Simulation in terminated system

Output Analysis for a
terminating simulators

Presented By :Saleem
Almaqashi
1

Introduction
– One realization (run) does not necessarily give
the “correct” answer(s).
– Variance exists in simulation results so we must
be cautious about how we interpret results
◦ Output from our model {Y1, Y2, Y3, …}
– {Y1, Y2, Y3, …} may not be independent;
– {Y1, Y2, Y3, …} may have different distributions,
depending on a number of different factors;
– Estimators, confidence interval, and so on, must
be constructed.

2

Types of Simulations with Regard to Output
Analysis
Terminating

Simulation Steady-state parameters
Systems

Steady-state cyclic parameters

Non-terminating Dynamic parameters

“Other
parameters” Transient parameters

3

Types of Simulations with
Regard to Output Analysis

 Terminating: Parameters to be estimated
are defined relative to specific initial and
stopping conditions that are part of the
model
 There is a “natural” and realistic way to
model both the initial and stopping
conditions
 Output performance measures generally
depend on both the initial and stopping
conditions
4

Terminating: Examples
 Natural event “E” that specifies the length of each
run.
 Ex.1: Bank opens from 9 to 5, E = {8 hours of
simulation (9-5)}
 Ex.2: Military exercise Red vs. Blue, E = {Either
Red or Blues wins}
 Ex.3: Manufacturing 100 products (random
process), E = {Completion of the 100th product}
 Ex.4: Reliability simulation of a car (quality
problems random), E = {First major quality problem
occurs}

5

Non terminating
 Non terminating: There is no natural and realistic
event that terminates the model Interested in “long-
run” behavior characteristic of “normal” operation

 If the performance measure of interest is a
characteristic of a steady-state distribution of the
process, it is a steady-state parameter of the model
.

 Theoretically, does not depend on initial conditions
 Practically, must ensure that run is long enough so
that initial-condition effects have dissipated
6

Non terminating Examples
 Ex. 1 (Cyclic): An inventory system, say a retail
store, replenishments are cyclic (e.g., monthly
orders and arrivals of goods), but demand is
random. The performance measure can be
average monthly inventory level.
 Ex . 2 (Dynamic ) : Airport security system
simulation (case study). Passenger volume varies
by hour of day, day of the week. Performance
measure – passenger waiting time (to get through
security check point) is hourly.
 Ex.3 (Transient ) : Transportation (case study).
Multiple-lane highway has one lane blocked (e.g.,
accidents). Cars need to merge and traffic slows
down. Performance measure: drive time during the
traffic jam.
7

Terminating vs. Steady state
simulation

8

Terminating Simulations
 Statistics based on Observations
◦ Let Di = delay of ith customer in queue
d = (the true value of) average delay in
queue
For m customer
m m
Di Estim ate
Di
i 1 i 1
D ( m) d lim (true mean)
m m m
◦ In general
xj = measured performance from the run
= E[x]
Confidence Interval (of n runs):
2
S 2 (n) x
x (n) tn 1,1 /2 (Prisker used )
n I
9

 Replication Approach
◦ Independent runs with different RN streams/seeds but
same I.C.’s, also need same number of observations
for each run (conflict with statistics based on time-
persistent variables)
◦ Ex: Run model 10 times with identical initial conditions
xj - 1.051 6.438 2.646 0.805 1.505
0.546 2.281 2.822 0.414 1.307
Each one is the average of 35 observations
35 D
j
x j D j (35)
j 1 35
Compute:
1 10 (x j X )2
X (10) x j (35) 1.982 S 2 (10) 3.172
10 j 1 n 1
10

 Replication Approach
◦ Ex: (Continue)
From the statistics table: use = 0.10 (t9,0.95 = 1.833)
So, confidence interval of delay times
3.172
1.982 1.833 1.982 1.032
10
0.950 d 3.014 witha 90% probabilit y
◦ Interpretation
 90% chance (probability) the true value is between
0.950 and 3.014 (not “falls into”) or 90% that 0.950
to 3.014 covers the true value
◦ Better answer increased number of runs

11

 Measure of Precision (Accuracy)
◦ Absolute vs. relative
 Half width of actual C.I. absolute precision
 Ratio of the half width to the magnitude of estimate
relative precision of C.I.
◦ Determine how many runs are needed
 Assumption: S2(n) is not changing (within each run)
 Absolute Precision ( )

* S 2 (n)
na ( ) min{i n : ti 1,1 /2 }
i
 Relative Precision ( )

S 2 ( n)
ti 1,1 /2
*
n( ) min{i n: i }
r
X ( n)
12

 Measure of Precision (Accuracy)
◦ In the previous Ex:
 Absolute precision ( ) = 1.032
If we want C.I. = 1.982 0.5
* 3.172
na (0.5) min{i 10 : ti 1,0.95 0.5}
i
* 2
i.e., na ( ) f ( S (n))
Continue to make additional runs i = 37 (27 more runs)
 Relative precision ( ) = 1.032/1.982 = 0.521
(i.e., 90% of prob. value is within 52% of the true mean)
If we want = 0.15 3.172
ti 1,0.95
*
nr (0.15) min{i 10 : i 0.15}
1.982
Lead to i = 99 (89 more runs)
13

 Sequential Procedure (to make multiple replications)
1. Select the initial replication n0 2
2. Compute (based on a selected )
S 2 (n)
(n, ) tn 1,1 /2
n

3. Compute
(n, )
? is determined - relative precision
X ( n)

4. If yes stop
If not n = n + 1 (one additional run)
and repeat steps 2, 3, and 4
14


 Sequential Procedure
◦ In practice, need to obtain statistics at the
end of each replication (run)
Replicate 1: Estimate: x1, S12(x)

clear stats - use new RN
streams
2: Estimate: x2, S22(x)

n0 2: Estimate: xn0, Sno2(x)
Compute X(n0), S2(n0)
 Until

S 2 ( n)
tn 1,1 /2
n:
n
X ( n)

15

Conclusion
 Statistical methods are used to
analyze the results of simulation
experiments.
 Terminating: A simulation where there
is a specific starting and stopping
condition that is part of the model.

16

Simulation in terminated system

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