Fluctuations Around the Mean-Field for a Large Scale Erlang Loss System Under the SQ(d) Load Balancing

Abstract

In this paper, we study the fluctuations of the transient and stationary empirical distributions around the meanheld for a large scale multi-server Erlang Loss system that has N servers. Jobs arrive according to a Poisson process with rate Nλ and each incoming job is dispatched by a central job dispatcher to the server with the minimum occupancy among d randomly chosen servers with ties broken uniformly at random. Previous works have studied the mean-held limit of this model and characterized the asymptotic behavior of the system when N → ∞. In this paper, we focus on quantifying the resulting error when we approximate the transient and stationary behavior of the system when N is large by the mean-held of the system. We obtain functional central limit theorems (FCLTs) by studying the limit of a suitably scaled fluctuation process of the stochastic empirical process of the model with index N around the mean-held limit when N → ∞. We show that for both the transient and stationary regimes, the limiting process is characterized by an OrnsteinUhlenbeck (OU) process. We also show that the interchange of limits lim <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N→∞</sub> lim <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t→∞</sub> = lim <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t→∞</sub> lim <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N→∞</sub> is valid under the CLT scaling. Finally, we exploit the FCLT to show that the gap between the exact average blocking probability of a job in the system with the number of servers N and the limiting average blocking probability which is a function of the hxed-point of the mean-held, is of the order o(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> ) and thus establish the accuracy of the mean-held approximation for hnite N.