Queues with Correlated Service Times -- the $M/M_D/c$ Model

Abstract

This paper studies multi-server queueing systems with correlated service times, modeled as the $M/M_D/c$ queue, which is a natural extension of the recent work by Thapa and Zhao \cite{Thapa-Zhao:2026}. In this model, arrivals follow a Poisson process, while service times across servers exhibit dependence captured by the Marshall--Olkin multivariate exponential distribution (MO-MVED). We first develop a rigorous sample-path construction of the system and establish that the resulting queueing process is a continuous-time Markov chain. We then analyze the stationary behavior of the $M/M_D/c$ model. In the homogeneous case, we derive a complete solution via geometric tail structure and explicit boundary equations, recovering a tractable one-dimensional representation. In the heterogeneous case, we establish a general framework combining a geometric tail with a finite boundary system, and prove existence, uniqueness, and nonnegativity of the stationary distribution. The above results provide a unified analytic framework extending classical $M/M/c$ theory to correlated-service settings, and reveal how dependence among service times fundamentally affects system performance and structure. Beyond the $M/M_D/c$ model, We next study the interplay between Marshall--Olkin service dependence and queue-state Markovianity. On the one hand, Marshall--Olkin dependent service completions are shown to preserve Markovianity for a broad class of queueing systems. On the other hand, if a queueing process admits a Markovian state description without tracking service ages, residual service times, or service phases, then its service mechanism must satisfy a weak multivariate lack-of-memory property and consequently belongs to the Marshall--Olkin family. These results provide a probabilistic foundation for the use of Marshall--Olkin multivariate exponential service times in Markovian queueing models.