For a CTMC (continuous time Markov chain) with S= N0 (0,1,....).

For the question a, i think the answer is the rate for the state 0. P_x(X_h=x) = 1- rate(x)h +o(h)

but I'm blocked on question b (the right part of the inequality) and the question c.

Thanks for your help.

5.10. Let P(1) = Poo(t). (a) Find lim 1 - P(1) 1-0 (b) Show that P(1) P(s) = P(1 + s) s 1 - P(s) + P(s) P(1) (c) Show | P(1) - P(s) | = 1 - P(1 - s), s