Question 4. [9 Marks]. Recall that the exponential PDF has the form f (t; A) 2 )le"\"5 where t, A > 0 and A is known as the rate parameter. Consider a job shop that consists of 3 identical machines and 2 technicians. Suppose that, the amount of time each machine operates before breaking down is exponentially distributed with rate parameter 0.1 and, a technician takes to x a machine is exponentially distributed with rate parameter 0.4. Sup- pose that all the times to breakdown and times to repair are independent random variables and let X (t) be the number of machines which are operating at time t. (3) Determine the Q-matrix for this Markov process. (b) Write the forward equations involving the P5j(t), j = 0, 1, 2, 3, in terms of pj(t) :2 P0j(t) = IP(X(t) = j|X(0) = O). (c) Obtain the equilibrium probabilities pj = limHoo pj(t). (d) What is the average number of busy technicians in the long-run? [Note: do not expect your answer to be an integer]