Many-Server Asymptotics for Join-the-Shortest-Queue: Large Deviations and Rare Events

TL;DR

This paper establishes a large deviation principle for the join-the-shortest-queue system with many servers, providing explicit exponential decay rates for rare events like long queues, in a complex infinite-dimensional setting.

Contribution

It introduces a large deviation framework for the join-the-shortest-queue model with many servers, handling technical challenges of infinite-dimensional dynamics and discontinuous statistics.

Findings

Derived explicit exponential decay rates for probabilities of long queues.

Established a large deviation principle in an infinite-dimensional path space.

Analyzed rare events and their decay rates in a many-server queueing system.

Abstract

The Join-the-Shortest-Queue routing policy is studied in an asymptotic regime where the number of processors $n$ scales with the arrival rate. A large deviation principle (LDP) for the occupancy process is established, as $n\to \infty$, in a suitable infinite-dimensional path space. Model features that present technical challenges include, Markovian dynamics with discontinuous statistics, a diminishing rate property of the transition probability rates, and an infinite-dimensional state space. The difficulty is in the proof of the Laplace lower bound which requires establishing the uniqueness of solutions of certain infinite-dimensional systems of controlled ordinary differential equations. The LDP gives information on the rate of decay of probabilities of various types of rare events associated with the system. We illustrate this by establishing explicit exponential decay rates for probabilities of long queues. In particular, denoting by $E_j^n(T)$ the event that there is at least one queue with $j$ or more jobs at some time instant over $[0,T]$, we show that, in the critical case, for large $n$ and $T$, $\mathbb{P}(E^n_j(T)) \approx \exp\left [-\frac{n (j-2)^2}{4T}\right].$