Uniform stability of a class of large-scale parallel server networks

In this paper we show that a large class of parallel server networks, with $\sqrt{n}$ safety staffing, and no abandonment, in the Halfin-Whitt regime are exponentially ergodic and their invariant probability distributions are tight, uniformly over all stationary Markov controls. This class consists of all networks of tree topology with a single non-leaf server pool, such as the 'N' and 'M' models, as well as networks of any tree topology with class-dependent service rates. We first define a parameter which characterizes the spare capacity (safety staffing) of the network. If the spare capacity parameter is negative, we show that the controlled diffusion is transient under any stationary Markov control, and that it cannot be positive recurrent when this parameter is zero. Then we show that the limiting diffusion is uniformly exponentially ergodic over all stationary Markov controls if this parameter is positive. As well known, joint work conservation, that is, keeping all servers busy unless all queues are empty, cannot be always enforced in multiclass multi-pool networks, and as a result the diffusion limit and the prelimit do not "match" on the entire state space. For this reason, 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. 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. We use a unified approach in which the same Lyapunov function is used in the study of the prelimit and diffusion limit.