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Heavy-traffic Optimality of Skip-the-Longest-Queues in Heterogeneous Service Systems

Yishun Luo, Martin Zubeldia

TL;DR

The paper tackles scalable load balancing in heterogeneous parallel queues with ultra-low communication overhead by introducing the $k$-Skip-the-$d$-Longest-Queues policy. It establishes a stability region that is independent of the sampling parameter $k$ and shows that, in classical heavy-traffic, the policy is delay-optimal for any fixed $k$ under a suitable $d$, with the overhead vanishing as $k$ grows. State Space Collapse is used to connect the multi-queue dynamics to a one-dimensional projection, enabling a sharp delay analysis and showing a pathway to vanishing communication while preserving performance. In the many-server regime, it derives necessary conditions on $d$ for throughput and provides tight delay-optimality criteria, including phase-transition insights in homogeneous systems, thereby offering a practical load-balancing scheme with tunable communication cost.

Abstract

We consider a discrete-time parallel service system consisting of $n$ heterogeneous single server queues with infinite capacity. Jobs arrive to the system as an i.i.d. process with rate proportional to $n$, and must be immediately dispatched in the time slot that they arrive. The dispatcher is assumed to be able to exchange messages with the servers to obtain their queue lengths and make dispatching decisions, introducing an undesirable communication overhead. In this setting, we propose a ultra-low communication overhead load balancing policy dubbed $k$-Skip-the-$d$-Longest-Queues ($k$-SLQ-$d$), where queue lengths are only observed every $k(n-d)$ time slots and, between observations, incoming jobs are sent to a queue that is not one of the $d$ longest ones at the time that the queues were last observed. For this policy, we establish conditions on $d$ for it to be throughput optimal and we show that, under that condition, it is asymptotically delay-optimal in heavy-traffic for arbitrarily low communication overheads (i.e., for arbitrarily large $k$).

Heavy-traffic Optimality of Skip-the-Longest-Queues in Heterogeneous Service Systems

TL;DR

The paper tackles scalable load balancing in heterogeneous parallel queues with ultra-low communication overhead by introducing the -Skip-the--Longest-Queues policy. It establishes a stability region that is independent of the sampling parameter and shows that, in classical heavy-traffic, the policy is delay-optimal for any fixed under a suitable , with the overhead vanishing as grows. State Space Collapse is used to connect the multi-queue dynamics to a one-dimensional projection, enabling a sharp delay analysis and showing a pathway to vanishing communication while preserving performance. In the many-server regime, it derives necessary conditions on for throughput and provides tight delay-optimality criteria, including phase-transition insights in homogeneous systems, thereby offering a practical load-balancing scheme with tunable communication cost.

Abstract

We consider a discrete-time parallel service system consisting of heterogeneous single server queues with infinite capacity. Jobs arrive to the system as an i.i.d. process with rate proportional to , and must be immediately dispatched in the time slot that they arrive. The dispatcher is assumed to be able to exchange messages with the servers to obtain their queue lengths and make dispatching decisions, introducing an undesirable communication overhead. In this setting, we propose a ultra-low communication overhead load balancing policy dubbed -Skip-the--Longest-Queues (-SLQ-), where queue lengths are only observed every time slots and, between observations, incoming jobs are sent to a queue that is not one of the longest ones at the time that the queues were last observed. For this policy, we establish conditions on for it to be throughput optimal and we show that, under that condition, it is asymptotically delay-optimal in heavy-traffic for arbitrarily low communication overheads (i.e., for arbitrarily large ).

Paper Structure

This paper contains 20 sections, 7 theorems, 107 equations, 1 table.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Basic Notations
  4. Queueing model and new dispatching policy
  5. Discrete-time queueing model
  6. $k$-Round-Robin Skip-the-Longest-$d$ Policy ($k$-SLQ-$d$)
  7. Communication overhead
  8. Main results
  9. Stability region
  10. State Space Collapse
  11. Delay optimality in classical heavy-traffic
  12. Join-the-Shortest vs Skip-the-Largest queue for homogeneous systems
  13. Many-server heavy-traffic
  14. Throughput optimality
  15. Delay optimality
  16. ...and 5 more sections

Key Result

Theorem 3.1

Fix $k \geq 1$ and $d\in \{1,\dots,n-1 \}$. Under the $k$-SLQ-$d$ policy, the Markov Chain $\{\mathbf{X}(t)\}_{t\geq 0}$ is positive recurrent if and only if:

Theorems & Definitions (8)

  • Theorem 3.1
  • Corollary 3.2
  • Theorem 3.3
  • Remark 3.4
  • Theorem 3.5
  • Corollary 3.6
  • Corollary 3.7
  • Corollary 4.1