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Steady-State Convergence of the Continuous-Time Routing System with General Distributions in Heavy Traffic

Jin Guang, Yaosheng Xu, J. G. Dai

TL;DR

This work analyzes a continuous-time load-balancing network with a single arrival stream feeding $J$ parallel stations under JSQ or Po2, with general interarrival and service-time distributions. Using the Basic Adjoint Relationship (BAR) framework and a state-space-collapse (SSC) analysis, the authors show that in heavy traffic the scaled queue lengths converge to a common exponential limit with mean $m=\frac{1}{2J}\sum_{j=1}^J \mu_j (c_e^2+c_{s,j}^2)$, under a finite $(2+\delta_0)$th moment condition for some $\delta_0>0$. The methodology hinges on carefully crafted BAR-based test functions and a two-tier induction to bound the perpendicular component of the queue-length vector, establishing SSC and enabling weak convergence results. The results relax prior bounded-support moment assumptions, broadening applicability to continuous-time routing with general distributions and informing heavy-traffic approximations for large-scale load-balancing networks.

Abstract

This paper examines a continuous-time routing system with general interarrival and service time distributions, operating under the join-the-shortest-queue and power-of-two-choices policies. Under a weaker set of assumptions than those commonly found in the literature, we prove that the scaled steady-state queue length at each station converges weakly to an identical exponential random variable in heavy traffic. Specifically, our results hold under the assumption of the $(2 + δ_0)$th moment for the interarrival and service distributions with some $δ_0 > 0$. The proof leverages the Palm version of the basic adjoint relationship (BAR) as a key technique.

Steady-State Convergence of the Continuous-Time Routing System with General Distributions in Heavy Traffic

TL;DR

This work analyzes a continuous-time load-balancing network with a single arrival stream feeding parallel stations under JSQ or Po2, with general interarrival and service-time distributions. Using the Basic Adjoint Relationship (BAR) framework and a state-space-collapse (SSC) analysis, the authors show that in heavy traffic the scaled queue lengths converge to a common exponential limit with mean , under a finite th moment condition for some . The methodology hinges on carefully crafted BAR-based test functions and a two-tier induction to bound the perpendicular component of the queue-length vector, establishing SSC and enabling weak convergence results. The results relax prior bounded-support moment assumptions, broadening applicability to continuous-time routing with general distributions and informing heavy-traffic approximations for large-scale load-balancing networks.

Abstract

This paper examines a continuous-time routing system with general interarrival and service time distributions, operating under the join-the-shortest-queue and power-of-two-choices policies. Under a weaker set of assumptions than those commonly found in the literature, we prove that the scaled steady-state queue length at each station converges weakly to an identical exponential random variable in heavy traffic. Specifically, our results hold under the assumption of the th moment for the interarrival and service distributions with some . The proof leverages the Palm version of the basic adjoint relationship (BAR) as a key technique.
Paper Structure (22 sections, 10 theorems, 111 equations)

This paper contains 22 sections, 10 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Model
  3. Main Results
  4. Basic Adjoint Relationship
  5. Proof of State Space Collapse in Proposition \ref{['prop: ssc']}
  6. Sketch of Proof
  7. Induction hypotheses.
  8. Mathematical induction.
  9. Proof of (S1)
  10. Proof of (S2) to (S4)
  11. Proof of Base Step When $n=0$
  12. Proof of (S1)
  13. Proof of (S2)
  14. Proof of (S3)
  15. Proof of (S4)
  16. ...and 7 more sections

Key Result

Theorem 1

Suppose Assumptions ass: moment condition and ass: stability hold. Under the JSQ or Po2 policy, as $r\to 0$, we have where $Z^*$ is an exponential random variable with mean Here, $c_e^2$ is the squared coefficient of variation (SCV) of the interarrival time, and $c_{s,j}^2$ is the SCV of the service time at station $j$.

Theorems & Definitions (17)

  • Theorem 1
  • Proposition 2: State-Space Collapse
  • Remark 1
  • Remark 2
  • Proposition 3
  • Lemma 4: Lemma 6.3 of BravDaiMiya2023
  • Proposition 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • ...and 7 more