solution

Consider a very large data center, comprising tens of thousands of servers, as
is common in companies like Google, Facebook, and Microsoft. Such a data
center might be approximated by an M/M/∞ system, where there is no queue
and all jobs are immediately served (see Section 15.2). To save power, when a
server goes idle, we assume that it is immediately shut off. When a job arrives,
it needs to turn on a server, requiring setup time I, where I ∼ Exp(α). If a
server, s1, is in setup mode and another server, s2, becomes free, then the job
waiting for s1 goes to s2. At this point, server s1 is shut off.
The effect of setup times for the M/M/∞ was first derived in [68], where
the following beautiful decomposition property was observed. Here λ is theoutside arrival rate, μ is the service rate at each server, and R = λμ .

P {i servers are busy & j servers are in setup}
= P {i servers are busy} · P {j servers are in setup}

That is, the number of busy servers follows a Poisson distribution with mean
R, just as in an M/M/∞ without setup, and is independent of the number of
servers in setup.
In this problem, you will verify the above result:
(a) Draw the CTMC for the M/M/∞ with setup, where the states are (i, j) as
defined above.
(b) Write the balance equation for state (i, j) and verify that the above formulas
satisfy the balance equation.
(c) Equation (27.21) makes intuitive sense because the long-run number of
busy servers should not be affected by the fact that servers first need a
setup; hence a Poisson(R) distribution is reasonable. What is the intuition
for equation (27.22)? [Hint: Assume that there are always exactly R servers
that are busy and draw a birth-death chain representing the number of
servers that are in setup, given that assumption.]