## 5.7 SPATIAL DISTRIBUTION OF BUSY SERVERS

In some situations we would like to know the spatial distribution of servers busy at the scene of service requests. For instance, we may wish to analyze a dispatching policy which may interrupt a server busy on low-priority service in order to send him or her to a nearby higher-priority request; in such a case we would need to know the distribution of travel time to the nearest busy server. Or, in police applications, since police presence is said to deter crime, we may wish to know the spatial distribution of busy servers (as well as available servers) because a parked patrol car also acts as a visible deterrent; we may wish to alter this distribution, if possible, by adjusting our spatial prepositioning policies (e.g., beat designs).

We examine this question using the theory of M / G / queues (cf. Section 4.8). The assumption of an infinite number of servers implies that the actual number of servers is sufficiently large or that workload is sufficiently small so that queues almost never form.

Consider, then, a spatially distributed service system in which:

1. Service requests are generated in a Poisson manner from a region of area Ao at a rate Ao requests per hour.

2. Service requests are distributed uniformly in space.

3. -1 = average time to service a request (general service time pdf).

4. There are infinitely many servers.

Then, according to (4.87), in the steady state the probability that there are k busy service units in any subregion of area A Ao is

 Pk(A) = k = 0, 1, 2, . . . (5.58)

This result says that, regardless of the method of prepositioning the units, the busy servers are distributed as a spatial Poisson process with parameter (/) busy servers per unit area. This simple result allows us to use "nearest neighbor theory" of spatial Poisson processes to develop probability laws of the travel times to the kth closest busy server (see Example 15 in Chap. 3).