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Derandomized Load Balancing using Random Walks on Expander Graphs

Dengwang Tang, Vijay G. Subramanian

TL;DR

The paper tackles efficient load balancing in large-scale systems by replacing independent uniform sampling with non-backtracking random walks on high-girth expander graphs. It proves that, under mild spectral and girth assumptions, the NBRW-Po$d$ scheme yields the same fluid-limit ODE as the classical power-of-$d$-choices, with stability and convergence of the stationary distribution to the ODE's fixed point. The authors develop novel probabilistic tools, including precise sampling bounds and a Foster-Lyapunov-type drift condition, to establish both fluid-limit and steady-state results. Simulations corroborate the theoretical findings, showing accurate ODE approximation for tens of thousands of servers and illustrating the impact of graph structure on performance.

Abstract

In a computing center with a huge amount of machines, when a job arrives, a dispatcher need to decide which machine to route this job to based on limited information. A classical method, called the power-of-$d$ choices algorithm is to pick $d$ servers independently at random and dispatch the job to the least loaded server among the $d$ servers. In this paper, we analyze a low-randomness variant of this dispatching scheme, where $d$ queues are sampled through $d$ independent non-backtracking random walks on a $k$-regular graph $G$. Under certain assumptions of the graph $G$ we show that under this scheme, the dynamics of the queuing system converges to the same deterministic ordinary differential equation (ODE) for the power-of-$d$ choices scheme. We also show that the system is stable under the proposed scheme, and the stationary distribution of the system converges to the fixed point of the ODE.

Derandomized Load Balancing using Random Walks on Expander Graphs

TL;DR

The paper tackles efficient load balancing in large-scale systems by replacing independent uniform sampling with non-backtracking random walks on high-girth expander graphs. It proves that, under mild spectral and girth assumptions, the NBRW-Po scheme yields the same fluid-limit ODE as the classical power-of--choices, with stability and convergence of the stationary distribution to the ODE's fixed point. The authors develop novel probabilistic tools, including precise sampling bounds and a Foster-Lyapunov-type drift condition, to establish both fluid-limit and steady-state results. Simulations corroborate the theoretical findings, showing accurate ODE approximation for tens of thousands of servers and illustrating the impact of graph structure on performance.

Abstract

In a computing center with a huge amount of machines, when a job arrives, a dispatcher need to decide which machine to route this job to based on limited information. A classical method, called the power-of- choices algorithm is to pick servers independently at random and dispatch the job to the least loaded server among the servers. In this paper, we analyze a low-randomness variant of this dispatching scheme, where queues are sampled through independent non-backtracking random walks on a -regular graph . Under certain assumptions of the graph we show that under this scheme, the dynamics of the queuing system converges to the same deterministic ordinary differential equation (ODE) for the power-of- choices scheme. We also show that the system is stable under the proposed scheme, and the stationary distribution of the system converges to the fixed point of the ODE.

Paper Structure

This paper contains 13 sections, 22 theorems, 178 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model
  3. Notations
  4. Preliminary Results for NBRW-Po$d$
  5. Large Deviation Results
  6. A Precise Estimate of Sampling Probability
  7. A Variant of the Foster-Lyapunov Theorem
  8. Main Results
  9. Fluid Limit Approximation for NBRW-Po$d$
  10. Stability of NBRW-Po$d$
  11. Convergence of Stationary Distributions
  12. Simulation Results
  13. Conclusions

Key Result

Lemma 1

Let $\{Z_j\}_{j=1}^\infty$ be a process adapted to the filtration $\{\mathcal{F}_j\}_{j=-1}^\infty$. Let $N> 0$ be even. If $0\leq Z_j\leq B$ and $\mathrm{E}[Z_j|\mathcal{F}_{j-2}]\leq m$ a.s. for all $j\geq 1$, then for any $\lambda\geq 2Nm$, we have

Figures (6)

  • Figure 1: Queue length statistics evolution for NBRW-Po$d$ algorithm with $d=2$ and $\lambda = 0.95$. System is empty at time $0$.
  • Figure 2: Queue length statistics evolution for NBRW-Po$d$ algorithm with $d=2$ and $\lambda = 0.95$. Each queue has a length of $5$ at time $0$.
  • Figure 3: Queue length statistics evolution for NBRW-Po$d$ algorithm with $d=2$ and $\lambda = 0.95$. System is empty at time $0$.
  • Figure 4: Queue length statistics evolution for NBRW-Po$d$ algorithm with $d=2$ and $\lambda = 0.95$. Each queue has a length of $5$ at time $0$.
  • Figure 5: Queue length statistics evolution for NBRW-Po$d$ algorithm with $d=2$ and $\lambda = 0.95$. System is empty at time $0$.
  • ...and 1 more figures

Theorems & Definitions (51)

  • Definition 1: Expander Graph
  • Remark 1
  • Lemma 1: Bernstein
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4: Moment Bound
  • proof
  • ...and 41 more