Uniform stability of some large-scale parallel server networks

Hassan Hmedi (), Ari Arapostathis () and Guodong Pang ()
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Hassan Hmedi: The University of Texas at Austin
Ari Arapostathis: The University of Texas at Austin
Guodong Pang: Rice University

Queueing Systems: Theory and Applications, 2022, vol. 102, issue 3, No 5, 509-552

Abstract: 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 $$\sqrt{n}$$ 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.

Keywords: Uniform exponential ergodicity; Parallel server (multiclass multi-pool) networks; Halfin–Whitt regime; Spare capacity; System-wide work conservation; Primary 90B22; Secondary 60K25; 49L20; 90B36 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11134-022-09850-y

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