34. 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 , and each “on the phone” period lasts for an exponentially distributed time with rate , 2, 3, 4.

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

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

Let Image.

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

(c) Is Image 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”?