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.