the system enters the
empty state. Regeneration points identified with time instances when the system enters
another state, can be also obtained if the service time is exponentially distributed. For
instance, the time instances when the system contains 5 customers are regeneration
points. Due to the memoryless property of the service time, the process repeats itself
probabilistically.
In the case of the machine interference problem, regeneration points related to the
repairman's queue-length probability, or waiting time in the queue, can be identified with
the time instances when the repairman is idle. Such time instances occur when the last
machine is repaired and the system enters the state where all machines are operational.
Due to the assumption that operational times are exponentially distributed, the process


107

repeats itself probabilistically following the occurrence of this state. Other states can be
used to identify regeneration points, assuming exponentially distributed repair times.
Now, let us assume that we are interested in estimating the mean value of a
random variable X. Let (xi1, xi2, ..., xini) be a sequence of realizations of X during the ith
regeneration cycle occurring between ti and ti+1 regeneration points. Let ni be the total
number of observations. This sequence of observations will be independent from the one
obtained during the (i+1)st cycle. Thus, the output is partitioned into independent
sequences. In the batch means case, the output was partitioned into approximately
independent sequences. However, the number of observations in each batch was constant
equal to batch size. In this case, the number of observations in each cycle is a random
variable.
Due to the nature of the regeneration method, there is no need to discard
observations at the beginning of the simulation run. The simulation starts by initially
setting the model to the regeneration state. Then the simulation is run for several cycles,
say M cycles. Let

ni
Zi = ∑xij
j=1

be the sum of all realizations of X in the ith cycle. Then a point estimate of E(X) can be
obtained using the expression:

1 M
M ∑ Zi

_
i=1
X = 1 M
.
ni
M∑
i=1

That is, E(X) is estimated using the ratio E(Z)/E(N), where Z is a random variable
indicating the sum of all observations of

X in a cycle, and N is a random variable

indicating the number of observations in a cycle.

!

108


X is not an unbiased estimator of E(X). That is E( X ) ≠ E(X). However, it is
consistent. That is,

!

interval of

X → E(X) as M → ∞. Now, in order to construct a confidence

X , we note that X is obtained as the ratio of two random variables. The
!

following two methods can be used to construct confidence intervals for a ratio of two

!

random variables: the central limit theorem, and the jackknife method.

!

!

The central limit theorem
2
2
Let σ12 = Cov (Z, N), " 11= Var(Z), and " 22 = Var(N). Also, let V=Z-NE(X). That is, for

cycle i we have Vi=Zi-niE(X). Then, the Vi's are independent and identically distributed
with a mean E(V) = 0. This because
!
!
E(V)

= E(Z - NE(X))
= E(Z) - E(N)E(X)
= 0.

The variance σ2 of the Vi's is as follows:
v

E(V - E(V))2

= E(V2)
= E(Z - NE(X))2
= E(Z2 - 2ZNE(X) + N2E(X)2)
= E(Z2) - 2E(X)E(ZN) + E(N2)E(X)2.

Since Var(Y) = E(Y2) - E(Y)2 we have that
2
2
σ2 = σ11 + E(Z)2 - 2E(X)E(ZN) + E(X)2 σ22 + E(N)2E(X)2
v
2
2
= σ11 + E(X)2 σ22 - 2(E(X)E(ZN) + E(Z)2 + E(N)2E(X)2.


109

Given that E(Z) = E(N)E(X) we have
2
2
σ2 = σ11 + E(X)2 σ22 - 2E(X)E(ZN) + 2E(N)2E(X)2
v
2
2
= σ11 + E(X)2 σ22 - 2E(X)[E(ZN) - E(X)E(N)2

2
2
= σ11 + E(X)2 σ22 - 2E(X)[E(ZN) - E(Z)E(N)]
or
2
2
σ2 = σ11 + E(X)2 σ22 - 2E(X)σ12
v
since
Cov(Y1, Y2)

= E(Y1 - E(Y1))(Y2 - E(Y2))
= E(Y1Y2 - E(Y1)Y2 - E(Y2)Y1 + E(Y1)E(Y2))
= E(Y1Y2) - E(Y1)E(Y2) - E(Y2)E(Y1) + E(Y1)E(Y2)
= E(Y1Y2) - E(Y1)E(Y2).

_
By the central limit theorem, we have that as M increases V becomes normally
distributed with a mean equal to 0 and a standard deviation equal to

_
1 M
V = M ∑Vi
i=1

1 M
= M ∑ (Zi - niE(X))
i=1

!

2
" v / M , where

110


1 M
1 M
= M ∑ Zi - E(X) M ∑ ni
i=1
i=1

_
_
= Z - E(X)N .
Hence,

_
_
Z - E(X)N ~ N(0,

σ2
v
M )

or
_
_
Z - E(X)N
~ N(0, 1)
σ2
v
M

Dividing by N we obtain that

_ _
Z/N - E(X)
~ N(0, 1).
_
σ2
v
(1/N)
M

Therefore, we obtain the confidence interval

_
Z
_ ± 1.96
N

σ2 / M
v
.
_
N


111

_
_
2
2
Now, it can be shown that s2 = sZ - 2X RZ,N + X 2sN is an estimate of σ2. Therefore, it
v
can be used in place of σ2 in the above confidence interval.
v
The above method can be summarized as follows:
1. Run the simulation for M cycles and obtain
Z1, Z2, Z3, ..., ZM
n1, n2, n3, ..., nM.
_ _
_
_ _
2. Estimate Z , N and set X = Z / N .
3. Estimate
M

s2 =
Z

1
# (zi "Z 2 )
M "1 i=1

2
sN =

1 M
# (ni "N 2 )
M "1 i=1

2
sZ,N =

1
# (zi "Z)(ni "N) .
M "1 i=1

!
!

M

The jacknife method
!
This method constitutes an alternative method to obtaining a confidence interval of a
ratio of two random variables. It also provides a means of obtaining a less biased point
estimator of the mean of the ratio of two random variables, sine the classical point
estimator is usually biased.
Let X,Y be two random variables. Suppose we want to estimate φ=E(Y)/E(X)
from the data y1, y2, ..., yn and x1, x2, ..., xn, where the yi's are i.i.d., the xi's are i.i.d. and
Cov(yi,xj)≠0 for i≠j. The classical point estimator of φ is φc =

Y / X and its confidence

interval can be obtained as shown above. The jacknife point and interval estimator can be
constructed as follows. Let

! !

112


∑yj

j≠g
^
θg = n φ C - (n - 1)
, g = 1,2, ...,n.
∑xj
j≠g

The jacknife point estimator of φ is
n
^
φ J = ∑θg/n .
g=1
^
This is, in general, less biased than φ C. Let

σ2 =
J

n

"
g=1

(# g $% j ) 2
n $1

.

Then, it can be shown that
!
^
φ J ~ N(φ,

σ2
J
n ), as n → ∞.

That is, we can obtain the confidence interval at 95%
σJ
σJ
^
^
(φ J -1.96
, φ J +1.96
).
n
n

The difficulty in using the regenerative method is that real-world simulations may
not have regeneration points. If they do happen to have regeneration points, the expected
regeneration cycle may be too large so that only a few cycles may be simulated.
5.4.2

Estimation of other statistics of a random variable

So far we considered estimation techniques for constructing a confidence interval of the
mean of an endogenously created random variable. Other interesting statistics related to
the probability distribution of a random variable are:


113

a. Probability that a random variable lies within a fixed interval.
b. Percentiles of the probability distribution.
c. Variance of the probability distribution.
Below, we examine ways of estimating the above statistics.
a)

Probability that a random variable lies within a fixed interval

The estimation of this type of probability can be handled exactly the same way as the
estimation of the mean of a random variable. Let I be the designated interval. We want to
estimate
p = Pr (X ∈ I)
where X is an endogenously created random variable. We generate M replications of the
simulation. For each replication i we collect N observations of X. Let vi be the number of
times X was observed to lie in I. Then, pi = vi/N is an estimate of probability p. Thus,

_ 1 M
p = M ∑pi
i=1
and
s2 =

1 M
# ( pi " p)2 .
M "1 i=1

_
As before, (p -p)/(s/ M ) ~ N(0,1) if M is large.
!
We observe, that the estimation of p requires M independent replications, each
giving rise to one realization of p. Other methods examined in section 4.1 of this Chapter
can be used in order to remove unwanted autocorrelations. For instance, instead of
replications, the batch means method or the regenerative method can be used.
Alternatively, the estimation of p can be seen as estimating a population mean.
Let Yi=1 if ith realization of X belongs to I. Otherwise, Yi=0. Then, in one replication we

114


have that (1/N)ΣYi is equal to Vi/N or pi. Thus, the techniques developed in section 4.1
of this Chapter can be employed to estimate p.

!

x!
Figure 5.9: Percentile xβ .

b)

Percentile of a probability distribution

This is a very important statistic that is often ignored in favour of the mean of a random
variable X. Management, sometimes, is not interested in the mean of a particular random
variable. For instance, the person in charge of a computer real-time system may not be
interested in its mean response time. Rather, he or she may be interested in "serving as
many as possible as fast as possible". More specifically, he or she may be interested in
knowing the 95th percentile of the response time. That is, a value such that the response
time of the real time system is below it 95% of the time.
In general, let us consider a probability density function f(x). The 100βth
percentile is the smallest value xβ such that f(xβ) < β. That is, the area from -∞ to xβ
under f(x) is less or equal to β as shown in figure 5.9. Typically, there is interest in the
50th percentile (median) x0.50 or in extreme percentiles such as x0.90, x0.95, x0.99.
We are interested in placing a confidence interval on the point estimator of xβ of a
distribution of a random variable X. Let us assume independent replications of the
simulation. Each replication yields N observations, having allowed for the transient
period. For each replication i, let xi1, xi2, ..., xiN be the observed realizations of X. Now,
let us consider a reordering of these observations Yil, Yi2, ..., YiN so that Yij<Yi,j+1.
Then, the 100βth percentile x(i) for the ith replication is observation Yik where
β


115

k=

N


∈βN


+1

if Nβ is integer
otherwise

(x means the largest integer which is less or equal to x). For instance, if we have a
sample of 50 observations ordered in an ascending order, then the 90th percentile is the
observation number 0.90.50 = 45. The 95th percentile is 50x.95+1=47.5+1=47+1=48.
Hence
_
1 M (i)
xβ=
"x
M i=1 β
and
2
s! =

M

1
# (x(i)"x $ ) 2.
M "1 i=1

Confidence intervals can now be constructed in the usual manner.
!
The estimation of extreme percentiles requires long simulation runs. If the runs
are not long, then the estimates will be biased. The calculation of a percentile requires
that a) we store the entire sample of observations until the end of the simulation, and b)
that we order the sample of observations in an ascending order. These two operations can
be avoided by constructing a frequency histogram of the random variable on the fly.
When a realization of the random variable becomes available, it is immediately classified
into the appropriate interval of the histogram. Thus, it suffices to keep track of how many
observations fall within each interval. At the end of the replication, the 100βth percentile
can be easily picked out from the histogram. Obviously, the accuracy of this
implementation depends on the chosen width of the intervals of the histogram.
Finally, we note that instead of independent replications of the simulation, other
methods can be used such as the regeneration method.
c)

Variance of the probability distribution

Let us consider M independent replications of the simulation. From each replication i we
obtain N realizations of a random variable xi1,xi2,...,xiN, after we allow for the transient

116


period. We have
_
1 N
µ i = N ∑ xij
j=1

and
_
_
1 M
µ = M ∑µ i
i=1

is the grand sample mean. Hence,

s2 =
i

1 N
" (x ij # µ) )2
N j=1

_ _ _
_ _
1 N 2
= N [ ∑ xij - 2µ iµ +µ 2].
j=1

!

As the point estimate of the variance σ2 we take

s2 =
i

1 M 2
"s
M i=1 i

=

1 M 1 N 2
2
1 2 M
( " x ij ) " N
" µi µ + M µ 2
M i=1
M N i=1
j=1

=

_
1 M 1 N 2 _2
" ( N " x ij) - µ
M i=1 ! j=1
!

!

!
2

The estimates of si are all functions of

µ . Thus, they are not independent. A confidence

interval can be constructed by jacknifing the estimator s2. Let " 2 be an estimate of σ2
!
i
with the ith replication left out. That is,

!

_ ! 1
_
1
1 N
( $ x2) - M! 1 µj 2 .
σ2 =
$ N
- j≠i
i
M "1 j#i
i=1
∑

!


117

_
Define zi = Ms2 - (M-1) σ2 . Then,
i

1
Z=M

M Zi
i=1
∑

_
1
2
sZ = M-1 M (Zi - Z) 2
i=1
∑

and a confidence interval can be constructed in the usual way.
Alternatively, we can obtain a confidence interval of the variance by running the
simualtion only once, rather than using repications, and then calculating the standard
deviation assuming that the successive observations are independent! This approach is
correct when the sample of observations is extremely large.
5.5

Estimation techniques for transient-state simulation

The statistical behaviour of a simulation during its transient state depends on the initial
condition. In order to estimate statistics of a random variable X during the transient state
one needs to be able to obtain independent realizations of X. The only way to get such
independent observations is to repeat the simulation. That is, employ the replications
technique discussed above in section 4.1. Each independent simulation run has to start
with the same initial condition.
As an example, let us assume that we want to estimate the 95th percentile of the
probability distribution of a random variable X. This can be achieved by simply
replicating the simulation times. For each replication we can obtain an estimate x.95. A
confidence interval can be obtained as shown in 4.2. Each replication will be obtained by
starting the simulation with the same initial condition, and running the simulation during
its transient period. This, of course, requires advanced knowledge of the length of the
transient period. Furthermore, the pseudo-random numbers used in a replication have to
be independent of those used in previous replications. This can be achieved by using a

118


different seed for each replication. The difference between two seeds should be about
10,000.
5.6

Pilot experiments and sequential procedures for achieving a required
accuracy

So far, we discussed techniques for generating confidence intervals for various statistics
of an endogenously generated random variable. The expected width of the confidence
interval is, in general, proportional to 1/ N , where N is the number of i.i.d. observations
used. Obviously, the larger the value of N, the smaller the width confidence interval.
(This is defined as half the confidence interval.) For instance, in order to halve the width
of the confidence interval of the mean of a random variable, N has to be increased by four
times so that 1/ 4N =(1/2)(1/ N ). In general, the accuracy of an estimate of a statistic
depends on the width of the confidence interval. The smaller the width, the higher is the
accuracy.
Quite frequently one does not have prior information regarding the value of N that
will give a required accuracy. For instance, one does not know the value of N that will
yield a width equal to 10% of the estimate. Typically, this problem is tackled by
conducting a pilot experiment. This experiment provides a rough estimate of the value of
N that will yield the desired confidence interval width. An alternative approach is the
sequential method. That is, the main simulation experiment is carried out continuously.
Periodically, a test is carried out to see if the desired accuracy has been achieved. The
simulation stops the first time it senses that the desired accuracy has been achieved.
Below, we examine the methods of independent replications and batch means.
5.6.1

Independent replications

This discussion is applicable to transient and steady state estimation. Let us assume that
we want to estimate a statistic θ of a random variable using independent replications. A

ˆ
pilot experiment is first carried out involving N1 replications. Let "1 be a point estimate

!


119

of θ and let Δ1 be the width of its confidence interval. We assume that the width of the
^
ˆ
confidence interval is required to be approximately less or equal to 0.1θ . If Δ1≤0.1 "1 ,

ˆ
then we stop. If Δ1>0.1 "1 , then we need to conduct a main experiment involving N2
ˆ
replications where, N2 = (Δ1/0.1 "1 )2N1 appropriately rounded off to the nearest integer.
!
ˆ
Let " 2 and Δ2 be the new estimate and width based on N2 replications. Due to
!
randomness, the N2 observations will not exactly yield a confidence interval width equal
!

ˆ
to 0.1 " 2 . We note that it is not necessary to run a new experiment involving N2

!

replications. Rather, one can use the N1 replications from the pilot experiment, and then

!

carry out only N2- N1 additional replications.
The above approach can be implemented as a sequential procedure as follows.
The simulation program is modified to conduct N* independent replications. The point

ˆ
ˆ
estimate "1 and the width Δ1 of its confidence interval are calculated. If Δ1≤0.1 "1 , then
the simulation stops. Otherwise, the simulation proceeds to carry out N* additional

ˆ
replications. The new point estimate " 2 and width Δ2 are constructed !
based on the total
!
ˆ
2N* replications. The simulation will stop if Δ ≤0.1 " . Otherwise, it will proceed to
2

2

generate another N* replication and so on. N* can be set equal to 10.
!

!
5.6.2

Batch means

This discussion is obviously applicable to steady state estimations. Using the pilot
experiment approach, one can run the simulation for k1 batches. From this, one can

ˆ
ˆ
obtain "1 and Δ1. If Δ1≤0.1 "1 , then the desired accuracy has been achieved. Otherwise,
ˆ
k2-k1 additional batches have to be simulated, where k2=(Δ1/0.1 "1 )2k1. These additional
! batches can be obtained by simply re-running the simulator for k2-k1 batches.
!
Alternatively, the simulation program can be enhanced to include the following
!
ˆ
sequential procedure. The simulation first runs for k* batches. Let "1 and Δ1 be the point
estimate and the width of its confidence interval respectively. The simulation stops if

ˆ
Δ1≤0.1 "1 . Otherwise, it runs for k* additional batches. Now, based on the 2k* batches
!

!

120


ˆ
ˆ
generated to this point, " 2 and Δ2 are constructed. If Δ2≤0.1 " 2 , then the simulation stops.
Otherwise, it runs for another k* batches, and so on.

!
1.

!

Consider the machine interference problem. Modify the simulation model to carry
out the following tasks:
a.

Set-up a data structure to collect information regarding the amount of time
each machine spends being broken down, i.e., waiting in the queue and also
being repaired (see section 2 of Chapter 3).

b.

Augment your simulation program to calculate the mean and standard
deviation. Then, run your simulation for 550 observations (i.e., repairs).
Discard the first 50 observations to account for the transient state. Based on
the remaining 500 observations, calculate the mean and the standard
deviation of the time a machine is broken down.

c.

The calculation of the standard deviation may not be correct, due to the
presence of autocorrelation. Write a program (or use an existing statistical
package) to obtain a correlogram based on the above 500 observations. (You
may graph the correlogram by hand!) Based on the correlogram, employ the
batch means approach to construct a confidence interval of the mean time a
machine is broken down. (You may need more than 500 observations!)

d.

Using the batch means approach, implement a sequential procedure for
estimating the mean time a machine is broken down (see section 5.6.2).

e.

Implement a scheme to estimate the 95th percentile of the time a machine is
broken down.

2.

Likewise for the token-based access scheme.

3.

Likewise for the to-stage manufacturing system.

CHAPTER 6:
VALIDATION OF A SIMULATION MODEL

Validation of a simulation model is a very important issue that is often neglected. How
accurately does a simulation model (or, for that matter, any kind of model) reflect the
operations of a real-life system? How confident can we be that the obtained simulation
results are accurate and meaningful?
In general, one models a system with a view to studying its performance. This
system under study may or may not exist in real-life. Let us consider, for instance, a
communications equipment manufacturer who is currently designing a new
communications device, such as a switch. Obviously, the manufacturer would like to
know in advance if the new switch has an acceptable performance. Also, the
manufacturer would like to be able to study various alternative configurations of this new
switch so that to come up with a good product. The performance of such a switch can be
only estimated through modelling since the actual system does not exist. The question
that arises here is how does one make sure that the model that will be constructed is a
valid representation of the system under study?
Let us consider another example involving a communication system already in
operation. Let us assume that it operates nearly at full capacity. The management is
considering various alternatives for expanding the system's capacity. Which of these
alternatives will improve the system's performance at minimum cost?

Now, these

alternative configurations do not exist. Therefore, their performance can be only
evaluated by constructing a model, say using simulation techniques. The standard method
is to construct a model of the existing system. Then, change the model appropriately in
order to analyze each of the above alternatives. The model of the existing system can be

122


validated by comparing its results against actual data obtained from the system under
investigation. However, there is no guarantee that when altering the simulation model so
that to study one of the above alternatives, this new simulation model will be a valid
representation of the configuration under study!
The following checks can be carried out in order to validate a simulation model.
1. Check the pseudo-random number generators. Are the pseudo-random numbers
uniformly distributed in (0,1) and do they satisfy statistical criteria of
independence? Usually, one takes for granted that a random number generator is
valid.
2. Check the stochastic variate generators. Similar statistical tests can be carried out
for each stochastic variate generator built into a simulation model.
3. Check the logic of the simulation program. This is a rather difficult task. One way
of going about it is to print out the status variables, the future event list, and other
relevant data structures each time an event takes place in the simulation. Then,
one has to check by hand whether the data structures are updated appropriately.
This is a rather tedious task. However, using this method one can discover
possible logical errors and also get a good feel about the simulation model.
4. Relationship validity. Quite frequently the structure of a system under study is not
fully reflected down to its very detail in a simulation model. Therefore, it is
important that the management has the opportunity to check whether the model's
assumptions are credible.
5. Output validity. This is one of the most powerful validity checks. If actual data are
available regarding the system under study, then these data can be compared with
the output obtained from the simulation model. Obviously, if they do not compare
well, the simulation model is not valid.


123

1.

Consider the machine interference problem. This problem can be also analyzed
using queuing theory. Set up the following validation scheme. Obtain exact values
of the mean time a machine is broken down using queueing theory results for
various values of the mean operational time. Let it vary, for instance, from 1 to 50
so that to get a good spread. For each of these values, obtain an estimate of the
mean down time of a machine using your simulation model. Graph both sets of
results. Be sure to indicate the confidence interval at each simulated point.
Compare the two sets of results.

CHAPTER 7:
VARIANCE REDUCTION TECHNIQUES

NC S U

Variance
Reduction

7.1

Introduction

In Chapter 6, it was mentioned that the accuracy of an estimate is proportional to 1/ n ,
where n is the sample size. One way to increase the accuracy of an estimate (i.e., reduce
the width of its confidence interval) is to increase n. For instance, the confidence interval
width can be halved if the sample size is increased to 4n. However, large sample sizes
required long simulation runs which, in general, are expensive. An alternative way to
increasing the estimate's accuracy is to reduce its variance. If one can reduce the variance
of an endogenously created random variable without disturbing its expected value, then
the confidence interval width will be smaller, for the same amount of simulation.

126


Techniques aiming at reducing the variance of a random variable are known as Variance
Reduction Techniques. Most of these techniques were originally developed in connection
with Monte Carlo Techniques.
Variance reduction techniques require additional computation in order to be
implemented. Furthermore, it is not possible to know in advance whether a variance
reduction technique will effectively reduce the variance in comparison with
straightforward simulation. It is standard practice, therefore, to carry out pilot simulation
runs in order to get a feel of the effectiveness of a variance reduction technique and of the
additional computational cost required for its implementation.
In this Chapter, we will examine two variance reduction techniques, namely, a)
the antithetic variates technique and (b) the control variates technique.
7.2

The Antithetic Variates Technique

This is a very simply technique to use and it only requires a few additional instructions in
order to be implemented. No general guarantee of its effectiveness can be given. Also, it
is not possible to know in advance how much of variance reduction can be achieved.
Therefore, a small pilot study may be useful in order to decide whether or not to
implement this technique.
Let X be an endogenously created random variable. Let
x(1) , x(1) , ..., x(1)
1
2
n
be n i.i.d. realizations of X obtained in a simulation run. Also, let
x(2) , x(2) , ..., x(2)
1
2
n
be n i.i.d. observations of X obtained in a second simulation run. Now, let us define a
new random variable

Variance Reduction Techniques

127

x(1) + x(2)
i
i
zi =
, i = 1, 2, ..., n.
2

(7.1)

More specifically, let Z=(X(1)+X(2))/2, where X(i), i=1,2, indicates the random variable X
as observed in the ith simulation run. We have

X(1)+X(2)

E(Z)= E
2
1
= 2 [E(X(1)) + E(X(2))]
= E(X)
seeing that the expected value of X(1) or X(2) is that of X. Thus, the expected value of this
new random variable Z is identical to that of X. Now, let us examine its variance. We
have

X(1)+X(2)

Var(Z) = Var
2

1
= 4 [Var(X(1)) + Var(X(2)) + 2Cov(X(1), X(2))].

Remembering that Var(X(1))=Var(X(2))=Var(X), we have that

1
Var(Z) = 2 (Var(X) + Cov (X(1), X(2))) .

Since Cov(X,Y)=ρ Var(X)Var(Y) , we have that

1
Var(Z) = 2 Var(X) (1 + ρ),

(7.2)

128


where ρ is the correlation between X(1) and X(2).
In order to construct an interval estimate of E(X), we use random variable Z.
Observations zi are obtained from (7.1) and the confidence interval is obtained using
(7.2). As will be seen below, by appropriately constructing the two samples
x(1) , x(1) , ..., x(1) and x(2) , x(2) , ..., x(2) ,
1
2
n
1
2
n
we can cause Var(Z) to become significantly less than Var(X). This is achieved by
causing ρ to become negative. In the special case where the two sets of observations
x(1) , x(1) , ..., x(1) and x(2) , x(2) , ..., x(2)
1
2
n
1
2
n
are independent of each other, we have that ρ=0. Hence, Var(Z)=Var(X)/2.
The antithetic variates technique attempts to introduce a negative correlation
between the two sets of observations. As an example, let us consider a simulation model
of a single server queue, and let X and Y indicate the waiting time in the queue and the
interarrival time respectively. If Y is very small, then customers arrive faster and,
therefore, the queue size gets larger. The larger the queue size, the more a customer has
to wait in the queue, i.e. X is larger. On the other hand, if Y is large, then customers
arrive slower and, hence, the queue size gets smaller. Obviously, the smaller the queue
size, the less a customer has to wait in the queue, i.e., X is small. Therefore, we see that X
and Y can be negatively correlated.
This negative correlation between these two variables can be created in a
systematic way as follows. Let F(t) and G(S) be the cumulative distribution of the interarrival and service time respectively. Let ri and vi be pseudo-random numbers. Then,
ti=F-1(ri) andsi=G-1(vi) are an interarrival and a service variate. These two variates can be
associated with the ith simulated customer. An indication of whether the queue is tending
to increase or decrease can be obtained by considering the difference di=ti-si. This
difference may be positive or negative indicating that the queue is going through a busy


129

or slack period respectively. Now, let us consider that in the second run, we associate
pseudo-random number r´ and v´ with the ith simulated customer, so that
i
i
d´ = t1 - s´ (where t´ =F-1(r´) and s´ = G-1(v´))
´ i
i
1
i
i
i
has the opposite sign of di. That is, if the queue was going through a slack (busy) period
in the first run at the time of the ith simulated customer, now it goes through a busy
(slack) period. It can be shown that this can be achieved by simply setting r´ = 1-ri and
i
v´ = 1-vi.
i
In this example, we make use of two controllable variables, Y1 and Y2, indicating
the interarrival time and the service time respectively. These two random variables are
strongly correlated with X, the waiting time in the queue. Yj(1) and Yj(2), j=1,2 can be
negatively correlated by simply using the compliment of the pseudo-random numbers
used in the first run.
This technique can be implemented as follows. Simulate the single server queue,
and let
x(1) , x(1) , ..., x(1)
1
2
n
be n i.i.d observations of X. Re-run the simulation, thus replicating the results, using
pseudo-random numbers (ri,vi)=(1-r,1-vi). Let
x(2) , x(2) , ..., x(2) ,b
1
2
n
e realizations of X. Construct the interval estimate of E(X) using the random variable Z
as described above. Obviously, the correlation between the two samples of observations
is as good as the correlation between Y

(1)
j

and Y

(2)
,
j

j=1,2.

The antithetic variates technique, as described above, was implemented in a
simulation of an M/M/1 queue. The random variable X is the time a customer spends in

130


the system. The i.i.d. observations of X were obtained by sampling every 10th customer.
The results given in table 7.1 were obtained using straight simulation. Using the antithetic
variates technique, we obtained a confidence interval of 13.52±1.76. We note that the
antithetic variates techniques were employed using two sets of observations each of size
equal to 300, i.e., a total of 600 observations. From table 7.1, we see that a similar width
was obtained using a sample size of n=1800. Figure 7.2 shows the actual values for the
original sample, the sample obtained using the antithetic variates and Z. We see, that the
two samples of observations are fairly negatively correlated. Also, we observe that the Z
values are all close to the mean, indicating that their variance is small.

Sample size n

Confidence interval

600

13.86 + 3.46
_

900

13.03 + 2.70
_

1200

13.11 + 2.30
_

1500

12.82 + 1.99
_

1800

12.86 + 1.84
_

Table 7.1: Straight simulation of an M/M/1 queue.

In the above example, the antithetic variates technique worked quite well.
However, this should not be construed that this method always works well. In particular,
in the following example, an M/M/2 queuing system was simulated. Table 7.2 and figure
7.2 show that in this case there is little benefit to be gained from using the antithetic
variates technique.


131
Waiting Time Per Customer
55.0

45.0

35.0

25.0

15.0

5.0

1
5
10
15
20

Simulated Number of Customers

25
Original Sample
Antithetic Variates
Mean of Two Samples

30
Figure 7.1:

Antithetic variates technique applied to an M/M/1 queue.

132


Type of Simulation

Sample
Size

Mean

Standard
Dev.

Conf.
Interval

Standard
Error

Cost

1.69
1.06
1.17
0.84
0.74
0.69
0.64
0.61

Standard
Error as a
% of
Mean
8.96
6.61
6.79
5.22
4.69
4.21
4.01
3.81

Straight
Straight
Standard Antithetic
Straight
Standard Antithetic
Straight
Standard Antithetic
Straight

400
800
800
1600
1600
2400
2400
3000

18.85
16.00
17.23
16.04
15.69
16.48
15.84
15.92

9.70
8.60
6.73
9.67
5.98
9.82
6.36
9.59

+/-3.31
2.07
2.30
1.64
1.44
1.36
1.25
1.19

Standard Antithetic

3000

16.11

6.96

1.17

0.60

3.72

1.12

Table 7.3: Straight simulation and antithetic variates techniques for an M/M/2 queue.

$

Standard Error

1.6

1.2

$!Straight Simulation
#!Standard Antithetic Variate Technique
#
$
$

0.8

$
$
#

$
#

$
#

0.4

800

Figure 7.2:

1600
2400
Sample Size

3200

Standard error plotted against sample size for straight simulation and antithetic
variates techniques for an M/M/2 queue.

7.3

The Control Variates Technique

This method is otherwise known as the method of Concomitant Information. Let X be an
endogenously created random variable whose mean we wish to estimate. Let Y be
another endogenously created random variable whose mean is known in advance. This is

$.68
.72
.97
.79
1.03
.88
1.09
.93


133

known as the control variable. Random variable Y is strongly correlated with X. We have
the following two cases.
a)

X and Y are negatively correlated:
Define a new random variable Z so that Z=X+Y-E(Y). We have
E(Z)=E(X+Y-E(Y))=E(X),
Var(Z) = Var(X) + Var(Y) + 2 Cov (X, Y).
Since X and Y are negatively correlated, we have that Cov(X,Y)<0. Therefore, if
Var(Y) - 2 |Cov(X,Y)| < 0 then, a reduction in the variance of Z has been
achieved.

b)

X and Y are positively correlated:
Define Z=X-Y+E(Y). Then
E(Z) = E(X - Y + E(Y)) = E(X).
Var(Z) = Var(X) + Var(Y) - 2|Cov(X,Y)|

Therefore, if Var(Y) - 2|Cov(X,Y)| < 0 then a reduction in the variance of Z has
been achieved.

Let us consider the example of a single server queue mentioned in section 7.2. Let
X be the time a customer spends in the system. Then X is negatively correlated with the
random variable Y representing the inter-arrival time. (It is also positively correlated with
the random variable representing the service time.)

Let x1, x2, ..., xn be n i.i.d.

observations of X. Likewise, let y1, y2, ..., yn be n observations of Y. (These observations

134


are i.i.d. by nature of the simulation model.) yi is the inter-arrival time associated with
the xi observation. Let
zi = xi + yi - E(Y), i=1,2,..., n.

_
sz
Then we can construct a confidence interval for the estimate of E(X) using Z +1.96
n
where

_ 1 n
Z = n ∑ zi
i=1

and
n

1
sz =
# (zi "Z) 2 .
n "1 i=1
2

More generally, random variable Z can be obtained using
!
Z = X - a (Y - E(Y)),
where a is a constant to be estimated and Y is positively or negatively correlated to X.
Again, we have E(Z)=E(X). Also,
Var(Z) = Var(X) + a2 Var(Y) - 2a Cov(X,Y)
so that Z has a smaller variance than X if
a2 Var(Y) - 2a Cov(X,Y) < 0.
We select a so that to minimize the right-hand side given in the above expression. We
have


135

2a Var(Y) - 2Cov(X,Y) = 0
or
Cov (X,Y)
a* = Var (Y)

Now, substituting into the expression for Var(Z) we have

Var(Z)

= Var(X) -

[Cov(X,Y)]2
Var(Y)

2
= (1-ρXY ) Var(X).

Thus, we always get a reduction in the variance of Z for the optimal value of a, provided
that X and Y are correlated. The determination of a* requires a priori knowledge of the
Var(Y) and Cov (X,Y). Sample estimates can be used in order to approximately obtain
a*.
The definition of Z can be further generalized using multiple control variables, as
follows

m

Z = X - ∑ ai(Yi - E(Yi)) ,
i=1

where ai, i=1,2,...,m, are any real numbers. In this case,

m

Var(Z) = Var(X) + m a2 Var(Yi) - 2 m ai Cov(X,Yi) + 2 ∑ i-1 aiaj Cov(Yi,Yj).
i=1 i
i=1
j=1
i=2
∑

∑

∑

136


1.

Consider the machine interference problem. Carry out the following tasks:
a.

Implement the antithetic variance reduction technique.

b.

Implement the control variates technique.

c.

Compare the above two variance reduction techniques against straight
simulation. In particular, compare the three techniques for various sample
sizes. Observe how the variance and the standard error expressed as a
percentage of the mean, educe as the sample size goes up. Contrast this
against the execution cost (if available).

138


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