Join the shortest queue: stability and exact asymptotics

Abstract

We consider the stability of a network serving a patchwork of overlapping regions where customers from a local region are assigned to a collection of local servers.These customers join the queue of the local server with the shortest queue of waiting customers.We then describe how the backlog in the network overloads.We do this in the simple case of two servers each of which receives a dedicated stream of customers in addition to customers from a stream of smart customers who join the shorter queue. There are three distinct ways the backlog can overload. If one server is very fast, then that server takes all the smart customers along with its dedicated customers and keeps its queue small while the dedicated customers at the other server cause the overload.We call this the unpooled case. If the proportion of smart customers is large, then the two servers overload in tandem.We call this the strongly pooled case. Finally, there is the weakly pooled case where both queues overload but in different proportions. The fact that strong pooling can be attained based on a local protocol for overlapping regions may have engineering significance. In addition, this paper extends the methodology developed in McDonald (to appear The Annals of Applied Probability) to cover periodicities. The emphasis here is on sharp asymptotics, not rough asymptotics as in large deviation theory. Moreover, the limiting distributions are for the unscaled process, not for the fluid limit as in large deviation theory. In the strongly pooled case, for instance, we give the limiting distribution of the difference between the two queues as the backlog grows.We also give the exact asymptotics of the mean time until overload.