Question: A continuous time Markov chain on states {a, b, c} has jump rates q ab =1 q ba =2 q ca =2 q ac =1

A continuous time Markov chain on states {a, b, c} has jump rates qab =1 qba =2 qca =2

qac =1 qbc =1 qcb =2.

Suppose X0 = a. We wish to find the expected time to reach state b, denoted Tb.

(a) What is the expected time for the first jump out of a?

(b) Once the chain leaves a, what is the distribution of the next state?

(c) Let Mi be the expected time to reach b if we start at state i. (So Mb = 0). Use parts (a),(b) to write

an equation for Ma in terms of Mb and Mc.

(d) Write a similar equation for Mc.

(e) Solve the equations to find M.

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