Week08.pdf

Static - Dynamic
Anna Maria Sri Asih
Department of Mechanical & Industrial Engineering
Gadjah Mada University

Introduction
 Deterministic models:
 expressed in terms of differential equations
 can exactly predict the development of a system based
on initial and boundary conditions
 Stochastic models
 use random variables
 outcomes are uncertain
 can only compute the probabilities of possible outcomes
Example:
throwing dice ( random experiment)
 outcome of a toss: initial conditions & external forces
 uncertain ?

How to study system?
 Analytic solution
 know how the model will behave iner any circumstances.
 closed form solution
 simple models
 Numerical methods
 More complex system
 example: Runge-Kutta method
Starts with the initial values of the variables, then use the
equations to figure out the changes In these variables over a very brief
time period
 repetition / iteration
 result: long list numbers, NOT equation
 Simulation
 even more complex system
 deals with uncertainty
 ”Calculate when you can, simulate when you can’t!”

Simulation models
 Model of the real system
 Faster, cheaper, or safer to perform
experiments on the model
 Computer simulation may use formulas,,
programming statements, or other means
to express math relationships between
inputs and outputs.
 Dealing with uncertainty  include
uncertain variables  random values from
a distribution.
 Simulation run includes many trials

Advantages of simulation
 Allows the study of complex, real-world systems (which
otherwise cannot be studied)
 Estimates performance of existing system under ‘projected’ /
different operating conditions
 Compares alternative proposed system designs
 Better control over experimental conditions
 Compress time, expand time
 Overall, if done correctly, simulation gives planners unlimited
freedom to try out different ideas for design and improvement
Disadvantages and pitfalls of
simulation
 Failure to produce exact results (only estimates)
 Costs of developing simulation models can be
expensive and time consuming

Simulation methodology
 Estimate probabilities of
future events
 Assign random number
ranges to percentages
(probabilities)
 Obtain random numbers
 Use random numbers to
“simulate events”

Simulation life cycle
Ayani, R., 2003

Building a valid and credible
simulation model
 Picture
verification asks “ Was the model made right? ”
validation asks “ Was the right model made? ”
accreditation asks “ Is the model the right one for the job? ”

Simulation methods – Monte Carlo
It is a method that
methods for solving
various kinds of
computational problems
by using random numbers

“Find the value of ”
 Use “hit and miss” method
 The area of square = (2r)2
 The area of circle = r2

𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑞𝑢𝑎𝑟𝑒
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒
=
4𝑟2
𝑟2 =
4

  = 4 ∗
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑞𝑢𝑎𝑟𝑒
=
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑜𝑡𝑠 𝑖𝑛𝑠𝑖𝑑𝑒 𝑐𝑖𝑟𝑐𝑙𝑒
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑜𝑡𝑠
r

“Birthday problem”
 Out of a group of 100 people, 2 people share a brithday
 Use “hit and miss” method
1. Pick 30 random numbers in the rang [1,365], each number
represent a day in a year
2. Check to see if any of the 30 random numbers are equal
3. Go back to step [1] and repeat 10,000 times
4. Report the fraction of trials that have matching birthdays

“Birthday problem”
 Out of a group of 100 people, 2 people share a brithday
 Use “sampling from distribution” method
1. Suppose we have the cdf of people’s birthday, F(x)
2. People’s birthday is represented as x = [1, 365]
3. Generate a random values, z, from U [0,1]
4. Compute x = F-1(z)
5. Check to see if any of the 30 random numbers are equal
6. Go back to step [3,4] and repeat 10,000 times
7. Report the fraction of trials that have matching birthdays
Many kinds of sampling, e.g.:
• Simple sampling
• Importance sampling
• Stratified sampling
• Non-stratified sampling
• Cluster sampling
• Latin hypercube
Output from Monte Carlo
can form a distribution too !

Simulation methods –
Discrete event Simulation (DES)
 Discrete time systems: system changes with
time, in discrete steps
 Stochastic, dynamic, discrete
 Uncertainty (modelled by probability)
 Example:
 Traffic problem: given locations of cars and
destinations and traffic rules  the expected time for
a specific car to reach its destination?
 In a system with n machines, the system will crash
when fewer than n machines are available  what is
the expected time for the system to crash?

 Example: Single-server queue
 Estimate expected average delay in queue (line, not service)
 State variables
○ Status of server (idle, busy) – needed to decide what to do with an
arrival
○ Current length of the queue – to know where to store an arrival that
must wait in line
○ Time of arrival of each customer now in queue – needed to compute
time in queue when service starts
 Events
○ Arrival of a new customer
○ Service completion (and departure) of a customer
○ Maybe – end-simulation event (a “fake” event) – whether this is an
event depends on how simulation terminates (a modeling decision)

Status
shown is
after all
changes
have
been
made in
each
case …
Interarrival times: 0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9, …
Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …

Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
Final output performance measures:
Average delay in queue = 5.7/6 = 0.95 min./cust.
Time-average number in queue = 9.9/8.6 = 1.15 custs.
Server utilization = 7.7/8.6 = 0.90 (dimensionless)

• Example: Single Stage Process with Two Servers and Queue
1
2
Arrivals
…
d1
d2
s = (s1, s2 , s3) where
0
1
2
3 4 5
(0,0,0)
(0,1,0)
(1,1,0) (1,1,1) (1,1,2)
(1,0,0)
• • •
a
a
a
a a
d1
d2
d2
d2
d2
d1
d1
d1
, ,
State-
transition
network
3
0, if server is idle
1, if server is busy
number in the queue
i
i
s
i
s

 


i = 1, 2

 Example: repair problem
n machines are needed to keep an operation.
There are s spare machines.
The machines in operation fail according to some unknown distribution
(e.g. exponential, Poisson, uniform, etc., with a known mean).
When a machine fails it is sent to repair shop and the time to fix is a
random variable that follows a known distribution.
System crashes when fewer than n machines are available
What is the expected time for the system to crash?

n = number of working machines needed
S = number of spares available

Simulation methods – Continuous
 Continuous time model:
 state variables may change
continuously, e.g. temperature in a class
room, the level of water in a tank
 used extensively in mechanical, chemical,
and electrical engineering
 Example:
 the level of temperature of a liquid
 the level of temperature in a classroom
 the level of water in a tank
 the level of drug concentration in body

Simulation methods – Hybrid
 Hybrid : combined discrete / continuous
model
 Some variables in the system model are
discrete and some continuous
 Example: unloading dock where tankers
queue up to unload their oil through a
pipeline:
 discrete : tanker arrivals
 continuous : flow of oil

Random variables
& Stochastic Process
 Random variables:
 variable whose possible values are numerical
outcomes of a random phenomenon
 Types:
Discrete random var: countable number of outcomes
(dead/alive, dice, etc.)
Continuous random var: infinite continuum of
possible values ( weight, speed, etc.)
 Stochastic process:
 a family of random variables

Probability functions
 A probability function maps the possible
values x against their respective
probabilities of occurrence, p(x)
 p(x)  [0,1]
 The area under a probability function is
always 1

pmf & pdf
Discrete random var: countable
number of outcomes (dead/alive,
dice, etc.)
 pmf : roll of a dice
Continuous random var: infinite
continuum of possible values
(weight, speed, etc.)
 pdf: weight of adult females in
Indonesia
x
p(x)
1/
6
1 4 5 6
2 3
 
x
all
1
P(x)
Pr( ) ( )
X x p x

 
Pr( ) ( )
b
a
a X b p x dx

   

 The probability that a real-valued random variable x with a given
probability distribution f(x) will be found at a value less than or
equal to x
F(x) = P(X < x) =
it gives the are under the pdf from minus infinity to x
F(x) = P(X < x) =
with properties:






x
t
x
-
t
f untuk
)
(
1. 0 < F(x) < 1
2. F(x) is nondecreasing function
3. F(-) = 0
4. F() = 1
Cummulative Distribution Function
(CDF)


x
dt
t
f )
(
x
P(x)
1/
6 1 4 5 6
2 3
1/
3
1/
2
2/
3
5/
6
1.
0

Stochastic process
 The process has a strong element of
random behaviour
 If the time set T is countable  discrete
time process : {Xn, n=0,1,2,…}
 If the time set T is an interval of the real
line  continuous time process: {X(t),
t0}

Types of formulations
 Static formulations
 involves functions with one or
more variables being random
 Dynamic formulations
 involves stochastic process with
independent var t (time) to model
uncertain dynamic systems

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