Matrix-analytic methods for $M/M_D/c$ queues with Marshall--Olkin dependent servers

Abstract

We study multiserver queues with Marshall--Olkin dependent service mechanisms, heterogeneous servers, and fixed idle-set assignment. An explicit matrix-analytic solution links the stationary queue distribution to departure and customer-delay laws. Waiting time conditional on delay is exactly exponential, and the stationary \chg{sojourn law has a phase-type representation with at most one more phase than the number of servers.} A consistency condition characterizes when Markovian completions admit a Poisson shock representation. For two three-server families with identical bivariate residual-service laws, we prove a conditional waiting-time ordering and identify when their unconditional waiting tails cross. The results show how higher-order dependence affects performance beyond marginal rates and pairwise service dependence.

arXiv comment47 pages. This is a significantly updated version using a different method (MAM), and extended studies to departure process, and virtual waiting time