As seen in Chapter 4, the number of queueing models (even with indistinguishable servers) is enormous. Each apparently minor change in assumptions can bring about a host of new analytical problems, thus requiring new analysis. This problem is much more severe in a spatial setting where the spatial distribution of servers and/or customers adds to the already nearly limitless number of possible system configurations and operating rules. Thus, we are motivated to choose a representative class of urban spatial queueing systems to illustrate the methods and approaches involved in a spatial setting.

In selecting a class of models our objectives are fivefold:

  1. To demonstrate the linkage between the geometrical probability concepts of Chapter 3 and the queueing concepts of Chapter 4.

  2. To maintain analytical tractability, at least for small numbers of servers.

  3. To illustrate how a complicated urban environment can be modeled in a queueing framework and analyzed numerically on the computer.

  4. To illustrate the derivation of simple rules of thumb, based on "back of-the-envelope" calculations.

  5. To select a model that has been used in cities to affect policy decisions.

Fortunately, all of these objectives can be satisfied by selecting from the class of "demand-responsive" queueing systems that operate as follows:

  1. Requests for service (customers) are generated as a Poisson process in time from throughout the city or part of the city.

  2. In response to each request, either

    a) a mobile server (called service unit or response unit) is assigned (dispatched) to travel to the customer and provide on-scene service (and perhaps related off-scene service)


    b) the customer travels to a nearby service facility to obtain service.

The first type of system, called a server-to-customer system, could be an emergency service system, for instance police, fire, ambulance, or emergency repair; a social service system involving demand-responsive home visits, for instance by a social worker, physician, or practical nurse; or a demandresponsive delivery system, delivering for instance parcels, pizzas, or parts for automobiles. The second type of system, called a customer-to-server system, could correspond to outpatient clinics, police precinct stations, little city halls," libraries, and so on.

Typically, the geometrical considerations are quite similar for the two types of systems, but their queueing characteristics are not. In both types of system, server proximity plays a key role, typically with the closest server being the one utilized. However, in the server-to-customer system, the fact that the first preferred (usually closest) server is busy does not necessarily imply a queueing delay for the customer; instead, server cooperation or workload sharing would usually result in the assignment of a nearby backup server. Thus, customers can have the dual benefits of a "neighborhood" server for most service requests and the small queueing delays associated with multiserver queues due to workload sharing. In the customer-to-server system, the customer is not likely to travel to the second or third closest service facility, even if such a policy would reduce the total time until service; this is due primarily to a lack of real-time knowledge of the queueing situation at each facility plus the fact that many neighborhood service systems allow use of only one service center (i.e., no planned workload sharing).

Because of the "standard" operating behavior of the customer-to-server system, most of the queueing techniques of Chapter 4 apply directly, perhaps modified due to geometrical considerations. However, the workload sharing nature of server-to-customer systems make them particularly attractive to study-and this will be our choice in this chapter. Most of the models we will examine will require further assumptions, which limit their applicability across the class of server-to-customer systems.

Our tour of server-to-customer queueing models proceeds from singleserver models (applying the theory of M / G / 1 queues); to simple two-server models. in which closed-form expressions can be obtained; to N-server models that may be analytically tractable or, more likely, require computer solution; to "infinite" or many-server models. We will find that this selection of models will fulfill each of our five objectives for the chapter. Several of the known extensions to this work are developed as problems at the end of the chapter. Given the vast number of urban problems that can be usefully approached from a queueing theory point of view, the results to date represent just a beginning. We expect this area of applied queueing theory to be a fertile one for research in the years ahead. For convenience, symbols frequently used in this chapter are collected together in Table 5-2.