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## Simulation Modeling and Analysis

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Outline

- Stochastic Nature of Output
- Taxonomy of Simulation Outputs
- Measures of Performance
- Point Estimation
- Interval Estimation
- Output Analysis in Terminating Simulations
- Output Analysis in Steady-state Simulations

2

Introduction

- Output Analysis
- Analysis of data produced by simulation
- Goal
- To predict system performance
- To compare alternatives
- Why is it needed?
- To evaluate the precision of the simulation performance parameter as an estimator

3

Introduction -contd

- Each simulation run is a sample point
- Attempts to increase the sample size by increasing run length may fail because of autocorrelation
- Initial conditions affect the output

4

Stochastic Nature of Output Data

- Model Input Variables are Random Variables
- The Model Transforms Input into Output
- Output Data are Random Variables
- Replications of a model run can be obtained by repeating the run using different random number streams

5

Example: M/G/1 Queue

- Average arrival rate Poisson with = 0.1 per minute
- Service times Normal with = 9.5 minutes and = 1.75 minutes
- Runs
- One 5000 minute run
- Five 1000 minute runs w/ 3 replications each

6

Taxonomy of Simulation Outputs

- Terminating (Transient) Simulations
- Runs until a terminating event takes place
- Uses well specified initial conditions
- Non-terminating (Steady-state) Simulations
- Runs continually or over a very long time
- Results must be independent of initial data
- Termination?
- What determines the type of simulation?

7

Examples: Non-terminating Systems

- Many shifts of a widget manufacturing process.
- Expansion in workload of a computer service bureau.

8

Measures of Performance: Point Estimation (Discrete-time Data)

- Point estimator of (of ) based on the simulation discrete-time output (Y1, Y2,.., Yn)

* = (1/n) i n Yi

- Unbiased point estimator

E(* ) =

- Bias

b = E(* ) -

10

Measures of Performance: Point Estimation (Continuous-time data)

- Point estimator of (of ) based on the simulation continuous-time output (Y(t), 0 < t < Te)

* = (1/ Te) 0 Te Y(t) dt

- Unbiased point estimator

E(* ) =

- Bias

b = E(* ) -

11

Measures of Performance: Interval Estimation (Discrete-time Data)

- Variance and variance estimator

2() = true variance of point estimator

2*() = estimator of variance of point estimator

- Bias (in variance estimation)

B = E(2*() )/ 2()

12

Measures of Performance: Interval Estimation - contd

- If B ~ 1 then t = ( - )/ 2*() has t/2,f distribution (d.o.f. = f). I.e.
- A 100(1 - )% confidence interval for is

- t/2,f2*() < < + t/2,f2*()

- Cases
- Statistically independent observations
- Statistically dependent observations (time series).

13

Measures of Performance: Interval Estimation - contd

- Statistically independent observations
- Sample variance

S2 = i n (Yi - )2/(n-1)

- Unbiased estimator of 2()

2*() = S2 /n

- Standard error of the point estimator

*() = S /n

14

Measures of Performance: Interval Estimation - contd

- Statistically dependent observations
- Variance of

2() = (1/n2) i n j n cov(Yi , Yj )

- Lag k autocovariance

k = cov(Yi , Yi+k )

- Lag k autocorrelation

k = k0

15

Measures of Performance: Interval Estimation - contd

- Statistically dependent observations (contd)
- Variance of

2() = (0 /n) [ 1 + 2k=1 n-1 (1- k/n) k] = (0 /n) c

- Positively autocorrelated time series (k > 0)
- Negatively autocorrelated time series (k < 0)
- Bias (in variance estimation)

B = E(S2/n )/ 2() = (n/c - 1)/(n-1)

16

Measures of Performance: Interval Estimation - contd

- Statistically dependent observations (contd)
- Cases
- Independent data k = 0, c = 1, B = 1
- Positively correlated data k > 0, c > 1, B < 1, S2/n is biased low (underestimation)
- Negatively correlated data k < 0, c < 1, B > 1, S2/n is biased high (overestimation)

17

Output Analysis for Terminating Simulations

- Method of independent replications
- n = Sample size
- Number of replications r=1,2,…,R
- Yji i-th observation in replication j
- Yji, Yjk are autocorrelated
- Yri, Ysk are statistically independent
- Estimator of mean (r =1,2,…,R)

r(1/nr) i nrYri

18

Output Analysis for Terminating Simulations - contd

- Confidence Interval (R fixed; discrete data)
- Overall point estimate

* = (1/R) 1 Rr

- Variance estimate

* (*) = [1/(R-1)R] 1 R(r

- Standard error of the point estimator

*() = * (*)

19

Output Analysis for Terminating Simulations - contd

- Estimator and Interval (R fixed; continuous data)
- Estimator of mean (r =1,2,…,R)

r(1/Te) 0 Te Yr(t) dt

Overall point estimate

* = (1/R) 1 Rr

- Variance estimate

* (*) = [1/(R-1)R] 1 R(r

20

Output Analysis in Terminating Simulations - contd

- Confidence Intervals with Specified Precision
- Half-length confidence interval (h.l.)

h.l. = t/2,f2*() = t/2,f S/ R <

- Required number of replications

R* > ( z /2 So/ )2

21

Output Analysis for Steady State Simulations

- Let (Y1, Y2,.., Yn) be an autocorrelated time series
- Estimator of the long run measure of performance (independent of I.C.s)

= lim n => (1/n) i n Yi

- Sample size n (or Te) is design choice.

22

Output Analysis for Steady State Simulations -contd

- Considerations affecting the choice of n
- Estimator bias due to initial conditions
- Desired precision of point estimator
- Budget/computer constraints

23

Output Analysis for Steady State Simulations -contd

- Initialization bias and Initialization methods
- Intelligent initialization
- Using actual field data
- Using data from a simpler model
- Use of phases in simulation
- Initialization phase (0 < t < To; for i=1,2,…,d)
- Data collection phase (To < t < Te; for i=d+1,d+2,…,n)
- Rule of thumb (n-d) > 10 d

24

Output Analysis for Steady State Simulations -contd

- Example M/G/1 queue
- Batched data
- Batched means
- Averaging batch means within a replication (I.e. along the batches)
- Averaging batch means within a batch (I.e. along the replications).

25

Steady State Simulations: Replication Method

- Cases

1.- Yrj is an individual observation from within a replication

2.- Yrj is a batch mean of discrete data from within a replication

3.- Yrj is a batch mean of continuous data over a given interval

26

Steady State Simulations: Replication Method -contd

- Sample average for replication r of all (nondeleted) observations

Y*r(n,d) = Y*r = [1/(n-d)] j=d+1n Yrj

- Replication averages are independent and identically distributed RV’s
- Overall point estimator

Y*(n,d) = Y* = [1/R] r=1R Yr(n,d)

27

Steady State Simulations: Replication Method -contd

- Sample Variance

S2 = [1/(R-1)] r=1R (Y*r - Y*)

- Standard error = S/ R
- 100(1-)% Confidence interval

Y* - t /2,R-1 S/ R < < Y* + t /2,R-1 S/ R

28

Steady State Simulations: Sample Size

- Greater precision can be achieved by
- Increasing the run length
- Increasing the number of replications

29

Steady State Simulations: Batch Means for Interval Estimation

- Single, long replication with batches
- Batch means treated as if they were independent
- Batch means (continuous)

Y*j = (1/m) (j-1)mjm Y(t) dt

- Batch means (discrete)

Y*j = (1/m) i=(j-1)mjm Yi

30

Steady State Simulations: Batch Size Selection Guidelines

- Number of batches < 30
- Diagnose correlation with lag 1 autocorrelation obtained from a large number of batch means from a smaller batch size
- For total sample size to be selected sequentially allow batch size and number of batches grow with run length.

31

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