6.17 Facility location with queueing Consider two small towns which are one unit distance apart, as shown in Figure P6.17. (Each town is represented as a single point (node) on this simple "network," i.e., intra-town distances can be considered insignificant).

A hospital equipped with a single ambulance is located at some point between the two towns which is a distance x away from the halfway point between the two towns.

Calls from the two towns that require dispatching of the ambulance occur in a Poisson manner at a combined rate = 1/4 calls/unit time. A fraction fA of these calls come from Town A and a fraction fBfrom Town B (fA + fB = 1)

In responding to each call the ambulance travels to the appropriate city at a constant speed v, spends a constant amount of time on the scene (picking up a patient) and returns to the hospital (with the patient) at the same constant speed v. Let v = 1 distance unit/time unit and = 1 time unit.

Calls for ambulance dispatching queue up in a first-come, first-served manner until the ambulance eventually serves them. We define the "total response time" of the ambulance to a patient as the time interval between the instant when that patient calls for the ambulance and the instant when the patient arrives at the hospital.