Asymptotic independence of queues under randomized load balancing

Abstract

Randomized load balancing greatly improves the sharing of resources while being simple to implement. In one such model, jobs arrive according to a rate-αN Poisson process, with α < 1, in a system of N rate-1 exponential server queues. In Vvedenskaya et al. [19], it was shown that when each arriving job is assigned to the shortest of D, D ≥ 2, randomly chosen queues, the equilibrium queue sizes decay doubly exponentially in the limit as N → ∞. This is a substantial improvement over the case D = 1, where queue sizes decay exponentially.The reasoning in [19] does not easily generalize to jobs with nonexponential service time distributions. A modularized program for treating randomized load balancing problems with general service time distributions was introduced in Bramson et al. [5]. The program relies on an ansatz that asserts that, for a randomized load balancing scheme in equilibrium, any fixed number of queues become independent of one another as N → ∞. This allows computation of queue size distributions and other performance measures of interest.In this article, we demonstrate the ansatz in several settings. We consider the least loaded balancing problem, where an arriving job is assigned to the queue with the smallest workload. We also consider the more difficult prob- lem, where an arriving job is assigned to the queue with the fewest jobs, and demonstrate the ansatz when the service discipline is FIFO and the service time distribution has a decreasing hazard rate. Last, we show the ansatz always holds for a sufficiently small arrival rate, as long as the service distribution has 2 moments.

Block-structured supermarket models

“…We call that f ∈ L depends only on the first K subvectors if for g (1) , g (2) ∈ Ω, it follows from g…”

“…Since a simple supermarket model was discussed by Mitzenmacher [23], Vvedenskaya et al [32] and Turner [30] through queueing theory as well as Markov processes, subsequent papers have been published on this theme, among which, see, Vvedenskaya and Suhov [33], Jacquet and Vvedenskaya [8], Jacquet et al [9], Mitzenmacher [24], Graham [5,6,7], Mitzenmacher et al [25], Vvedenskaya and Suhov [34], Luczak and Norris [20], Luczak and McDiarmid [18,19], Bramson et al [1,2,3], Li et al [17], Li [13] and Li et al [15]. For the fast Jackson networks (or the supermarket networks), readers may refer to Martin and Suhov [22], Martin [21] and Suhov and Vvedenskaya [29].…”

On the Approximation Error of Mean-Field Models

“…For example, in Mitzenmacher (1996) and Vvedenskaya et al (1996), the mean-field analysis has been used to show that routing each incoming task to the shorter of two randomly sampled servers can significantly reduce queueing delays, a phenomenon called the power of two choices (Po2). The result has been extended to heavy-tailed service-time distributions (Bramson et al 2012), and to heterogeneous servers (Mukhopadhyay and Mazumdar 2013). In Tsitsiklis and Xu (2012), a mean-field model has been used to quantify the significant benefit of resource pooling.…”

“…This can be proved using different techniques including Kurtz's theorem (Kurtz 1971(Kurtz , 1981Mitzenmacher 1996;Ying et al 2015), propagation of chaos (Sznitman 1991, Anantharam and Benchekroun 1993, Bramson et al 2012, or the convergence of the transition semigroup of CTMCs (Vvedenskaya et al 1996, Mukhopadhyay andMazumdar 2013).…”

Randomized Load Balancing in Finite Regimes

“…The extension to general service times was addressed in Bramson et al [3], [4]. The corresponding case of the randomized longest-queue-first policy was addressed asymptotically in Alanyali and Dashouk [2] and Dieker and Suk [5].…”