We study an M|G|1 queue in which both the arrival rate and the service time distribution depend on the state of an underlying finite-state Markov chain. The solution is obtained by a matrix factorization method. This leads to results for waiting times and queue lengths both at arrival epochs and in continuous time. A numerical algorithm for the calculation of several quantities of interest is described and some numerical examples are given.
Regterschot, G. J. K., & de Smit, J. H. A. (1986). The queue M|G|1 with Markov modulated arrivals and services. Mathematics of operations research, 11(3), 465-483. https://doi.org/10.1287/moor.11.3.465