Phone calls arrive at a voice-mail system at a rate of 20 per hour according to a Poisson process (the voice-mail system routes calls to appropriate operators based on pushing buttons on the phone;...

Phone calls arrive at a voice-mail system at a rate of 20 per hour according to a Poisson process (the voice-mail system routes calls to appropriate operators based on pushing buttons on the phone; understanding this is not important for working the problem). The voice-mail system can handle only one call at a time (in other words, it is a one-server system with no queueing). If a second call arrives while the voice-mail system is busy, then it is routed to a human operator; there is only one human operator. If the human operator is also busy, then the call is lost. The mean time for the voice-mail system to route a call is 2 minutes (once it has routed the call, it is free to take another call). The human operator processes calls at a rate of 10 per hour. All times are exponentially distributed.

(a) Derive a Markov process model capable of answering the questions below. Be sure to define your state space, time index and generator matrix.

(b) Over the long run, what fraction of the time is the human operator busy?

(c) What is the probability that an arriving call will be routed by the voice-mail system?