Queue-length balance equations in multiclass multiserver queues and their generalizations

Marko A.A. Boon*, Onno J. Boxma, Offer Kella, Masakiyo Miyazawa

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


A classical result for the steady-state queue-length distribution of single-class queueing systems is the following: The distribution of the queue length just before an arrival epoch equals the distribution of the queue length just after a departure epoch. The constraint for this result to be valid is that arrivals, and also service completions, with probability one occur individually, i.e., not in batches. We show that it is easy to write down somewhat similar balance equations for multidimensional queue-length processes for a quite general network of multiclass multiserver queues. We formally derive those balance equations under a general framework. They are called distributional relationships and are obtained for any external arrival process and state-dependent routing as long as certain stationarity conditions are satisfied and external arrivals and service completions do not simultaneously occur. We demonstrate the use of these balance equations, in combination with PASTA, by (1) providing very simple derivations of some known results for polling systems and (2) obtaining new results for some queueing systems with priorities. We also extend the distributional relationships for a nonstationary framework.

Original languageAmerican English
Pages (from-to)277-299
Number of pages23
JournalQueueing Systems
Issue number3-4
StatePublished - 1 Aug 2017

Bibliographical note

Publisher Copyright:
© 2017, The Author(s).


  • Balance equations
  • Distributional relationship
  • Nonstationary framework
  • Palm distribution
  • Queue length
  • Steady-state distribution


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