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Sample Path Moderate Deviation Principle for Queues with Waiting-time Dependent Interarrival and Service Times

Chang Feng, John J. Hasenbein, Guodong Pang

TL;DR

This paper analyzes sample-path moderate deviations for a single-server queue with waiting-time dependent interarrival and service times, modeled by a Lindley-type recursion with random weights $C_i^n$ and increments $X_i^n$. By embedding the model in a linearly recursive Markov framework with $C_i^n=1- rac{1}{n} heta_i$, the authors develop aMDP analysis via linear SDE representations driven by two independent random walks, exponential tightness, and the contraction principle. The fluid analysis reveals regimes with stable fixed points at either $0$ or $ rac{ m mu}{ heta}$, which in turn yield different MD rate functions; explicit rate functions are derived for the nonzero-centering and zero-centering cases. The results connect MD principles to the underlying Skorokhod-type reflections and provide explicit rate-function formulas, enriching the understanding of workload fluctuations in state-dependent queueing systems with random perturbations and reflections.

Abstract

We consider a single-server queue where interarrival and service times depend linearly and randomly on customer waiting times, and establish a sample-path moderate deviation principle (MDP) for the waiting time process. The waiting times for the queue can be written as a modified Lindley recursion with a random weight coefficient. Under a natural scaling of the random coefficients, we analyze the fluid behavior of the workload process and derive the stable equilibrium point, which can be zero or a positive value. The moderate-deviation-scaled process is centered around the stable equilibrium point and then represented as a linear stochastic differential equation driven by two random walks together with additional asymptotically negligible error terms and possibly a reflection at zero. The rate functions of MDPs in the two scenarios can be characterized explicitly, and they differ in that the case with zero centering term involves the linearly generalized Skorokhod reflection mapping while the case with positive centering term does not (similar to the corresponding diffusion limits). Our analysis involves the MDP for the associated linearly recursive Markov chains, invoking a perturbation of two independent random walks, and employing martingale techniques to prove the asymptotically exponentially vanishing error terms.

Sample Path Moderate Deviation Principle for Queues with Waiting-time Dependent Interarrival and Service Times

TL;DR

This paper analyzes sample-path moderate deviations for a single-server queue with waiting-time dependent interarrival and service times, modeled by a Lindley-type recursion with random weights and increments . By embedding the model in a linearly recursive Markov framework with , the authors develop aMDP analysis via linear SDE representations driven by two independent random walks, exponential tightness, and the contraction principle. The fluid analysis reveals regimes with stable fixed points at either or , which in turn yield different MD rate functions; explicit rate functions are derived for the nonzero-centering and zero-centering cases. The results connect MD principles to the underlying Skorokhod-type reflections and provide explicit rate-function formulas, enriching the understanding of workload fluctuations in state-dependent queueing systems with random perturbations and reflections.

Abstract

We consider a single-server queue where interarrival and service times depend linearly and randomly on customer waiting times, and establish a sample-path moderate deviation principle (MDP) for the waiting time process. The waiting times for the queue can be written as a modified Lindley recursion with a random weight coefficient. Under a natural scaling of the random coefficients, we analyze the fluid behavior of the workload process and derive the stable equilibrium point, which can be zero or a positive value. The moderate-deviation-scaled process is centered around the stable equilibrium point and then represented as a linear stochastic differential equation driven by two random walks together with additional asymptotically negligible error terms and possibly a reflection at zero. The rate functions of MDPs in the two scenarios can be characterized explicitly, and they differ in that the case with zero centering term involves the linearly generalized Skorokhod reflection mapping while the case with positive centering term does not (similar to the corresponding diffusion limits). Our analysis involves the MDP for the associated linearly recursive Markov chains, invoking a perturbation of two independent random walks, and employing martingale techniques to prove the asymptotically exponentially vanishing error terms.

Paper Structure

This paper contains 27 sections, 30 theorems, 200 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Preliminaries and notations
  4. Model Formulation and Main Results
  5. The model
  6. Moderate Deviation Results
  7. Fluid Analysis
  8. Fluid Limit
  9. Stability of the fluid limit equation
  10. Sample-path MDP for a Linearly Recursive Markov System
  11. Fluid analysis
  12. MDP results
  13. Exponential Tightness
  14. Proofs for MDP Results in Section \ref{['subsec:V_MDP_results']}
  15. Proofs for MDP Results in Section \ref{['subsec:MD_results']}
  16. ...and 12 more sections

Key Result

Theorem 2.4

Under Assumptions assump:model (i), (iii) and assump:MDP_W (ii), the $\mathcal{D}_T$-valued families of processes $\{\widetilde{R}_{X}^{n},\ n \in \mathbb{N}\}$ and $\{\widetilde{R}_{\Theta}^{n},\ n \in \mathbb{N} \}$ respectively satisfies an MDP in $\mathcal{D}_T$ with rate $b_n^{2}$ and rate func

Figures (3)

  • Figure 1: Fluid limits for overloaded systems ($\mu > 0$).
  • Figure 2: Fluid limits for critically loaded systems ($\mu = 0$).
  • Figure 3: Fluid limits for underloaded systems ($\mu < 0$).

Theorems & Definitions (53)

  • Remark 2.2
  • Theorem 2.4: MDP for Random Walks
  • Theorem 2.5
  • Theorem 2.6
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.3
  • ...and 43 more