We consider a large-scale flexible service system with two large server pools and two types of customers. 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." Our results hold for a more general class of systems as well.) The service rate of a customer depends both on its type and the pool where it is served. We study a priority service discipline, where type 2 has priority in pool 2, and type 1 prefers pool 1. 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( √ n). For this system we prove tightness of diffusion-scaled stationary distributions. Our approach relies on a single common Lyapunov function G(x), defined on the entire state space as a functional of the drift-based fluid limits (DFL). Specifically, G(x) = ∞ 0 g(y(t))dt, where y(·) is the DFL starting at x, and g(·) is a "distance" to the origin. The key part of the analysis is the study of the (first and second) derivatives of the DFLs and function G(x). Our approach, as well as many parts of the analysis, seem quite generic and may be of independent interest. 1. Introduction. In this paper we consider a large-scale service system in the so-called HalfinWhitt asymptotic regime. Such systems received a lot of attention in the literature, especially in the past 10-15 year, because they find a variety of applications, including, e.g., large customer contact centers [1, 9] and large computer farms in network clouds. The Halfin-Whitt regime, introduced originally in [12] , is such that the system capacity (roughly, number of servers) increases in proportion to a scaling parameter n, and exceeds the system load by O( √ n). It is attractive because it allows -in principle, under a good control algorithm -to achieve both good performance (e.g. waiting times) and high resource utilization.
We also note that the same method has been used in REF to prove the tightness of stationary distributions of a flexible-server system in the Halfin-Whitt regime, which proves the interchange of the limits and justifies the diffusion approximation.
Tightness of stationary distributions of a flexible-server system in the Halfin-Whitt asymptotic regime
arXiv: Probability
Mathematics