49. Let Y denote an exponential random variable with rate that is independent of the continuous-time

49. Let Y denote an exponential random variable with rate λ that is independent of the continuous-time Markov chain and let

(a) Show that where is 1 when and 0 when .

(b) Show that the solution of the preceding set of equations is given by where is the matrix of elements , I is the identity matrix, and R the matrix specified in Section 6.9.

(c) Suppose now that Image are independent exponentials with rate λ that are independent of . Show that is equal to the element in row i, column j of the matrix .

(d) Explain the relationship of the preceding to Approximation 2 of Section 6.9.