An exact root-free method for the expected queue length for a class of discrete-time queueing systems

Anna Oblakova*, Ahmad Al Hanbali, Richard Boucherie, Jan C.W. van Ommeren, Henk Zijm

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

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For a class of discrete-time queueing systems, we present a new exact method of computing both the expectation and the distribution of the queue length. This class of systems includes the bulk-service queue and the fixed-cycle traffic-light (FCTL) queue, which is a basic model in traffic-control research and can be seen as a non-exhaustive time-limited polling system. Our method avoids finding the roots of the characteristic equation, which enhances both the reliability and the speed of the computations compared to the classical root-finding approach. We represent the queue-length expectation in an exact closed-form expression using a contour integral. We also introduce several realistic modifications of the FCTL model. For the FCTL model for a turning flow, we prove a decomposition result. This allows us to derive a bound on the difference between the bulk-service and FCTL expected queue lengths, which turns out to be small in most of the realistic cases.
Original languageEnglish
Pages (from-to)257-292
Number of pages36
JournalQueueing systems
Issue number3-4
Early online date17 May 2019
Publication statusPublished - 1 Aug 2019



  • UT-Hybrid-D
  • Fixed-cycle traffic-light model
  • Roots
  • Contour integrals
  • Bulk-service queue

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