5.13 Server-dependent mean service times Suppose that in the three-server example of Section 5.4.3 we were told that the mean service times of the various servers were not identical, but were given as follows: ₁^-1 = 2/3, ₂^-1 = 1.0, and ₃^-1 = 2.0(units of time). Here includes both travel time and on-scene time.

1. Determine a requirement for A such that the system is not saturated (i.e., that the total required service does not exceed the total available capacity).

2. Assuming that = 1.5, write a set of balance-of-flow equations analogous to (5.22)-(5.29) whose solutions yield the system equilibrium state probabilities.

   Hint: This system, when unsaturated, does not collapse to an M / M / 3 birth-and-death model, so one does not know the sums of probabilities along certain hyperplanes.

3. Suppose we are given a numerical value for P₁₁₁. Find P_Q in terms of P₁₁₁ [see (5.21e)].

   Hint: The system is a birth-and-death process for saturated system states.

4. Argue that (5.30) for server workloads and (5.32) for "unsaturated" interatom dispatch frequencies remain unchanged.

5. Argue that (5.34) should be replaced with

Conclude that (5.36), (5.37), and (5.38) remain unchanged, assuming that f_nj^[2], as given by (*), is substituted for _jP'_Q/N in (5.35).

6. Mean travel times are more difficult and require approximations. Can you fill in the details?