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Diffusion parameters of flows in stable multi-class queueing networks

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Abstract

We consider open multi-class queueing networks with general arrival processes, general processing time sequences and Bernoulli routing. The network is assumed to be operating under an arbitrary work-conserving scheduling policy that makes the system stable. We study the variability of flows within the network. Computable expressions for quantifying flow variability have previously been discussed in the literature. However, in this paper, we shed more light on such analysis to justify the use of these expressions in the asymptotic analysis of network flows. Toward that end, we find a simple diffusion limit for the inter-class flows and establish the relation to asymptotic (co-)variance rates.

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Notes

  1. The simulation was carried out using a simulation package written in C++: PRONETSIM. See [32], Appendix A, for details about this software.

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Acknowledgements

YN is supported by Australian Research Council (ARC) grant DP180101602. Part of the work was carried out, while WS was supported by an Ethel Raybould Visiting Fellowship to the University of Queensland. SBM is supported by the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS) under grant number CE140100049. We thank an anonymous referee of this paper for their valuable comments and suggestions, particularly for highlighting the need for Assumption (A7). We also thank an anonymous referee of an earlier version of the paper for drawing our attention to the usefulness of the innovations method calculations.

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Authors and Affiliations

  1. School of Mathematical and Physical Sciences, Macquarie University, Sydney, Australia

    Sarat Babu Moka

  2. School of Mathematics and Physics, The University of Queensland, Brisbane, Australia

    Yoni Nazarathy

  3. Department of Applied Mathematics, University of Twente, Enschede, The Netherlands

    Werner Scheinhardt

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  1. Sarat Babu Moka
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  2. Yoni Nazarathy
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Correspondence to Sarat Babu Moka.

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Moka, S.B., Nazarathy, Y. & Scheinhardt, W. Diffusion parameters of flows in stable multi-class queueing networks. Queueing Syst 103, 313–346 (2023). https://doi.org/10.1007/s11134-022-09869-1

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