Page 219 - DCAP601_SIMULATION_AND_MODELING

Unit 12: Design and Evaluation of Simulation Experiments (II)

described in this paper to the Markovian queue-length process in order to estimate the required Notes
simulation run length. Alternatively, to capture the impact of the arrival and service processes
beyond their means, we might use heavy-tra±c limit theorems to approximate the queue-length
process of the bottleneck queue by a reflected Brownian motion (RBM); e.g., see Chen and Yao
(2001) and Whitt (2002). We then apply the techniques described in this paper to the limiting
RBM, which is also a Markov process.

Notes Methods described in these last two paragraphs, we can treat quite general queueing-
network models, albeit roughly.
12.3.1 The Standard Statistical Framework

Probability Model of a Simulation

We base our discussion on a probability model of a (stochastic) simulation experiment: In the
model, the simulation experiment generates an initial segment of a stochastic process, which
may be a discrete-time stochastic process X n : n   1 1g or a continuous-time stochastic process
 ( ):X t t   0 . We form the relevant sample mean

n t
s
X  n  1  X or X  t 1  X ( ) ds ,
n i t 0 
i 1
and use the sample mean to estimate the long-run average,
µ  lim X n or µ  lim X t ,
n   n  
which is assumed to exist as a proper limit with probability one (w.p.1). Under very general
regularity conditions, the long-run average coincides with the expected value of the limiting
steady-state distribution of the stochastic process.

These stochastic processes arise in both observations from a single run and from independent
replications. For example, in observations from a single run, a discrete-time stochastic process
[X : n  1] arises if we consider the waiting times of successive arrivals to a queue. The random
n
variable X might be the waiting time of the nth arrival before beginning service; then µ is the
n
long-run average waiting time of all arrivals, which usually coincides with the mean steady-
state waiting time. On the other hand, X might take the value 1 if the nth arrival waits less than
n
or equal to x minutes, and take the value 0 otherwise; then µ  µ (x) is the long-run proportion
of customers that wait less than or equal to x minutes, which usually corresponds to the probability
that the steady-state waiting time is less than or equal to x minutes.
Alternatively, in observations from a single run, a continuous-time stochastic process [x(t) : t  0]
arises if we consider the queue length over time, beginning at time 0. The random vari- able X(t)
might be the queue length at time t or X(t) might take the value 1 if the queue length at time t is
less than or equal to k, and take the value 0 otherwise.
With independent replications (separate independent runs of the experiment), we obtain a
discrete-time stochastic process {X : n  1}. Then X represents a random observation obtained
n n
from the n run. For example, X might be the queue length at time 7 in the n replication or X
th
th
n n
might be the average queue length over the time interval [0, 7] in the nth replication. Then the
limit  represents the long-run average over many independent replications, which equals the
expected value of the random variable in any single run. Such expected values describe the
expected transient (or time-dependent) behavior of the system.

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