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. 



Reference Request: $M_t/M_t/1/K$ queue length distributions
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.


