[Paper Review] Doubly Exponential Solution for Randomized Load Balancing Models with General Service Times

Quan‐Lin Li|arXiv (Cornell University)|Aug 23, 2010

Advanced Queuing Theory Analysis30 references3 citations

TL;DR

This paper proposes a novel supplementary variable method to analyze randomized load balancing models with general service times, formulating an infinite system of integral-differential equations via density-dependent jump Markov processes. It establishes a closed-form doubly exponential fixed point solution, proving it persists even under heavy-tailed service times, and confirms exponential convergence and Lipschitz continuity under general service assumptions.

ABSTRACT

In this paper, we provide a novel and simple approach to study the supermarket model with general service times. This approach is based on the supplementary variable method used in analyzing stochastic models extensively. We organize an infinite-size system of integral-differential equations by means of the density dependent jump Markov process, and obtain a close-form solution: doubly exponential structure, for the fixed point satisfying the system of nonlinear equations, which is always a key in the study of supermarket models. The fixed point is decomposited into two groups of information under a product form: the arrival information and the service information. based on this, we indicate two important observations: the fixed point for the supermarket model is different from the tail of stationary queue length distribution for the ordinary M/G/1 queue, and the doubly exponential solution to the fixed point can extensively exist even if the service time distribution is heavy-tailed. Furthermore, we analyze the exponential convergence of the current location of the supermarket model to its fixed point, and study the Lipschitz condition in the Kurtz Theorem under general service times. Based on these analysis, one can gain a new understanding how workload probing can help in load balancing jobs with general service times such as heavy-tailed service.

Motivation & Objective

• To address the open problem of whether heavy-tailed service times disrupt the doubly exponential structure in supermarket models.
• To develop a tractable analytical framework for supermarket models with general (non-exponential) service times, which are more realistic but analytically challenging.
• To establish the existence and structure of the fixed point in the infinite-population limit under general service time distributions.
• To prove exponential convergence of the system state to the fixed point and verify the Lipschitz condition required for Kurtz-type diffusion limits.
• To demonstrate that workload probing remains effective even with heavy-tailed service times, offering new insights for real-world load balancing in data centers and multi-core systems.

Proposed method

• Applies the supplementary variable method to model the joint dynamics of queue length and residual service times in the supermarket model.
• Constructs an infinite-size system of integral-differential equations by modeling the system as a density-dependent jump Markov process.
• Derives the fixed point of the system by solving a nonlinear system of equations, showing it admits a product-form decomposition into arrival and service information.
• Establishes Lipschitz continuity of the drift function in the fluid limit, enabling application of Kurtz's theorem for diffusion approximations.
• Uses the fluid limit and exponential convergence analysis to bound the expected sojourn time of jobs in the system.
• Validates the approach with numerical examples, demonstrating its effectiveness for non-exponential service time distributions.

Experimental results

Research questions

• RQ1 Does the doubly exponential structure of the fixed point in the supermarket model persist when service times are heavy-tailed?
• RQ2 Can a closed-form solution be derived for the fixed point in supermarket models with general (non-phase-type) service time distributions?
• RQ3 How does the system’s state converge to the fixed point under general service time assumptions?
• RQ4 What conditions ensure the Lipschitz continuity of the drift function in the fluid limit for general service time distributions?
• RQ5 To what extent does workload probing remain effective in balancing loads when service times are heavy-tailed?

Key findings

• The fixed point of the supermarket model with general service times exhibits a doubly exponential structure, even when the service time distribution is heavy-tailed.
• The fixed point is decomposed into a product form of arrival and service information, distinguishing it from the tail of the stationary queue length distribution in an M/G/1 queue.
• The doubly exponential solution is not unique in more general supermarket models, indicating structural sensitivity to system parameters.
• Exponential convergence of the system state to the fixed point is established under general service time distributions.
• The drift function in the fluid limit satisfies the Lipschitz condition, enabling rigorous diffusion approximation via Kurtz's theorem.
• The expected sojourn time of a job in an initially empty system is bounded above by an expression involving the doubly exponential fixed point, confirming performance gains under general service times.