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I am investigating functionals defined over sequences of discrete probability distributions related to dynamical/stochastic system performance. As an initial step, I am searching for references that discuss deriving the time-dependent distribution of queue length in $M_t/M_t/1/K$ queues.

Ideally, I'd like to find formulas that describe the distribution of the number of persons in the queue for a given time (i.e., $F_t$). I have found some numerical studies of various related queues, but not much in my open-source searches of the analytica/mathematical properties of these systems.

Any pointers to papers is appreciated.

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Go for William Massey (who had a paper on Mt/Mt/1 queue asymptotics) or Ward Whitt's page (like 400 papers to explore). The latter is working on time-varying single server recently with someone called Ni Ma.

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