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.
a. Assuming steady-state conditions, find the expected total response time to a random patient. Your answer should be in terms of x, fa and fb only.

Hint: To keep the algebra simple, write your answer in terms of x and of (fa - fb).

b. If the objective is to minimize the expected total response time per patient, what is the optimal value of x when fa = fb = 1/2?
c. Does your answer in (b) agree with or violate Hakimi's theorem for the location of a median on networks? Please explain briefly.
d. In the general case (arbitrary , , v, fa and fb), does the question of whether steady-state is reached depend on the location of the hospital/ambulance? Please explain briefly (no mathematics).
e. Suppose now that all calls that find the ambulance busy, i.e. away from the hospital, are lost (e.g., the patients are transported to the hospital by taxi). Where should the hospital be located in this case if fa = 0.8 and fb = 0.2 (and, as before, = 1/4, = 1, v = 1) and the objective is still to minimize expected total response time for those patients who are served by ambulance? (Note that no queueing ever occurs in this case.)
f. Repeat part (e) assuming that an additional constraint is that no total response time should ever exceed 2.6 units (induding the time on the scene).