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Moderate deviations of many-server queues in the Halfin-Whitt regime and weak convergence methods

Anatolii Puhalskii

TL;DR

The paper develops logarithmic (moderate) deviation asymptotics for the number-in-system process $Q_n(t)$ in many-server queues under the Halfin–Whitt regime with general interarrival and service times. It proves a Large Deviation Principle with rate $b_n^2$ for the scaled process $X_n(t)=\frac{\sqrt{n}}{b_n}(\frac{Q_n(t)}{n}-1)$, and expresses the deviation function via a Fredholm equation of the second kind, using an idempotent-process weak-convergence framework and exponential tightness. The trajectory-level results include a variational rate function $I^Q(q)$ and a limit idempotent process $X$ driven by Brownian-bridge and Kiefer idempotent components, with explicit integral equations; special cases with renewal arrivals yield a Wiener-idempotent limit. The work provides a practical route to compute moderate-deviation exponents and demonstrates the utility of weak-convergence methods for LDP proofs in complex queueing systems. The findings advance understanding of probability tails for performance metrics in many-server queues, with potential applications to staffing and reliability analyses.

Abstract

This paper obtains logarithmic asymptotics of moderate deviations of the stochastic process of the number of customers in a many--server queue with generally distributed interarrival and service times in the Halfin--Whitt heavy traffic regime. The deviation function is expressed in terms of the solution to a Fredholm equation of the second kind. The proof uses characterisation of large deviation relatively compact sequences of probability measures as exponentially tight ones.

Moderate deviations of many-server queues in the Halfin-Whitt regime and weak convergence methods

TL;DR

The paper develops logarithmic (moderate) deviation asymptotics for the number-in-system process in many-server queues under the Halfin–Whitt regime with general interarrival and service times. It proves a Large Deviation Principle with rate for the scaled process , and expresses the deviation function via a Fredholm equation of the second kind, using an idempotent-process weak-convergence framework and exponential tightness. The trajectory-level results include a variational rate function and a limit idempotent process driven by Brownian-bridge and Kiefer idempotent components, with explicit integral equations; special cases with renewal arrivals yield a Wiener-idempotent limit. The work provides a practical route to compute moderate-deviation exponents and demonstrates the utility of weak-convergence methods for LDP proofs in complex queueing systems. The findings advance understanding of probability tails for performance metrics in many-server queues, with potential applications to staffing and reliability analyses.

Abstract

This paper obtains logarithmic asymptotics of moderate deviations of the stochastic process of the number of customers in a many--server queue with generally distributed interarrival and service times in the Halfin--Whitt heavy traffic regime. The deviation function is expressed in terms of the solution to a Fredholm equation of the second kind. The proof uses characterisation of large deviation relatively compact sequences of probability measures as exponentially tight ones.
Paper Structure (3 sections, 4 theorems, 38 equations)

This paper contains 3 sections, 4 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Trajectorial moderate deviation limit theorems
  3. Proofs of Theorems \ref{['the:ldp']} and \ref{['the:heavy_traffic']}

Key Result

Theorem 2.1

Suppose, in addition, that $A_n$ is a renewal process of rate $\lambda_n$ . Let $\rho_n=\lambda_n/(n\mu)$ , $\beta\in\mathbb R$ , $q_0\in\mathbb R$ and $\sigma>0$ . Suppose that, as $n\to\infty$, and the sequence of random variables $\sqrt{n}/b_n\,(Q_n(0)/n-1)$ obeys the LDP in $\mathbb R$ for rate $b_n^2$ with deviation function $I_{q_0}(y)$ such that $I_{q_0}(q_0)=0$ and $I_{q_0}(y)=\infty$ , f

Theorems & Definitions (6)

  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Lemma 3.1
  • proof