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Asymptotic Optimality of the Speed-Aware Join-the-Shortest-Queue in the Halfin-Whitt Regime for Heterogeneous Systems

Sanidhay Bhambay, Burak Büke, Arpan Mukhopadhyay

TL;DR

This paper shows that the SA-JSQ scheme is also asymptotically optimal for heterogeneous systems in the Halfin-Whitt traffic regime where the normalised arrival rate scales as $1-O(1/\sqrt{n})$.

Abstract

The Join-the-Shortest-Queue (JSQ) load balancing scheme is known to minimise the average response time of jobs in homogeneous systems with identical servers. However, for {\em heterogeneous} systems with servers having different processing speeds, finding an optimal load balancing scheme remains an open problem for finite system sizes. Recently, for systems with heterogeneous servers, a variant of the JSQ scheme, called the {\em Speed-Aware-Join-the-Shortest-Queue (SA-JSQ)} scheme, has been shown to achieve asymptotic optimality in the fluid-scaling regime where the number of servers $n$ tends to infinity but the normalised the arrival rate of jobs remains constant. {In this paper, we show that the SA-JSQ scheme is also asymptotically optimal for heterogeneous systems in the {\em Halfin-Whitt} traffic regime where the normalised arrival rate scales as $1-O(1/\sqrt{n})$.} Our analysis begins by establishing that an appropriately scaled and centered version of the Markov process describing system dynamics weakly converges to a two-dimensional reflected {\em Ornstein-Uhlenbeck (OU) process}. We then show using {\em Stein's method} that the stationary distribution of the underlying Markov process converges to that of the OU process as the system size increases by establishing the validity of interchange of limits. {Finally, through coupling with a suitably constructed system, we show that SA-JSQ asymptotically minimises the diffusion-scaled total number of jobs and the diffusion-scaled number of waiting jobs in the steady-state in the Halfin-Whitt regime among all policies which dispatch jobs based on queue lengths and server speeds.}

Asymptotic Optimality of the Speed-Aware Join-the-Shortest-Queue in the Halfin-Whitt Regime for Heterogeneous Systems

TL;DR

This paper shows that the SA-JSQ scheme is also asymptotically optimal for heterogeneous systems in the Halfin-Whitt traffic regime where the normalised arrival rate scales as .

Abstract

The Join-the-Shortest-Queue (JSQ) load balancing scheme is known to minimise the average response time of jobs in homogeneous systems with identical servers. However, for {\em heterogeneous} systems with servers having different processing speeds, finding an optimal load balancing scheme remains an open problem for finite system sizes. Recently, for systems with heterogeneous servers, a variant of the JSQ scheme, called the {\em Speed-Aware-Join-the-Shortest-Queue (SA-JSQ)} scheme, has been shown to achieve asymptotic optimality in the fluid-scaling regime where the number of servers tends to infinity but the normalised the arrival rate of jobs remains constant. {In this paper, we show that the SA-JSQ scheme is also asymptotically optimal for heterogeneous systems in the {\em Halfin-Whitt} traffic regime where the normalised arrival rate scales as .} Our analysis begins by establishing that an appropriately scaled and centered version of the Markov process describing system dynamics weakly converges to a two-dimensional reflected {\em Ornstein-Uhlenbeck (OU) process}. We then show using {\em Stein's method} that the stationary distribution of the underlying Markov process converges to that of the OU process as the system size increases by establishing the validity of interchange of limits. {Finally, through coupling with a suitably constructed system, we show that SA-JSQ asymptotically minimises the diffusion-scaled total number of jobs and the diffusion-scaled number of waiting jobs in the steady-state in the Halfin-Whitt regime among all policies which dispatch jobs based on queue lengths and server speeds.}
Paper Structure (23 sections, 24 theorems, 263 equations, 4 figures)

This paper contains 23 sections, 24 theorems, 263 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Literature
  4. Notation
  5. System Model
  6. Main Results
  7. State Space Collapse: Proof of Proposition \ref{['prop:state_space_collapse']}
  8. Transient Analysis: Proof of Theorem \ref{['thm:SA-JSQ_diffusion_limit']}
  9. Martingale Convergence
  10. Stationary Analysis: Proof of Theorem \ref{['thm:stationary_convergence']}
  11. Construction of the Lyapunov Function for Lemma \ref{['lem:solution_PDE']}
  12. Asymptotic optimality of the SA-JSQ scheme: Proof of Theorem \ref{['thm:sajsq_optimality']}
  13. A Modified System with Free Servers and Blocking
  14. A Coupling Construction: Proof of Proposition \ref{['prop:coupling_modf']}
  15. Diffusion Limit of the Modified System: Proof of Proposition \ref{['prop:modified_diffusion_limit']}
  16. ...and 8 more sections

Key Result

Proposition 3.1

Suppose that eqn:heavy_traffic_limit and eq:proportion_assumption hold and $Y^{(n)}_{[1,M-1],1}(0) \Rightarrow Y_{[1,M-1],1}(0)$, where $Y_{[1,M-1],1}(0)$ is a proper random variable Then for any $\epsilon>0$, $t_0>0$ and $T\geq t_0$ we have Moreover, if $Y^{(n)}_{[1,M-1],1}(0)\Rightarrow 0$, then eqn:ssc_1 also holds for $t_0=0$.

Figures (4)

  • Figure 3.1: Sample paths for $Y_{1,1}^{(n)}$ for varying $n$ with $M=2$, $\gamma_1=1/5$, $\gamma_2=4/5$, $\mu_1=20/8$, $\mu_2=5/8$ and $\beta=2$.
  • Figure 3.2: Sample paths for $Y_{j,i}^{(n)}$ with $n=700$, $M=2$, $\gamma_1=1/5$, $\gamma_2=4/5$, $\mu_1=20/8$, $\mu_2=5/8$ and $\beta=2$.
  • Figure 6.1: Any trajectory that begins below the dashed curve representing the curve $\Gamma^{\beta/\mu_1}$ will not intersect the vertical axis, while trajectories starting above the curve will eventually reach the axis and move downward until arriving at the point $(0, \beta/\mu_1\sqrt{n})$.
  • Figure 7.1: A modified system with 2 type 1 servers, two type 2 servers and one type 3 server.

Theorems & Definitions (27)

  • Definition 2.1
  • Proposition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Remark 3.4
  • Theorem 3.5
  • Corollary 3.6
  • Theorem 3.7
  • Lemma 4.1: Lemma 5.8 in Pang2007
  • Proposition 5.1
  • ...and 17 more