Download - Output Analysis for Simulation Written by: Marvin K. Nakayama Presented by: Jennifer Burke MSIM 752
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Output Analysis for Simulation
Written by:Marvin K. Nakayama
Presented by:Jennifer BurkeMSIM 752
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Outline
Performance measures Output of a transient simulation Techniques for steady-state
simulations Estimation of multiple performance
measures Other methods for analyzing
simulation output
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Example
Automatic Teller Machine (ATM)
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Performance Measures
Measure how well the simulation runs
Different types of simulations require different statistical techniques to analyze the results Terminating (or transient) Steady-state (or long run)
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Terminating Performance Measures
Terminating simulation Simulation will finish at a given event Initial conditions have a large impact
Ex: Queue starts with no customers present
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ATM example (Terminating)
Open 9:00am – 5:00pm X = # of customers using ATM in a
day E(X) P(X 500)
C = queue is empty
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Output of a Terminating Simulation
Goal: calculate E(X)
Approach: n 2 i.i.d duplications
X1,X2,…,Xn
find the average of those duplications
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Output of a Terminating Simulation
calculate the sample variance of X1,X2,…,Xn
and the sample standard deviation
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Output of a Terminating Simulation
Central Limit Theorem
confidence interval for E(X)
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Output of a Terminating Simulation
the confidence interval provides a form of error bound
Hn is the half-width of the confidence interval
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ATM example (Terminating)
Expected daily withdraw within $500
ε = 500 S(n) = sample standard
deviation
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Steady-state Performance Measures
Steady-state simulation Simulation that stabilizes over time
Initial condition C Fi(y|C)
Fi(y|C) → F(y) as i →
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ATM example (Steady-state)
Open 24 hours a day Yi = number of customers served on
the ith day of operation E(Y) P(Y 400)
C = queue is empty
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Output of a Steady-state Simulation
Case 1: discrete-time process Y1,Y2,…,Yn
estimate v, as m →
Case 2: continuous-valued time index Y(s)
estimate v, as m →
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ATM (Continuous)
Y(s) = number of customers waiting in line at time s
Assume Y(s) has a steady-state Calculate v
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Difficulties of Steady-state analysis
Discrete-time process
if m is large, then is a good approximation of v
Confidence interval
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Simplifications to Steady-state Analysis
Multiple replications Initial-data deletion Single-replicate algorithm
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Method of Multiple Replications
Estimate
r i.i.d replications, length k = m/r 10 r 30
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Method of Multiple Replications
Average of jth row
Using find the sample mean
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Method of Multiple Replications
Sample variance
Confidence interval
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Problem with Multiple Replication Method
Simple estimation of variance
can be contaminated by initialization bias
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Initial-Data Deletion
Partial solution Delete first c observations Replication mean
sample mean sample variance
confidence interval
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Single-Replicate Algorithm
Single simulation of length m + c Divide the m observations into n
batches
10 n 30 Batch mean
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Single-Replicate Algorithm
Sample mean
Sample variance
Confidence interval
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Estimating Multiple Performance Measures
Terminating simulations
Confidence interval for each performance measure
Joint confidence interval
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ATM example (Terminating)
Open 9:00am – 5:00pm μ1 = expected # of customers
served in a day μ2 = probability # served in a day is
at least 1000 μ3 = expected amount of $
withdrawn in a day
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Conclusions
Basics of analyzing simulation output
Application potential is high Not state of the art Benefit Lacked comparison