Problem 1. (40 points) A single repair person is responsible from repairing two machines:

Machines 1 and 2. Each time it is repaired, machine i stays up for an exponential time with

rate i, for i = 1; 2. When machine i fails, it requires an exponentially distributed amount of

work with rate i to complete its repair, for i = 1; 2. The repair person always gives priority

to machine 1 and starts repairing it as soon as it fails. This means that if machine 1 fails

while 2 is being repaired, then the repair person will immediately stop work on machine 2

and start repairing machine 1.

(a) Model this system as a continuous-time Markov chain (CTMC). Dene the state of the

process clearly, provide the state space, and obtain either the transition rate diagram or the

transition rate matrix.

(b) Write down the balance and normalizing equations for this CTMC.

(c) Suppose that i = 1 and i = 1 for i = 1; 2. What is the proportion of time machine 2

is down in the long-run?

(d) Again suppose that i = 1 and i = 1 for i = 1; 2. Suppose also that the prot from

machine i is $200=i per unit time for i = 1; 2. A machine generates prot only when it is

up. The cost of repair of machine i is $50=i per unit time for i = 1; 2. What is the long-run

average net revenue generated by these two machines?

(e) The repair person realizes that repair times for machine 2 have an Erlang distribution

with two phases and mean 1=2. Can you still model this system as a CTMC? If yes, than

answer part (a) under this new information. If no, then explain the reason.