Tightness of invariant distributions of a large-scale flexible service system under a priority discipline

We consider large-scale service systems 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, we show that (a) the system under LAP discipline is stochastically stable for all sufficiently large r and (b) the family of the invariant distributions is tight on scales r1/2 + ε for all ε > 0. (More precisely, the sequence of invariant distributions, centered at the equilibrium point and scaled down by r (1/2 + ε), is tight.)