## 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.
In working with queueing theory one must, first of all, take the particular real-world system of interest, study this system, and create (or simply choose from the list of models in queueing theory) a mathematical model to represent it. Through the analysis of this mathematical model, one then obtains the answers, which supposedly apply to the original system as well. Inherent to the procedure of creating a mathematical model are the notions of simplification and approximation: The analyst must necessarily disregard many details which he or she considers superfluous (or of minor importance) to the central points of interest. In most cases, approximations must also be made in transforming raw and often incomplete data into mathematical quantities that will make the analysis of the model possible. Finally, it is not unusual for an analyst to make many assumptions about certain aspects of the behavior of the real system-assumptions based mostly on intuition and experience rather than on any real evidence that the system indeed behaves in this way. Under the circumstances it would then be fair to state that, in most applications, the estimates of quantities of interest which can be obtained through a queueing analysis should only be viewed as approximate indicators of the size of these quantities in the real world. Consequently, the application of queueing theory is most useful in pointing out the inadequacies of existing operating systems, the directions in which to proceed for improving these systems, and the approximate values that some of the controllable variables of the system must assume to achieve a satisfactory level of performance.
A second major point that should be realized is that queueing theory does not offer a full menu of answers. The state-of-the-art after nearly three decades of intensive research can be summarized roughly as follows:

1. 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).
2. 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.
3. 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.
4. 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.
In the following sections, our attention will be focused on answering questions related to estimating such quantities as the fraction of time service facilities are idle (or busy); the expected values (and occasionally the variance and simple moments) of the time spent by queueing system users while waiting to gain access to servers; the expected duration of periods during which a server is continually busy; and the number of other users that an arriving user can expect to find in a queueing system. As we have just indicated, these are precisely the types of questions that queueing theory has been most successful with. In the process we shall present, whenever available, other results which cast additional light on the queueing phenomena that we wish to explore.