This document discusses statistics from simulation runs and analyzing simulation results. It provides an example of estimating the mean waiting time in a single server queueing system where arrivals and services are exponentially distributed. There are two main issues discussed: 1) observations may not be independent or the distribution may not be stationary, violating assumptions for confidence levels, and 2) the sample mean is biased for estimating the true mean waiting time, especially with early samples, as the distribution is not initially stationary. As the sample size increases, the bias diminishes but may not fully converge even with thousands of samples.

1. Simulation Run Statistics
Sachin dheke 013-333
Sadiksha kafle 013-334
Sagar khadka 013-335

2. INTRODUCTION
 Method to handle problems that arise in
measuring statistics from simulation runs
 Method used to analyze simulation results.
 Two assumption made to establish the
confidence levels
i) Observation are independent.
ii) Distribution is stationary.
 Many statistics do not meet these condition.

3. Example
• Single server system(denoted by MM1)
M= inter-arrival time is distribution exponentially
M= the service time is distributed exponentially
1- one server.
• First-in , first-out with no priority.
Objective:
• To measure the mean waiting time.

4. Mean Waiting Time
• Simplest approach to estimate mean waiting time.
• Usual formula to estimate mean value:
- Sample Mean
- individual waiting times
• Calculated waiting time is dependent
• Data are autocorrelated
• Varience of autocorrelated data is not related to
population variance
• Positive term is added for autocorrelation but
may be negetive for other system.
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i
n i
xnx
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i
x

5. Another Problem
• Distribution not stationary.
• Early arrival obtain service quickly so sample mean
including it will be biased
• Biased die out as simulation length extends and
sample size increases.

7. Description
• Figure based on theoretical results
• Show dependency between expected value of
sample mean and sample length for M/M/1 system
• Server utilization=0.9
• Steady state mean=8.1
• Mean value biased below steady state mean
• As sample size increase bias diminishes but even
sample =2000 mean only reached 95% of steady
state value.

8. THANK YOU