Four workers share an office that contains four telephones. At any time, each worker is either “working” or “on the phone.” Each “working” period of worker i lasts for an exponentially distributed time with rate λi , and each “on the phone” period lasts for an exponentially distributed time with rate μi, i = 1, 2, 3, 4.

(a) What proportion of time are all workers “working”?

Let Xi (t) equal 1 if worker i is working at time t , and let it be 0 otherwise.

Let X(t) = (X1(t ),X2(t ),X3(t ),X4(t)).

(b) Argue that {X(t ), t ≥ 0} is a continuous-time Markov chain and give its infinitesimal rates.

(c) Is {X(t)} time reversible? Why or why not?
Suppose now that one of the phones has broken down. Suppose that a worker who is about to use a phone but finds them all being used begins a new “working”
period.

(d) What proportion of time are all workers “working”?