48. A taxi alternates between three different locations. Whenever it reaches location i, it stops and spends a random time having mean ti before obtaining another passenger, i = 1, 2, 3. A passenger entering the cab at location i will want to go to location j with probability Pij . The time to travel from i to j is a random variable with mean mij . Suppose that t1 = 1, t2 = 2, t3 = 4, P12 = 1, P23 = 1, P31 = 23

= 1 − P32, m12 = 10, m23 = 20, m31 = 15, m32 = 25. Define an appropriate semi-Markov process and determine

(a) the proportion of time the taxi is waiting at location i, and

(b) the proportion of time the taxi is on the road from i to j, i, j = 1, 2, 3.