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

A. Oblakova*, A. Al Hanbali, R. J. Boucherie, J. C.W. van Ommeren, W. H.M. Zijm

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


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
StatePublished - 14 Aug 2019

Bibliographical note

Funding Information:
This paper is part of the project Dynafloat which was supported by Topconsortia voor Kennis en Innovatie Logistiek and Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant Number 438-13-206].

Publisher Copyright:
© 2019, The Author(s).


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

ASJC Scopus subject areas

  • Statistics and Probability
  • Computer Science Applications
  • Management Science and Operations Research
  • Computational Theory and Mathematics


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