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

7. 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.