A Refined Mean Field Approximation

In multi-server distributed queueing systems, the access of stochastically arriving jobs to resources is often regulated by a dispatcher, also known as load balancer. A fundamental problem consists in designing a load balancing algorithm that minimizes the delays experienced by jobs. During the last two decades, the power-of-d-choice algorithm, based on the idea of dispatching each job to the least loaded server out of d servers randomly sampled at the arrival of the job itself, has emerged as a breakthrough in the foundations of this area due to its versatility and appealing asymptotic properties. In this paper, we consider the power-of-d-choice algorithm with the addition of a local memory that keeps track of the latest observations collected over time on the sampled servers. Then, each job is sent to a server with the lowest observation. We show that this algorithm is asymptotically optimal in the sense that the load balancer can always assign each job to an idle server in the large-system limit. This holds true if and only if the system load λ is less than 1 − 1 d . If this condition is not satisfied, we show that queue lengths are tightly bounded by − log(1−λ) log(λd+1) . This is in contrast with the classic version of the power-of-d-choice algorithm, where at the fluid scale a strictly positive proportion of servers containing i jobs exists for all i ≥ 0, in equilibrium. Our results quantify and highlight the importance of using memory as a means to enhance performance in randomized load balancing. * INRIA Bordeaux Sud Ouest, 200 av. de la Vieille Tour,

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“…but on the other hand we must have G i (t) = G i (x(t)), where G i is defined in (10). We have thus shown thaṫ…”

Section: Lemma 12mentioning

confidence: 77%

“…• Server selections can be made without replacement, instead of with replacement. In view of the results in Gast and Van Houdt [10], this may also improve the convergence speed of X N to fluid solutions.…”

Section: Global Stabilitymentioning

confidence: 96%

Power-of-d-Choices with Memory: Fluid Limit and Optimality

Anselmi

Dufour

2020

Mathematics of OR

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“…In Ying (2019, 2020), Liu et al (2022), the authors used Stein's method for mean-field analysis to obtain bounds on steady-state performance metrics of interest, like EQ 2 for instance, for the power-of-d system. Another line of work on power-of-d systems was by Gast (2017), Gast and Van Houdt (2017), Gast et al (2019), where the authors showed how to derive refined mean-field models for improved steady-state approximations. More recently, Hairi et al (2021) provide calculable error bounds for the mean-field approximation of the power-of-two-choices model.…”

Section: Literature Reviewmentioning

confidence: 99%

The join-the-shortest-queue system in the Halfin-Whitt regime: rates of convergence to the diffusion limit

Braverman¹

2022

Preprint

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We show that the steady-state distribution of the join-the-shortest-queue (JSQ) system converges, in the Halfin-Whitt regime, to its diffusion limit at a rate of at least 1/ √ n, where n is the number of servers. Our proof uses Stein's method, and, specifically, the recently proposed prelimit generator comparison approach.Being the first application of the prelimit approach to a non-trivial and high-dimensional model, this paper may be helpful to readers wishing to apply the approach to their own setting. For example, the approach has the potential to be applied to other heavy-traffic regimes or load-balancing policies.

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“…A key question is how well the mean-field limit describes the occupancy distribution and the blocking probabilities when N is finite but large. Recently, there have been a number of papers [6, 23] that have addressed this issue for M/M/1 queueing models for the case where the limiting stationary distribution can be characterized explicitly as a double-exponential distribution. They used an approach based on Stein’s method and showed that the rate of convergence of the empirical occupancy distribution to the mean-field distribution is .…”

Section: Introductionmentioning

confidence: 99%

“…They used an approach based on Stein’s method and showed that the rate of convergence of the empirical occupancy distribution to the mean-field distribution is . In [6] a refined term is also given. These approaches use Stein’s method and exponential stability of the underlying mean-field equation to study the mean-squared error between the empirical distribution and the mean-field limit to characterize the rate of convergence.…”

Section: Introductionmentioning

confidence: 99%

Sensitivity of mean-field fluctuations in Erlang loss models with randomized routing

Vasantam

Mazumdar

2021

J. Appl. Probab.

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In this paper we study a large system of N servers, each with capacity to process at most C simultaneous jobs; an incoming job is routed to a server if it has the lowest occupancy amongst d (out of N) randomly selected servers. A job that is routed to a server with no vacancy is assumed to be blocked and lost. Such randomized policies are referred to JSQ(d) (Join the Shortest Queue out of d) policies. Under the assumption that jobs arrive according to a Poisson process with rate $N\lambda^{(N)}$ where $\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}\,}$ , $\sigma\in\mathbb{R}_+$ and $\beta\in\mathbb{R}$ , we establish functional central limit theorems for the fluctuation process in both the transient and stationary regimes when service time distributions are exponential. In particular, we show that the limit is an Ornstein–Uhlenbeck process whose mean and variance depend on the mean field of the considered model. Using this, we obtain approximations to the blocking probabilities for large N, where we can precisely estimate the accuracy of first-order approximations.

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A Refined Mean Field Approximation

Cited by 50 publications

References 29 publications

Power-of-d-Choices with Memory: Fluid Limit and Optimality

Power-of-d-Choices with Memory: Fluid Limit and Optimality

The join-the-shortest-queue system in the Halfin-Whitt regime: rates of convergence to the diffusion limit

Sensitivity of mean-field fluctuations in Erlang loss models with randomized routing

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