4: Introduction to Queueing Theory and Its Applications

     Requests for service very often find some or all of the units associated with some urban service system busy. The performance of these systems on such occasions (i.e., under conditions of some "stress") plays a major role in shaping citizens' perceptions with regard to the level of service offered.
    That some deterioration in the level of service will occur during periods of intensive activity is clear. For instance, a fire alarm may occur at an instant when all fire companies stationed at the nearest fire houses have already been dispatched to another alarm elsewhere in a city. To the extent, then, that the response time to the new fire will be longer under these circumstances (because of the need to dispatch fire engines from remote fire stations), the perceived level of service will be lower. Similarly, a police dispatcher will often postpone service to medium-priority calls for police assistance during periods when nearly all police cars in a district are busy. In doing so, the dispatcher preserves the ability of the police to respond immediately to top priority calls while maintaining some level of police "visibility" through the patrolling activities of non busy cars. Under normal conditions such medium priority calls would have received prompt service.
    Busy period service delays must inevitably occur in the case of services that respond to unpredictable demands whose time and location of occurrence are governed by some type of (known or unknown) probabilistic law. The cost of providing sufficient capacity to avoid all delays under all circumstances would be insupportable. The proper role of analysis, therefore, is to design service systems that achieve an acceptable balance between system operating costs, on the one hand, and the delays suffered by users of that system, on the other. As to where this acceptable balance lies, it all depends on the nature of the service provided. In some cases-for example, in fire or ambulance services-the costs of delays are generally perceived to be very high. To assure a low probability of such delays, it is therefore necessary to design systems that are relatively underutilized and whose servers (e.g., fire engines, ambulances, etc.), experience long periods of idleness. In other instances (e.g., collection of solid refuse or the delivery of mail), delays of a few hours or even of days are not usually catastrophic. This, in turn, permits a high level of utilization for the servers used by the system.
    Queueing theory, the theory of congestion, is the branch of operations research which explores the relationships between demand on a service system and the delays suffered by the users of that system. Since almost all urban service systems can be viewed as queueing systems (as it will become clear in this chapter), queueing theory plays a central role in the analysis of and planning for urban services. This chapter will therefore deal with a review of some important results in queueing theory and with an introduction to the applications of these results to the problems on which this book focuses.