*7. 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 ì. Let Í((ß) denote the number of club members at time / that have passed through exactly / stages, / =

1 AT — 1. Also, let N(0 = (Nx(t),N2(t), ...,Nk_x(t)).

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

(b) If so, give the infinitesimal transition rates. That is, for any state

ç = («!,..., nk_x) give the possible next states along with their infinitesimal rates.