## 4.1 QUESTIONS AND ANSWERS IN QUEUEING THEORY It is important that those who wish to apply the results of queueing
theory have an appreciation for the kinds of questions that queueing theory can answer and
for the nature of and the assumptions behind these answers. - Few closed-form expressions exist for the transient and the nonstationary behavior of queueing systems. Almost all the existing important results of queueing theory are obtained for equilibrium conditions (i.e., with the queueing system operating in the "steady state," in engineering parlance).
- Even assuming equilibrium conditions, queueing theory runs into enormous mathematical difficulties in all but relatively few types of situations. Quite often, the choice facing an analyst is between, on the one hand, using a realistic mathematical model for which almost no results can be obtained and, on the other, using a simplified model that provides results of questionable validity for the problem at hand.
- Most of the exact results of queueing theory apply to queueing systems in which the
interarrival times or the service times or (ideally) both are negative exponential.
Fortunately, there are many realworld systems for which at least the interarrival times
are negative exponential. The main reason is that many arrival processes observable in
practice can be modeled as Poisson processes (which in turn implies negative exponential
interarrival times). This is especially true when one refers to
*urban*service systems, where Poisson (or nearly Poisson) arrival processes are abundant. - Queueing theory is "very good" at estimating the low moments and central moments of such important quantities as the waiting times or the "number of users present" in queueing systems but not nearly as good at computing the probability distributions for these quantities. Indeed, in all but a handful of cases, the only approach available for obtaining probability distributions for most of the quantities of interest in queueing theory is through the use of a combination of transform analysis and numerical analysis techniques. We shall see examples of the numerical analysis approach later in this chapter and in Chapter 5.
Items (1)-(4) above, discouraging as they may sound, are only meant
to provide some perspective and not to detract from the value of the results that queueing
theory has generated to date. In fact, some of these results are very powerful. They apply
to quite general queueing systems and provide important information about the queueing
phenomena that occur, while requiring only a minimum amount of knowledge about the
characteristics of interarrival times, service times, queue discipline, and so on. |