Tightness of Stationary Distributions of a Flexible-Server System in the Halfin-Whitt Asymptotic Regime

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 $O\left(\sqrt{n}\right)$.

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, $G^{\left(n\right)}\left(x\right)={\int }_0^{\infty }g\left(y^{\left(n\right)}\left(t\right)\right)dt$, where y⁽ⁿ⁾(·) is the DFL starting at x, and g(·) is a “distance” to the origin. (g(·) 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.