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

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

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 ¯Pij , 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 ¯Pn.

(d) Explain the relationship of the preceding to Approximation 2 of Section 6.8 42.

(a) Show that Approximation 1 of Section 6.8 is equivalent to uniformizing the continuous-time Markov chain with a value v such that vt = n and then approximating Pij(t) by P∗n ij .

(b) Explain why the preceding should make a good approximation.
Hint: What is the standard deviation of a Poisson random variable with mean n?

1 Pij - + ; + v; + k