[Paper Review] Heavy-traffic Delay Optimality in Pull-based Load Balancing Systems: Necessary and Sufficient Conditions

Xingyu Zhou, Jian Tan+1 more|arXiv (Cornell University)|Aug 21, 2018

Advanced Queuing Theory Analysis13 references4 citations

TL;DR

This paper establishes necessary and sufficient conditions for heavy-traffic delay optimality in pull-based load balancing systems with a dynamic reporting threshold. It proves that the threshold must grow logarithmically with the mean number of tasks to achieve optimality, resolving a conjecture by Kelly and Laws, while showing that constant thresholds like JIQ’s are strictly suboptimal in heavy traffic.

ABSTRACT

In this paper, we consider a load balancing system under a general pull-based policy. In particular, each arrival is randomly dispatched to one of the servers whose queue lengths are below a threshold, if there are any; otherwise, this arrival is randomly dispatched to one of the entire set of servers. We are interested in the fundamental relationship between the threshold and the delay performance of the system in heavy traffic. To this end, we first establish the following necessary condition to guarantee heavy-traffic delay optimality: the threshold will grow to infinity as the exogenous arrival rate approaches the boundary of the capacity region (i.e., the load intensity approaches one) but the growth rate should be slower than a polynomial function of the mean number of tasks in the system. As a special case of this result, we directly show that the delay performance of the popular pull-based policy Join-Idle-Queue (JIQ) lies strictly between that of any heavy-traffic delay optimal policy and that of random routing. We further show that a sufficient condition for heavy-traffic delay optimality is that the threshold grows logarithmically with the mean number of tasks in the system. This result directly resolves a generalized version of the conjecture by Kelly and Laws.

Motivation & Objective

• To characterize the fundamental relationship between the reporting threshold and delay performance in pull-based load balancing systems under heavy traffic.
• To determine the necessary conditions on the threshold growth rate for achieving heavy-traffic delay optimality.
• To establish sufficient conditions under which pull-based policies achieve steady-state delay optimality in heavy traffic.
• To resolve a generalized version of the conjecture by Kelly and Laws regarding threshold scaling for delay optimality.
• To provide a sharp characterization of the performance gap between JIQ (constant threshold) and optimal policies in heavy traffic.

Proposed method

• Proposes a general pull-based load balancing policy where arrivals are routed to a server with queue length below a dynamic threshold, or randomly if no such server exists.
• Uses a Lyapunov function approach to analyze system stability and delay performance, focusing on the drift of a projected state norm.
• Derives a necessary condition by showing that the threshold must grow to infinity as the load approaches capacity, but slower than any polynomial in the mean number of tasks.
• Establishes a sufficient condition by proving that logarithmic growth of the threshold with respect to the mean number of tasks guarantees heavy-traffic delay optimality.
• Employs a drift analysis involving the orthogonal projection of the queue length vector and uses a coupling argument to bound the expected change in the Lyapunov function.
• Validates the Lyapunov drift condition by showing negative drift for large system states, independent of the heavy-traffic parameter ε.

Experimental results

Research questions

• RQ1 What growth rate of the reporting threshold is necessary for a pull-based load balancing policy to achieve heavy-traffic delay optimality?
• RQ2 How does the delay performance of the Join-Idle-Queue (JIQ) policy compare to that of optimal and random routing policies in heavy traffic?
• RQ3 Is a logarithmic threshold growth rate sufficient for achieving steady-state delay optimality in pull-based systems?
• RQ4 Can the conjecture by Kelly and Laws on threshold scaling for delay optimality be formally resolved?
• RQ5 What is the precise characterization of the performance gap between JIQ and optimal policies in the heavy-traffic regime?

Key findings

• A necessary condition for heavy-traffic delay optimality is that the reporting threshold must grow to infinity as the load intensity approaches one, but at a rate slower than any polynomial function of the mean number of tasks in the system.
• The delay performance of the JIQ policy (with constant threshold r=1) lies strictly between that of any heavy-traffic delay optimal policy and random routing in heavy traffic.
• A sufficient condition for heavy-traffic delay optimality is that the threshold grows logarithmically with the mean number of tasks in the system.
• The paper resolves a generalized version of the conjecture by Kelly and Laws, confirming that logarithmic threshold growth ensures delay optimality.
• The Lyapunov drift analysis establishes a negative drift bound independent of the heavy-traffic parameter ε, proving stability and delay optimality under the sufficient condition.
• The results imply that a logarithmic threshold growth rate is both sufficient and likely necessary for delay optimality, suggesting a tight characterization of pull-based policies in heavy traffic.