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Summary: Homework 2
Due February 19, Tuesday
1. If {X(t) : t 0} and {Y (t) : t 0} are independent time-reversible
continuous time Markov chains, show that the process {(X(t), Y (t)), t 0}
is also time reversible.
2. Consider two queues with Poisson arrivals and single server with expo-
nentially distributed service times. Suppose that the arrival rate for queue
i is i and service rate is µi for i = 1, 2. Assume that the queues share the
same waiting room which has finite capacity N. That is whenever this room
is full, all potential arrivals to either queue are lost. Compute the limiting
probability that there will be n customers at the first queue and m at the
second queue.
3. N customers move among r servers. The service times at server i are
exponential with rate µi and when a customer leaves server i it joins the
queue (if there are others waiting or else it enters service) at server j, j = i,
with probability 1/(r - 1). Let the state be (n1, . . . , nr) when there are ni
customers at server i, i = 1, . . . , r. Show that the corresponding continuous
time Markov chain is time reversible and find the limiting probabilities.
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