I have a question about setting up the generator matrix of the problem below.
Suppose customers arrive to a processing center in accordance to a Poisson process with rate λ. This center consists of infinitely many servers, labeled $1, 2, 3, . . .$. An arriving customer always joins the lowest-numbered available server, and each customer brings an exponentially distributed amount of work with rate $\mu$, independently of everything else. This, however, is not a classical infinite-server queue, as the customers interact in the following way: whenever a customer at server i is completely served, it, along with all other customers at servers $i+1,i+2,i+3,...,$ leave the system.
If we let $Q(t)$ denote the number of customers in the system at time $t, t ≥ 0$, then $\{Q(t); t ≥ 0\}$ is a CTMC. What would be the rate this chain goes from state 3 to state 2, 1 and 0? My thought was that these rates are $\mu$, $2\mu$ and $3\mu$ respectively, since to go from 3 to 2 we need the last customer to finish first; from 3 to 1 we need the minimum of the next to the last customer to finish and from 3 to 0 we need the the first customer to finish. However, the solution I have says that these rates should be 0,0, $3\mu$ respectively. I'm hoping somebody will be able to explain to me why is this right?