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

41. Let Y denote an exponential random variable with rate λ that is independent of the continuous-time Markov chain {X(t)} and let P¯

ij = P{X(Y) = j |X(0) = i}

(a) Show that P¯

ij = 1 vi + λ



k qikP¯

kj +

λ

vi + λ

δij where δij is 1 when i = j and 0 when i = j .

(b) Show that the solution of the preceding set of equations is given by P¯ = (I − R/λ)−1 where P¯ is the matrix of elements P¯

ij , I is the identity matrix, and R the matrix specified in Section 6.8.

(c) Suppose now that Y1,...,Yn are independent exponentials with rate λ that are independent of {X(t)}. Show that P{X(Y1 +···+ Yn) = j |X(0) = i}

is equal to the element in row i, column j of the matrix P¯ n.

(d) Explain the relationship of the preceding to Approximation 2 of Section 6.8.