Sensitivity of Mean-Field Fluctuations in Erlang loss models with randomized routing

TL;DR

This paper analyzes the fluctuations in large-scale Erlang loss systems with randomized routing, deriving central limit theorems and Ornstein-Uhlenbeck process limits to improve blocking probability approximations.

Contribution

It establishes functional central limit theorems for mean-field fluctuations in Erlang loss models with JSQ(d) routing, providing new insights into their stochastic behavior.

Findings

Limit process is an Ornstein-Uhlenbeck process.

Derived accurate blocking probability approximations.

Quantified the impact of system parameters on fluctuations.

Abstract

In this paper, we study a large system of $N$ servers each with capacity to process at most $C$ simultaneous jobs and an incoming job is routed to a server if it has the lowest occupancy amongst $d$ (out of N) randomly selected servers. A job that is routed to a server with no vacancy is assumed to be blocked and lost. Such randomized policies are referred to JSQ(d) (Join the Shortest Queue out of $d$) policies. Under the assumption that jobs arrive according to a Poisson process with rate $N\lambda^{(N)}$ where $\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}}$, $\sigma\in\mb{R}_+$ and $\beta\in\mb{R}$, we establish functional central limit theorems (FCLTs) for the fluctuation process in both the transient and stationary regimes when service time distributions are exponential. In particular, we show that the limit is an Ornstein-Uhlenbeck process whose mean and variance depend on the mean-field of the considered model. Using this, we obtain approximations to the blocking probabilities for large $N$, where we can precisely estimate the accuracy of first-order approximations.