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 A,, and each "on the phone" period lasts for an exponentially distributed time with rate ,, i=1, 2, 3, 4.

(a) What proportion of time are all workers "working"? Let X() equal 1 if worker i is working at time t, and let it be 0 otherwise. Let X(r) (X,(1), X2(f), X(f), X(f)).

(b) Argue that (X(1), 1 0) is a continuous-time Markov chain and give its infinitesimal rates.

(c) Is (X()) 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"?