In this chapter we shall use the term queueing system to refer to a generic model (see Figure 4.1) which comprises three elements: a user source, a queue, and a service facility that contains one or more (including possibly an infinite number of) identical servers in parallel. Thus, each user of a queueing system is "generated" by a user source, passes through the queue where (s)he may remain for a nonnegative period of time (including possibly zero time), and then is processed by a single server-because of the parallel arrangement of servers. Once a user has left any one of the servers in the system, after obtaining service there, the user is considered to have left the queueing system as well.

In view of the description given above, a queueing network can now be defined as a set of interconnected queueing systems. Thus, in a queueing network, the user sources for some of the queueing systems in the network may be other queueing systems in the same network (see also Figure 4.2). It can also be inferred that the analysis of models of queueing systems, as we have defined them here, provides the building blocks for the analysis of queueing networks.

To describe a queueing system fully, information must be supplied about all three generic elements of the system (Figure 4.1):

  1. About the user generating process (i.e., the arrival process of users at the system).
  2. About the queue discipline [i.e., the order in which users obtain access to the service facility, once (and if) they join the queue].
  3. About the service process.

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    To describe a queueing network, further information must be provided (Figure 4.2) on how the queueing systems are interconnected, how they interact (e.g., do bottlenecks "downstream" ever affect the preceding servers?), and how users are assigned to the queueing systems.
    It should be obvious that there exist countless variations of queueing systems and networks. We shall have occasion to refer to some of these variations in following sections. At this point, however, it should be noted that a code has been used in queueing theory for years to describe some of

fig4.2.gif (24185 bytes)

the simplest (and best understood) queueing systems. This code is of the form A|B|m, where A and B are letter symbols and m is an integer constant. The letters A and B indicate the probability distribution of user interarrival times and of service times, respectively, and m is the number of identical parallel servers in the queueing system (thus, m can take values from 1 to ).
    The standard code letters used for probability distributions in queueing theory are:

M = Poisson (i.e., negative exponential pdf for user interarrival times or for service times; M stands for "memoryless")

D = deterministic (i.e., interarrival or service times are constant)

Ek = kth-order Erlang distribution [see (2.50) and (2.57)]

Hk = kth-order hyperexponential distribution (see Problem 4.6)

G = "general" distribution (i.e., any distribution at all)

The letters A and B can thus be any one of the five symbols above.
    To newcomers to queueing theory it always seems strange that, out of all possible probability distributions, only four (M, D, Ek, and Hk) have been assigned special symbols. The simple reason is that only these four distributions offer significant advantages in an abstract analysis, in the sense that when they are present in a queueing model, parts of the mathematical analysis of that model may become more tractable. Thus, all other distributions are lumped under the "general" (G) category (which, of course, also includes the M, D, E, and Hk cases). These comments will become more clear as we make our way through this chapter.
    The coded systems also assume independence of successive user arrival times and of successive service times at the queueing system. We shall see several examples of urban service systems where this assumption is not valid in practice.
    Some more-or-less standard abbreviations are also used to indicate the most commonly encountered queue disciplines. FIFO is used to indicate the first-in, first-out queueing arrangement, also known as FM first come, first served). Similarly, LIFO (= last in, first out) or LCFS last come, first served) indicate the situation in which the last user to join the queue becomes the next in line for entering service. The abbreviation SIRO is used to indicate "service in random order." Although LIFO and SIRO may at times appear offensive to our sense of fairnessespecially in connection with gaining access to public transportation vehicles during peak traffic hours-they, and their variations are all too common in real life and thus cannot be ignored.
    Another important parameter in the description of a queueing system is the system capacity. This indicates the maximum number of users that can at any time be in the service facility and in the queue. Queue capacity, on the other hand, indicates the maximum number of users that can be in the queue alone.
    Finally, it is important to emphasize that queueing systems need not adhere to the classical picture of a physically stationary server (a bank teller, a checkout counter, highway toll booths, etc.) where prospective users queue up to be served. It is very common to have systems in which the prospective users remain stationary, possibly at geographically widely separated points, while the server(s) associated with the queueing system visit them according to some (implicit or explicit) priority scheme and provide the requested service. We often use the term spatially distributed queues to refer to such geographically "spread-out" systems. They are of particular interest in urban operations research and Chapter 5 will deal with their specific characteristics. For some of these systems (e.g., fire departments or ambulance services), queues in the sense of a backlog of calls for service, rarely, if ever, occur. Yet queueing theory is most valuable in planning for such systems-in determining, for instance, the number of vehicles needed, the expected workload of service units, or the best deployment of the service units in a city.