Modern Stochastics: Theory and Applications

We study a level-dependent PH/PH/1 queue with an infinite waiting room and derive explicit density formulas for the associated Matrix Mittag-Leffler (MML) random variables. The motivation arises from the use of Phase Type (PH) and Mittag-Leffler type distributions in modeling heavy-tailed and non-Markovian arrival and service processes in queueing systems, where classical exponential assumptions may fail to capture realistic behavior. Our goal is to obtain explicit density expressions for the MML random variables associated with the truncated queueing systems PH/M/$1\sim n$ and M/PH/$1\sim n$, where n denotes the truncation level at which the queueing process is absorbed. We first analyze PH/M/1 queues in which the inter-arrival times are PH-distributed, focusing specifically on exponential and Erlang cases, and derive closed-form MML densities. We then consider M/PH/1 queues with PH-distributed service times and obtain analogous explicit density formulas. Several examples with PH-generators of different orders are presented to illustrate the results.