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

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?