Asymptotics of insensitive load balancing and blocking phases

Abstract We study a single class of traf?c acting on a symmetric set of processorsharing queues with ?nite buffers, and we consider the case where the load scales with the number of servers. We address the problem of giving robust performance bounds based on the study of the asymptotic behaviour of the insensitive load balancing schemes, which have the desirable property that the stationary distribution of the resulting stochastic network depends on the distribution of job-sizes only through its mean.Itwasshownforsmallsystemswithlossesthattheygivegoodestimatesofperformanceindicators,generalizinghenceforthErlangformula,whereasoptimalpolicies are already theoretically and computationally out of reach for networks of moderate size. We characterize the response of symmetric systems under those schemes at different scales and show that three amplitudes of deviations can be identi?ed according towhether?1.Acentrallimitscalingtakesplaceforasub-critical load; for ? =1, the number of free servers scales like n ? ?+1 (? being the buffer depth and n being the number of servers) and is of order 1 for super-critical loads. This further implies the existence of different phases for the blocking probability. Before a (re?ned) critical load ?c(n) = 1 ? an? ? ?+1, the blocking is exponentially small and becomes of order n? ? ?+1 at ?c(n). This generalizes the well-known quality-and