An infinite-server queue influenced by a semi-Markovian environment

We consider an infinite-server queue, where the arrival and service rates are
both governed by a semi-Markov process that is independent of all other aspects of
the queue. In particular, we derive a system of equations that are satisfied by various
"parts" of the generating function of the steady-state queue-length, while assuming
that all arrivals bring an amount of work to the system that is either Erlang or hyperexponentially distributed. These equations are then used to show how to derive all
moments of the steady-state queue-length. We then conclude by showing how these
results can be slightly extended, and used, along with a transient version of Little’s
law, to generate rigorous approximations of the steady-state queue-length in the case
that the amount of work brought by a given arrival is of an arbitrary distribution.