Stochastic Systems DIFFUSION SCALE TIGHTNESS OF INVARIANT DISTRIBUTIONS OF A LARGE-SCALE FLEXIBLE SERVICE SYSTEM

In this paper we consider the model studied in [17] – a large-scale service system with multiple customer classes and multiple server pools; interarrival and service times are exponentially distributed, and mean service times depend both on the customer class and server pool. It is assumed that the allowed activities (routing choices) form a tree (in the graph with vertices being both customer classes and server pools). We study the behavior of the system under a Leaf Activity Priority (LAP) policy, which assigns static priorities to the activities in the order of sequential “elimination” of the tree leaves. We consider the scaling limit of the system as the arrival rate of customers and number of servers in each pool tend to infinity in proportion to a scaling parameter r, while the overall system load remains strictly subcritical. Indexing the systems by parameter r, it was shown in [17] that the family of the invariant distributions is tight on scales r for any > 0. Namely, the sequence of invariant distributions, centered at the equilibrium point and scaled down by r−(1/2+ , is tight. In this paper we prove a stronger result: the invariant distributions are, in fact, tight on the diffusion, i.e. r, scale. (This is the strongest possible tightness property for the model and the asymptotic regime in this paper.) As a consequence, we obtain a limit interchange result: the limit of diffusion-scaled invariant distributions is equal to the invariant distribution of the limiting diffusion process.