Individuals join a club in accordance with a Poisson process with rate . Each new member must

Individuals join a club in accordance with a Poisson process with rate λ. Each new member must pass through k consecutive stages to become a full member of the club. The time it takes to pass through each stage is exponentially distributed with rate μ. Let Ni(t) denote the number of club members at time t who have passed through exactly i stages, i = 1,..., k − 1. Also, let N(t) =

(N1(t), N2(t), . . . , Nk−1(t)).

(a) Is {N(t), t 0} a continuous-time Markov chain?

(b) If so, give the infinitesimal transition rates. That is, for any state n = (n1,..., nk−1) give the possible next states along with their infinitesimal rates.