## Abstract

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 language | English |
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Pages (from-to) | 257-292 |

Number of pages | 36 |

Journal | Queueing Systems |

Volume | 92 |

Issue number | 3-4 |

DOIs | |

State | Published - 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).

## Keywords

- 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