Diffusion scale tightness of invariant distributions of a large-scale flexible service system

In this paper we consider the model studied in [6]. Namely, 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 $r$, it was shown in [6] that the family of the invariant distributions is tight on scales $r^{1/2+epsilon}$ for any $epsilon > 0$. Namely, the sequence of invariant distributions, centered at the equilibrium point and scaled down by $r^{-(1/2+epsilon)}$, is tight. In this paper we prove a stronger result: the invariant distributions are, in fact, tight on the diffusion, i.e. $r^{1/2}$, scale. (This is the strongest possible tightness property for the model and the asymptotic regime in this paper.) As a consequence, we obtain the "limit interchange" result: the limit of diffusion-scaled invariant distributions is equal to the invariant distribution of the limiting diffusion process.