Consider a Markov renewal process with the semi-Markov kernel Q given by Q =

0.6(1 − e−5t

) 0.4

− 0.4e−2t 0.5

− 0.2e−3t − 0.3e−5t 0.5

− 0.5e−2t − te−2t

a. Determine the state-transition probability matrix P for the Markov chain {Xn}.

b. Determine the conditional distributions Gij (t) of the waiting time in state i given that the next state is j for all i, j.