Consider a continuous-time Markov chain (X_t)_t0with state space S = {1, 2, 3, 4}. From state 1 the process is equally likely to either transition to 2 or 4; from state 2 and 3 the process is equally likely to transition to all other states; and from state 4 the process always transitions to state 1. Suppose further that the process spends on average sqrt(i) units of time in each state i S.

(a) Provide the transition probability matrix P^~ for the embedded chain.

(b) Provide the holding time parameters q_i.

(c) Provide the generator matrix Q. Justify your answer.

(d) Draw the transition rate graph.