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.