Page 218 - DCAP601_SIMULATION_AND_MODELING

Simulation and Modelling

Notes basic model data such as the arrival stochastic process and the service-time distribution. This
queueing experimental design problem is interesting and important primarily because a uniform
allocation of data over all cases (parameter values) is not nearly appropriate. Experience indicates
that, for given statistical precision, the required amount of data increases dramatically as the
traffic intensity  (arrival rate divided by the service rate) increases toward the critical level for
stability and as the arrival-and-service variability (appropriately quantified) increases.
For example, the required simulation run length to obtain 5% relative error (width of confidence
interval divided by the estimated mean) at a high traffic intensity such as 0:95 tends to be 100
times greater than at a lower traffic intensity such as 0:50. (The operative formula underlying
this rough estimate is f()  (1–) ; note that f(0:95)/f(0:50) = 400/4 = 100. If we consider the
– 2
more extreme case  = 0:995, then the factor is 10,000. If we used a criterion of absolute error
instead of relative error, then the operative formula becomes even more impressive: then f() 
(1 – ) –4.

In this queueing example, and throughout this paper, we use simulation time as our
characterization of computational effort. (For a theoretical discussion of this issue, see Glynn
and Whitt (1992).) Some computational experience or additional experiments on the selected
computer are needed to convert simulation time into computational effort. Since there is a
degree of freedom in choosing the measuring units for time, it is important to normalize these
time units. For example, in a queueing model we might measure time in terms of the number of
arrivals that enter the system or we might stipulate that a representative service-time distribu-
tion has mean 1. On the positive side, focusing on required simulation time has the advantage
that it yields characterizations of computational effort that are independent of the specific
computer used to conduct the simulation. It seems best to try to account for that important factor
separately.

We assume that the quantities of interest will be estimated by sample means. With sample
means, in great generality the required amount of simulation time can be determined by
computing quantities called the asymptotic variance and the asymptotic bias of the sample
means. Thus, we want to estimate these quantities before conducting the simulation. In general,
that is not so easy to do, but existing theory supports this step for a sample mean of a function of
a Markov process. However, the stochastic processes of interest in simulation models are rarely
Markov processes.
It is important to approach this approximation step with the right attitude. Remember that we
usually only want to obtain a rough estimate of the required simulation run length. Thus, we
may well obtain the desired insight with only a very rough approximation. We do not want this
analysis step to take longer than it takes to conduct the simulation itself. So we want to obtain
the approximation quickly and we want to be able to do the analysis quickly. Fortunately, it is
often possible to meet these goals.
For example, we might be interested in simulating a non-Markovian open network of single-
server queues.

We might be interested in the queue-length distributions at the different queues. To obtain a
rough estimate of the required simulation run length, we might first solve the traffic- rate
equations to find the net arrival rate at each queue. That step is valid for non-Markovian queueing
networks as well as Markovian queueing networks; e.g., see Chen and Yao (2001), Kelly (1979)
or Walrand (1988). Given the net arrival rate at each queue, we can calculate the tra±c intensity
at each queue by multiplying the arrival rate times the mean service time. Then we might focus
on the bottleneck queue, i.e., the queue with the highest traffic intensity. We do that because the
overall required run length is usually determined by the bottleneck queue. Then we analyze the
bottleneck queue separately (necessarily approximately). We might approximate the bottleneck
queue by the Markovian M=M=1 queue with the same traffic intensity, and apply the techniques

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