Page 198 - DCAP601_SIMULATION_AND_MODELING

Simulation and Modelling

Notes planning is our concern. One reason is that we have to expect something before we can recognize
the unexpected. The purpose of this paper is not so much to suggest that we look before we leap,
but to suggest a few things that we might look at. The context is a simulation of a stochastic
model to estimate steady-state quantities of interest. Our idea is to develop some expectations
and design the initial experiment by doing some preliminary mathematical analysis. We focus
on simulation run lengths. Of course, since we are going to simulate, the stochastic model of
interest presumably is relatively complicated, so that it is not easy to calculate the quantities of
interest analytically. Thus, we suggest approximating the stochastic model of interest by a more
elementary stochastic model that can be analyzed analytically. For the approximating model, in
addition to the steady-state quantity of interest, we calculate the asymptotic variance and the
asymptotic bias of the sample mean used to estimate the steady-state quantity of interest
(assuming that a sample mean will be used). Then we apply these quantities to estimate the
simulation run lengths required in the original model to obtain desired statistical precision. The
estimated simulation run lengths can then be used, before any data have been collected, to
design the experiment, i.e., to determine what cases to consider, what statistical precision to aim
for, what experimental budget is appropriate, or even whether to conduct the experiment at all.
There are two steps: First, we must find a suitable approximation for the given stochastic model
and, second, we must calculate the asymptotic quantities of interest for the approximating
model. Of course, we do not want the preliminary analysis to be harder than doing the experiment
itself. The preliminary analysis better be easy or we wouldn’t bother with it. Fortunately, there
is substantial literature supporting these two steps. In Whitt (1989a) we show how this program
can be carried out for a large class of stochastic models. The models considered are those for
which the stochastic process of interest can be approximated by reflecting Brownian motion
(RBM). Through much previous work on heavytraffic limit theorems and diffusion
approximations, it is known that many queueing processes can be approximated by RBM, at
least roughly, so that the class of models covered is relatively large. The class includes the
standard GI/G/1 queue, as was shown previously in related work by Blomqvist (1969), Moeller
and Kobayashi (1974) and Woodside, Pagurek and Newell (1980), but also applies to many other
models. In Whitt (1989a) we also show how to calculate the asymptotic quantities of interest for
RBM to obtain very simple approximate formulas. Moreover, we show that the scaling of time
in the heavy-traffic limit theorem plays an essential role in determining the form of the final
formulas.
Of course, RBM is by no means the only stochastic process that can be used as an approximation.
For example, the Ornstein-Uhlenbeck diffusion process is a natural candidate for infinite-server
queues, which also yields very simple formulas. In Whitt (1989b) we apply recurrent potential
theory for Markov processes to obtain asymptotic formulas for a large class of Markov processes,
including general birth and death processes and diffusion processes. In Heidelberger and Whitt
(1989) we compare the asymptotic formulas, and thus the simulation efficiency, for several
different queueing models. There we show that it usually is easier to obtain reliable estimates
for an infiniteserver queue than for a single-server queue. It usually is even easier for small
closed queueing networks. Roughly speaking, simulation efficiency increases (estimation
becomes easier) as the relaxation time (the time required to reach steady state; see Keilson
(1979)) decreases. Our purpose here is to provide a brief overview. We review the standard
statistical theory leading up to the large-sample formula for the required simulation run length
with the relative width criterion; i.e., the run length such that the width of the confidence
interval divided by the quantity being estimated (which is presumed to be positive) takes some
prescribed value. In Section 3 we review some of the RBM-type examples from Whitt (1989a),
including the M/M/1 queue, RBM, the GI/G/m queue and a packet queue model from Fendick,
Saksena and Whitt (1989). The packet queue model is relatively complicated, so that an exact
analysis is evidently not possible with current methodology. However, simple formulas to
determine appropriate simulation run lengths can be obtained from an RBM approximation.

192 LOVELY PROFESSIONAL UNIVERSITY