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Stability and Heavy-traffic Delay Optimality of General Load Balancing Policies in Heterogeneous Service Systems

Yishun Luo, Martin Zubeldia

Abstract

We consider a load balancing system consisting of $n$ single-server queues working in parallel, with heterogeneous service rates. Jobs arrive to a central dispatcher, which has to dispatch them to one of the queues immediately upon arrival. For this setting, we consider a broad family of policies where the dispatcher can only access the queue lengths sporadically, every $T$ units of time. We assume that the dispatching decisions are made based only on the order of the scaled queue lengths at the last time that the queues were accessed, and on the processing rate of each server. For these general policies, we provide easily verifiable necessary and sufficient conditions for the stability of the system, and sufficient conditions for heavy-traffic delay optimality. We also show that, in heavy-traffic, the queue length converges in distribution to a scaled deterministic vector, where the scaling factor is an exponential random variable.

Stability and Heavy-traffic Delay Optimality of General Load Balancing Policies in Heterogeneous Service Systems

Abstract

We consider a load balancing system consisting of single-server queues working in parallel, with heterogeneous service rates. Jobs arrive to a central dispatcher, which has to dispatch them to one of the queues immediately upon arrival. For this setting, we consider a broad family of policies where the dispatcher can only access the queue lengths sporadically, every units of time. We assume that the dispatching decisions are made based only on the order of the scaled queue lengths at the last time that the queues were accessed, and on the processing rate of each server. For these general policies, we provide easily verifiable necessary and sufficient conditions for the stability of the system, and sufficient conditions for heavy-traffic delay optimality. We also show that, in heavy-traffic, the queue length converges in distribution to a scaled deterministic vector, where the scaling factor is an exponential random variable.
Paper Structure (22 sections, 9 theorems, 137 equations, 2 tables)

This paper contains 22 sections, 9 theorems, 137 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Basic Notations
  4. Literature Review
  5. Load Balancing Policies
  6. Stability of load balancing policies
  7. Heavy-traffic Delay Optimality
  8. Queueing Model and General Dispatching Framework
  9. Discrete-time queueing model
  10. General Policy Framework
  11. Main Results
  12. Stability Region
  13. Heavy-traffic Optimality and Queue Length Distribution
  14. Delay optimality
  15. Queue length distribution
  16. ...and 7 more sections

Key Result

Theorem 4.1

For any load balancing policy $\pi \in \Pi$, if then the Markov Chain $\{\mathbf{X}(t)\}_{t\geq 0}$ is positive recurrent.

Theorems & Definitions (15)

  • Example 3.1
  • Example 4.1
  • Definition 4.1
  • Theorem 4.1
  • Corollary 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Theorem 4.5
  • Corollary 4.6
  • Theorem 4.7
  • ...and 5 more