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2015 Tightness of stationary distributions of a flexible-server system in the Halfin-Whitt asymptotic regime
Alexander L. Stolyar
Stoch. Syst. 5(2): 239-267 (2015). DOI: 10.1214/14-SSY139

Abstract

A large-scale flexible service system with two large server pools and two types of customers is considered. Servers in pool 1 can only serve type 1 customers, while server in pool 2 are flexible – they can serve both types 1 and 2. (This is a so-called “N-system.”) The customer arrival processes are Poisson and customer service requirements are independent exponentially distributed. The service rate of a customer depends both on its type and the pool where it is served. A priority service discipline, where type 2 has priority in pool 2, and type 1 prefers pool 1, is considered. We consider the Halfin-Whitt asymptotic regime, where the arrival rate of customers and the number of servers in each pool increase to infinity in proportion to a scaling parameter $n$, while the overall system capacity exceeds its load by $O(\sqrt{n})$.

For this system we prove tightness of diffusion-scaled stationary distributions. Our approach relies on a single common Lyapunov function $G^{(n)}(x)$, depending on parameter $n$ and defined on the entire state space as a functional of the drift-based fluid limits (DFL). Specifically, $G^{(n)}(x)=\int_{0}^{\infty}g(y^{(n)}(t))dt$, where $y^{(n)}(\cdot)$ is the DFL starting at $x$, and $g(\cdot)$ is a “distance” to the origin. ($g(\cdot)$ is same for all $n$). The key part of the analysis is the study of the (first and second) derivatives of the DFLs and function $G^{(n)}(x)$. The approach, as well as many parts of the analysis, are quite generic and may be of independent interest.

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Alexander L. Stolyar. "Tightness of stationary distributions of a flexible-server system in the Halfin-Whitt asymptotic regime." Stoch. Syst. 5 (2) 239 - 267, 2015. https://doi.org/10.1214/14-SSY139

Information

Received: 1 March 2014; Published: 2015
First available in Project Euclid: 23 December 2015

zbMATH: 1331.60180
MathSciNet: MR3442428
Digital Object Identifier: 10.1214/14-SSY139

Subjects:
Primary: 60F17, 60K25

Rights: Copyright © 2015 INFORMS Applied Probability Society

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Vol.5 • No. 2 • 2015
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