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Uniform stability of some large-scale parallel server networks

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Abstract

In this paper we study the uniform stability properties of two classes of parallel server networks with multiple classes of jobs and multiple server pools of a tree topology. These include a class of networks with a single nonleaf server pool, such as the ‘N’ and ‘M’ models, and networks of any tree topology with class-dependent service rates. We show that with \(\sqrt{n}\) safety staffing, and no abandonment, in the Halfin–Whitt regime, the diffusion-scaled controlled queueing processes are exponentially ergodic and their invariant probability distributions are tight, uniformly over all stationary Markov controls. We use a unified approach in which the same Lyapunov function is used in the study of the prelimit and diffusion limit. A parameter called the spare capacity (safety staffing) of the network plays a central role in characterizing the stability results: the parameter being positive is necessary and sufficient that the limiting diffusion is uniformly exponentially ergodic over all stationary Markov controls. We introduce the concept of “system-wide work conserving policies," which are defined as policies that minimize the number of idle servers at all times. This is stronger than the so-called joint work conservation. We show that, provided the spare capacity parameter is positive, the diffusion-scaled processes are geometrically ergodic and the invariant distributions are tight, uniformly over all “system-wide work conserving policies." In addition, when the spare capacity is negative we show that the diffusion-scaled processes are transient under any stationary Markov control, and when it is zero, they cannot be positive recurrent.

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Acknowledgements

This work is supported in part by the Army Research Office through Grant W911NF-17-1-0019, and in part by National Science Foundation Grants DMS-1715210, DMS-1715875 and DMS-2216765, and in part by the Office of Naval Research through Grant N00014-16-1-2956 and was approved for public release under DCN #43-5454-19.

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX, 78712, USA

    Hassan Hmedi & Ari Arapostathis

  2. Department of Computational Applied Mathematics and Operations Research, George R. Brown College of Engineering, Rice University, Houston, TX, 77005, USA

    Guodong Pang

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  1. Hassan Hmedi
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  2. Ari Arapostathis
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  3. Guodong Pang
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Correspondence to Guodong Pang.

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Hmedi, H., Arapostathis, A. & Pang, G. Uniform stability of some large-scale parallel server networks. Queueing Syst 102, 509–552 (2022). https://doi.org/10.1007/s11134-022-09850-y

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  • DOI: https://doi.org/10.1007/s11134-022-09850-y

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