Fluctuations around the mean-field for a large scale Erlang Loss system under SQ(d) load balancing
31th International Teletraffic Congress (ITC 31), Budapest, Hungary, (2019)Аннотация
In this paper, we study the fluctuations of the transient and stationary empirical distributions around the mean-field for a large scale multi-server Erlang Loss system that has $N$ servers. Jobs arrive according to a Poisson process with rate $Nłambda$ 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-field limit of this model and characterized the asymptotic behavior of the system when $N\toınfty$. 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-field 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-field limit when $N\toınfty$. We show that for both the transient and stationary regimes, the limiting process is characterized by an Ornstein-Uhlenbeck (OU) process. We also show that the interchange of limits $łim_N\toınftyłim_t\toınfty=łim_t\toınftyłim_N\toınfty$ 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 fixed-point of the mean-field, is of the order $o(N^-\frac12)$ and thus establish the accuracy of the mean-field approximation for finite $N$.