5.13 Serverdependent mean service times Suppose that
in the threeserver 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 onscene time.
 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).
 Assuming that = 1.5, write a set of
balanceofflow 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 birthanddeath model, so one does not know the sums of
probabilities along certain hyperplanes.
 Suppose we are given a numerical value for
P_{111}. Find P_{Q} in terms of
P_{111} [see (5.21e)].
Hint: The system is a birthanddeath process for saturated
system states.
 Argue that (5.30) for server workloads and (5.32) for
"unsaturated" interatom dispatch frequencies remain unchanged.
 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 _{j}P'_{Q}/N in (5.35).
 Mean travel times are more difficult and require
approximations. Can you fill in the details?
