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

  • Qihui Bu
  • Suman Thapa
  • Yiqiang Q. Zhao
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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.

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Cite

BibTeXRIS
BibTeX
@misc{bu2026matrixanalytic,
  title = {{Matrix-analytic methods for $M/M_D/c$ queues with Marshall--Olkin dependent servers}},
  author = {Qihui Bu and Suman Thapa and Yiqiang Q. Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.24881},
  eprint = {2606.24881},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.24881},
}
RIS
TY  - UNPB
TI  - Matrix-analytic methods for M/M_D/c queues with Marshall--Olkin dependent servers
AU  - Qihui Bu
AU  - Suman Thapa
AU  - Yiqiang Q. Zhao
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2606.24881
UR  - https://arxiv.org/abs/2606.24881
AB  - 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.
ER  -