Moderate Deviation Principle for Join-The-Shortest-Queue-d Systems

TL;DR

This work analyzes moderate deviation principles for the Join-the-Shortest-Queue($d$) (JSQ($d$)) policy in large parallel-server systems. It develops an infinite-dimensional LLN framework using Poisson random measures and derives MDPs for the occupancy process $Q^n$ (and its derived quantities), with rate functions expressed variationally as controlled deterministic trajectories. A buffered variant JSQ($d$) with buffer size $K$ is treated, yielding an MDP with a truncated state space and a corresponding rate function, and a case study examines the convergence of these rate functions as $K\to\infty$, linking finite-buffer approximations to the no-buffer limit. The results provide precise, operator-based characterizations of deviations from the mean-field limit, offering insight into the tail behavior of queue-length distributions in large-scale load-balancing networks and informing design choices in distributed systems where communication overhead and capacity constraints are present.

Abstract

The Join-the-Shortest-Queue-d routing policy is considered for a large system with $n$ servers. Moderate deviation principles (MDP) for the occupancy process and the empirical queue length process are established as $n\to \infty$. Each MDP is formulated in terms of a large deviation principle with an appropriate speed function in a suitable infinite-dimensional path space. Proofs rely on certain variational representations for exponential functionals of Poisson random measures. As a case study, the convergence of rate functions for systems with finite buffer size $K$ to the rate function without buffer is analyzed, as $K \to \infty$.

Theorems & Definitions (43)

• Lemma 2.1
• proof
• Proposition 2.1: LLN
• proof
• Theorem 3.1: MDP without buffer
• Theorem 3.2
• proof
• Corollary 3.1
• Remark 3.1
• Lemma 4.1