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

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

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

(a) Show that P¯

i j = 1 vi + λ

k qik P¯

kj +

λ

vi + λ

δi j where δi j 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¯

i j , I is the identity matrix, and R the matrix specified in Section 6.9.

(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 6.9.

*49.

(a) Show that Approximation 1 of Section 6.9 is equivalent to uniformizing the continuous-time Markov chain with a value v such that vt = n and then approximating Pi j(t) by P∗n i j .

(b) Explain why the preceding should make a good approximation.