Queues on a Dynamically Evolving Graph

Michel Mandjes, Nicos J. Starreveld*, René Bekker

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

This paper considers a population process on a dynamically evolving graph, which can be alternatively interpreted as a queueing network. The queues are of infinite-server type, entailing that at each node all customers present are served in parallel. The links that connect the queues have the special feature that they are unreliable, in the sense that their status alternates between ‘up’ and ‘down’. If a link between two nodes is down, with a fixed probability each of the clients attempting to use that link is lost; otherwise the client remains at the origin node and reattempts using the link (and jumps to the destination node when it finds the link restored). For these networks we present the following results: (a) a system of coupled partial differential equations that describes the joint probability generating function corresponding to the queues’ time-dependent behavior (and a system of ordinary differential equations for its stationary counterpart), (b) an algorithm to evaluate the (time-dependent and stationary) moments, and procedures to compute user-perceived performance measures which facilitate the quantification of the impact of the links’ outages, (c) a diffusion limit for the joint queue length process. We include explicit results for a series relevant special cases, such as tandem networks and symmetric fully connected networks.

Original languageEnglish
Pages (from-to)1-25
Number of pages25
JournalJournal of Statistical Physics
Volume2018
Issue number3-4
DOIs
Publication statusPublished - 24 Apr 2018

Funding

The research for this paper is partly funded by the NWO Gravitation Programme Networks, Grant Number 024.002.003 (Mandjes, Starreveld), and an NWO Top Grant, Grant Number 613.001.352 (Mandjes). The authors thank Melike Baykal-G?rsoy (Rutgers University, USA), Dieter Fiems (University of Ghent, Belgium), Brendan Patch (The University of Queensland, Australia & University of Amsterdam, the Netherlands), and Peter Taylor (The University of Melbourne, Australia) for helpful comments and discussions. The research for this paper is partly funded by the NWO Gravitation Programme Networks , Grant Number 024.002.003 (Mandjes, Starreveld), and an NWO Top Grant, Grant Number 613.001.352 (Mandjes).

FundersFunder number
Australia & University of Amsterdam
Rutgers University
University of Queensland
Nederlandse Organisatie voor Wetenschappelijk Onderzoek024.002.003, 613.001.352

    Keywords

    • Infinite-server systems
    • Link failures
    • Queueing networks
    • Randomly evolving graphs

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