output analysis for simulation written by: marvin k. nakayama presented by: jennifer burke msim 752
TRANSCRIPT
Output Analysis for Simulation
Written by:Marvin K. Nakayama
Presented by:Jennifer BurkeMSIM 752
Outline
Performance measures Output of a transient simulation Techniques for steady-state
simulations Estimation of multiple performance
measures Other methods for analyzing
simulation output
Example
Automatic Teller Machine (ATM)
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)
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
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
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
Output of a Terminating Simulation
calculate the sample variance of X1,X2,…,Xn
and the sample standard deviation
Output of a Terminating Simulation
Central Limit Theorem
confidence interval for E(X)
Output of a Terminating Simulation
the confidence interval provides a form of error bound
Hn is the half-width of the confidence interval
ATM example (Terminating)
Expected daily withdraw within $500
ε = 500 S(n) = sample standard
deviation
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 →
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
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 →
ATM (Continuous)
Y(s) = number of customers waiting in line at time s
Assume Y(s) has a steady-state Calculate v
Difficulties of Steady-state analysis
Discrete-time process
if m is large, then is a good approximation of v
Confidence interval
Simplifications to Steady-state Analysis
Multiple replications Initial-data deletion Single-replicate algorithm
Method of Multiple Replications
Estimate
r i.i.d replications, length k = m/r 10 r 30
Method of Multiple Replications
Average of jth row
Using find the sample mean
Method of Multiple Replications
Sample variance
Confidence interval
Problem with Multiple Replication Method
Simple estimation of variance
can be contaminated by initialization bias
Initial-Data Deletion
Partial solution Delete first c observations Replication mean
sample mean sample variance
confidence interval
Single-Replicate Algorithm
Single simulation of length m + c Divide the m observations into n
batches
10 n 30 Batch mean
Single-Replicate Algorithm
Sample mean
Sample variance
Confidence interval
Estimating Multiple Performance Measures
Terminating simulations
Confidence interval for each performance measure
Joint confidence interval
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
Conclusions
Basics of analyzing simulation output
Application potential is high Not state of the art Benefit Lacked comparison