Let Y denote an exponential random variable with rate that is independent of the continuous-time Markov chain (X()) and let P = P(X(Y) =j|X(0) = i}

(a) Show that 1 Py = v; +2 2 U; +2 where 5 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/)- where P is the matrix of elements P, I is the identity matrix, and R the matrix specified in Section 6.8.

(c) Suppose now that Y,..., Y, are independent exponentials with rate that are independent of (x(t)). Show that P{X(Y++ Y) =j|x(0) = i} is equal to the element in row i, column j of the matrix P".

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