Individuals join a club in accordance with a Poisson process with rate A. 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 u. Let N,(t) denote the number of club members at time t that have passed through exactly i stages, i = 1,..., k 1.

Also, let N(r) = (N(t), N(),..., Nx-1(t)).

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

(b) If so, give the infinitesimal transition rates. That is, for any state n(n-1) give the possible next states along with their infinitesi- mal rates.