(Continuation of continuous-time branching). This exercse views the continuous-time branching process of Exercise as a stopped random walk. Recall that the process was specified there as a Markov...

(Continuation of continuous-time branching). This exercse views the continuous-time branching process of Exercise as a stopped random walk. Recall that the process was specified there as a Markov process such that for each state j, j ≥ 0, the transition rate to j + 1 is j and to j ≥ 1 is jµ. There are no other transitions, and in particular, there are no transitions out of state 0, so that the Markov process is reducible. Recall that the embedded Markov chain is the same as the embedded chain of an M/M/1 queue except that there is no transition from state 0 to state 1.

a) To model the possible extinction of the population, convert the embedded Markov chain abve to a stopped random walk, {S_n; n ≥ 0}. The stopped random walk starts at S₀
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