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

Marko A. A. Boon (), Onno J. Boxma (), Offer Kella () and Masakiyo Miyazawa ()
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Marko A. A. Boon: Eindhoven University of Technology
Onno J. Boxma: Eindhoven University of Technology
Offer Kella: The Hebrew University of Jerusalem
Masakiyo Miyazawa: Tokyo University of Science

Queueing Systems: Theory and Applications, 2017, vol. 86, issue 3, No 5, 277-299

Abstract: Abstract 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.

Keywords: Queue length; Steady-state distribution; Balance equations; Distributional relationship; Palm distribution; Nonstationary framework; 60K25; 90B22 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s11134-017-9528-z

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