4 1 . Let Y denote an exponential random variable with rate that is independent of...

4 1 . Let Y denote an exponential random variable with rate ë that is independent of the continuous-time Markov chain [X(t)} and let Pu = PiX(Y)=j\X(0) = i}

(a) Show that where is 1 when / = j and 0 when / * j .

(b) Show that the solution of the preceding set of equations is given by

Ñ = (I - R/A)"1 where Ñ is the matrix of elements PU91 is the identity matrix, and R the matrix specified in Section 6.8.

(c) Suppose now that Yt,..., Yn are independent exponentials with rate ë

that are independent of {^(0). Show that P{jf(yi + - + yn)=y|jf(0) = /j is equal to the element in row /, column j of the matrix P \

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