4.6 Spatially distributed queue with random server location In this problem we examine a spatially distributed queueing system similar to that of Example 1, Section 4.7. In the process we shall examine how the "advantage of central location" (i.e., the benefits of positioning a facility/server at the geographical center of an urban area) behaves as a function of system utilization. In this problem the server will be randomly located in the district, in contrast to Example 1, in which it was centrally located.
      We shall examine here the case in which the rectangular district of Figure 4.10 is patrolled by a police car which is dispatched to incidents within the district. Incidents are served in a FCFS manner, with the patrol car traveling from incident to incident whenever there is a backlog of calls for police assistance. At times when no pending calls exist and the patrol car is free, it remains stationary at the location of the last incident that it served, waiting to be dispatched to the next call for assistance. All other assumptions in the problem (e.g., right-angle travel, uniformly distributed incidents in the district, Poisson demands, service time on the scene, travel speeds, etc.) are identical to those of Example 1. It will be assumed, however, that the dimensions of the district are now 2X₀ × 2Y₀ miles. This is done to make the results of this problem comparable to those of Example 1, where the ambulance must make round trips between the hospital and the incidents that it serves.

a. Obtain expressions for ,, _q, , and _q, in this case. To avoid any confusion, we specify that the patrol car "begins serving" a particular incident, as soon as the patrol car begins its trip toward that incident. You should also assume that successive service times (= travel time + time on the scene) are statistically independent. [In truth, there is a slight correlation between successive service times (why?).]

b. Assuming exactly the same numerical values as in Example 1 (numerical example), prepare a table for ,, _q, , and _q, as the rate of calls per hour, A, increases. Compare these values to those for the ambulance service of Example 1. How do these values differ as increases ( 1) ?