On System-Wide Safety Staffing of Large-Scale Parallel Server Networks

Hassan Hmedi (), Ari Arapostathis () and Guodong Pang ()
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Hassan Hmedi: Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, Texas 78712
Ari Arapostathis: Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, Texas 78712
Guodong Pang: Department of Computational and Applied Mathematics, George R. Brown College of Engineering, Rice University, Houston, Texas, 77005

Operations Research, 2023, vol. 71, issue 2, 415-432

Abstract: We introduce a “system-wide safety staffing” (SWSS) parameter for multiclass multipool networks of any tree topology, Markovian or non-Markovian, in the Halfin-Whitt regime. This parameter can be regarded as the optimal reallocation of the capacity fluctuations (positive or negative) of order n when each server pool uses a square-root staffing rule. We provide an explicit form of the SWSS as a function of the system parameters, which is derived using a graph theoretic approach based on Gaussian elimination. For Markovian networks, we give an equivalent characterization of the SWSS parameter via the drift parameters of the limiting diffusion. We show that if the SWSS parameter is negative, the limiting diffusion and the diffusion-scaled queueing processes are transient under any Markov control and cannot have a stationary distribution when this parameter is zero. If it is positive, we show that the diffusion-scaled queueing processes are uniformly stabilizable ; that is, there exists a scheduling policy under which the stationary distributions of the controlled processes are tight over the size of the network. In addition, there exists a control under which the limiting controlled diffusion is exponentially ergodic. Thus, we identified a necessary and sufficient condition for the uniform stabilizability of such networks in the Halfin-Whitt regime. We use a constant control resulting from the leaf elimination algorithm to stabilize the limiting controlled diffusion while a family of Markov scheduling policies that are easy to compute are used to stabilize the diffusion-scaled processes. Finally, we show that under these controls the processes are exponentially ergodic and the stationary distributions have exponential tails.

Keywords: Stochastic Models; parallel server networks; Halfin-Whitt regime; system-wide safety staffing; uniform stabilizability (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:inm:oropre:v:71:y:2023:i:2:p:415-432

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