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
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:
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.
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.