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

摘要

We consider large-scale service systems with multiple customer classes andmultiple server pools; interarrival and service times are exponentiallydistributed, and mean service times depend both on the customer class andserver pool. It is assumed that the allowed activities (routing choices) form atree (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 ofsequential "elimination" of the tree leaves. We consider the scaling limit of the system as the arrival rate of customersand number of servers in each pool tend to infinity in proportion to a scalingparameter r, while the overall system load remains strictly subcritical.Indexing the systems by parameter r, we show that (a) the system under LAPdiscipline is stochastically stable for all sufficiently large r and (b) thefamily of the invariant distributions is tight on scales $r^{1/2 + \epsilon}$for all $\epsilon > 0$. (More precisely, the sequence of invariantdistributions, centered at the equilibrium point and scaled down by $r^{-(1/2 +\epsilon)}$, is tight.)