## 5: Spatially Distributed Queues

Armed with the tools of probabilistic modeling, geometrical probability, and queueing theory, we are now prepared to study in more detail spatially distributed urban service systems. These systems include emergency services (e.g., police, fire, ambulance, emergency repair), door-to-door pickup and delivery services (e.g., mail delivery, solid waste collection), neighborhood service centers (e.g., outpatient clinics, 1ittle city halls," libraries, social work agencies), and transportation services (e.g., bus and subway services, taxicab services, "dial-a-ride" systems).

Most of these service systems share the following characteristics:

1. Elements of uncertainty appear in the time, location, and perhaps duration (or amount) of required service.

2. The spatial structure of the city plays a key role in the way service is provided. Particularly important are the spatial distribution of service requirements and the city's transit characteristics from point to point.

3. In the provision of service, congestion is likely to arise in various forms. This is due to the uncertainties in service requirements and the finite amount of resources dedicated to any particular service.

The uncertainties give rise to the need for a probabilistic analysis. The spatial nature of the problem often requires tools of geometrical probability (or more standard probabilistic analysis in a geometrical setting). Congestion often requires a queueing type of analysis. Thus the name of this chapter-spatially distributed queues.

In an urban setting, a spatially distributed queue is one in which the customers and/or servers are distributed over the city (or part of the city). Either the customer travels to the server for service or the server travels to the customer. The service may be provided "on the scene" (e.g., at a residence, at the scene of an emergency), at a service facility (e.g., clinic or social service center), or over a route (e.g., a bus route or path of a taxicab). Sometimes service may consist of a combination of these, as with emergency medical services. Table 5-1 contains several examples of spatially distributed queues found in a city.

Most spatially distributed queues are multiserver systems, so in analyzing these systems we must confront the complexities inherent in multiserver models. Moreover, unlike most multiserver queues analyzed so far, spatially distributed queues have distinguishable servers, each with different operating characteristics (such as different workloads and mean service times). So the state depiction of spatially distributed queues must retain the identities of busy and available servers.