Throwing Away Servers
Suppose your computer center currently consists of a single server of speed μ.
Jobs arrive according to a Poisson process with rate λ, and their service times
are Exponentially distributed.
Suppose the current response time is considered intolerable by the users. A
second, faster server, running at speed αμ (for α > 1), is purchased and added
to the system in a heterogeneous M/M/2 structure with a single queue as in
Exercise 14.5. Denote the load (utilization) of the M/M/2 system by ρ. Denote
the mean response time of the M/M/2 system by E [T].
(a) Use the result in (14.14) from Exercise 14.5 to derive a formula for E [T],
the mean response time of the M/M/2 system with heterogeneous servers.
(b) A hotshot consultant walks in and makes the radical proposal of disconnecting the original server entirely (i.e., simply letting the faster server run
by itself). Clearly this makes sense with respect to power, but the consultant claims this is also a win for E [T]. For $400/hr, what is the consultant
thinking? Come up with an instance, in terms of λ, μ2 , and μ1 for which
the consultant is right. Also, explain intuitively what is happening. [If you
find this problem interesting, you can think about a general criterion under
which the consultant would be right . . . Throwing away servers is fun!]