Steady-State Analysis of the Join-the-Shortest-Queue Model in the Halfin–Whitt Regime

Braverman, Anton

doi:10.1287/moor.2019.1023

Cited by 43 publications

(85 citation statements)

References 45 publications

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“…Recall the diffusion process {(Q 1 (t), Q 2 (t))} t 0 defined by Equation (1.1). As mentioned in the introduction, it is known [4] that for any β > 0, (Q 1 , Q 2 ) is an ergodic continuous-time Markov process. Let (Q 1 (∞), Q 2 (∞)) denote a random variable distributed as the unique stationary distribution π of the process.…”

Section: Resultsmentioning

confidence: 99%

“…where we used the fact that Q 2 (t) decreases exponentially for t τ 1 (0). By Lemma 4.6, there is β 0 1 such that for all β β 0 and all x (2β) 4 ,…”

Section: Claimmentioning

confidence: 96%

“…for some Brownian motion W. Therefore, for x (2β) 4 where β β 0 for sufficiently large β 0 , sup y x 1/4…”

Section: Claimmentioning

confidence: 99%

“…In fact, even establishing its ergodicity is non-trivial and was left as an open question in [7]. The tightness of the diffusion-scaled occupancy measure under the JSQ policy, exponential ergodicity of the diffusion process, and the interchange of limits were established by Braverman [4] via a sophisticated generator expansion framework using the Stein's method. There, it was shown that the steady state of the N-server systemQ N (∞) converges weakly to (Q 1 (∞), Q 2 (∞), 0, 0, .…”

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confidence: 99%

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Join-the-Shortest Queue diffusion limit in Halfin–Whitt regime: Sensitivity on the heavy-traffic parameter

Banerjee¹,

Mukherjee²

2020

Ann. Appl. Probab.

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Consider a system of N parallel single-server queues with unit-exponential service time distribution and a single dispatcher where tasks arrive as a Poisson process of rate λ(N). When a task arrives, the dispatcher assigns it to one of the servers according to the Join-the-Shortest Queue (JSQ) policy. Eschenfeldt and Gamarnik (2018) [7] identified a novel limiting diffusion process that arises as the weak-limit of the appropriately scaled occupancy measure of the system under the JSQ policy in the Halfin-Whitt regime, where (N − λ(N))/ √ N → β > 0 as N → ∞. The analysis of this diffusion goes beyond the state of the art techniques, and even proving its ergodicity is non-trivial, and was left as an open question. Recently, exploiting a generator expansion framework via the Stein's method, Braverman (2018) [4] established its exponential ergodicity, and adapting a regenerative approach, Banerjee and Mukherjee (2018) [3] analyzed the tail properties of the stationary distribution and path fluctuations of the diffusion.However, the analysis of the bulk behavior of the stationary distribution, viz., the moments, remained intractable until this work. In this paper, we perform a thorough analysis of the bulk behavior of the stationary distribution of the diffusion process, and discover that it exhibits different qualitative behavior, depending on the value of the heavy-traffic parameter β. Moreover, we obtain precise asymptotic laws of the centered and scaled steady state distribution, as β tends to 0 and ∞. Of particular interest, we also establish a certain intermittency phenomena in the β → ∞ regime and a surprising distributional convergence result in the β → 0 regime.

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Section: Resultsmentioning

confidence: 99%

“…where we used the fact that Q 2 (t) decreases exponentially for t τ 1 (0). By Lemma 4.6, there is β 0 1 such that for all β β 0 and all x (2β) 4 ,…”

Section: Claimmentioning

confidence: 96%

“…for some Brownian motion W. Therefore, for x (2β) 4 where β β 0 for sufficiently large β 0 , sup y x 1/4…”

Section: Claimmentioning

confidence: 99%

mentioning

confidence: 99%

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Join-the-Shortest Queue diffusion limit in Halfin–Whitt regime: Sensitivity on the heavy-traffic parameter

Banerjee¹,

Mukherjee²

2020

Ann. Appl. Probab.

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“…JIQ exists in keeping track of the ids of the idle queues and assigning incoming jobs to an idle queue whenever there exists an idle server and simply assigning it to a random server otherwise. is policy has vanishing delays when the number of servers tends to in nity in case of in nite memory [5,6,10,15]. e paper is structured as follows : In Section 2, the model is introduced and we shortly review previously obtained results for SQ(d) and LL(d).…”

Section: Introductionmentioning

confidence: 99%

Performance Analysis of Workload Dependent Load Balancing Policies

Hellemans

Bodas

Houdt

2019

SIGMETRICS Perform. Eval. Rev.

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Joining the shortest or least loaded queue among d randomly selected queues are two fundamental load balancing policies. Under both policies the dispatcher does not maintain any information on the queue length or load of the servers. In this paper we analyze the performance of these policies when the dispatcher has some memory available to store the ids of some of the idle servers. We consider methods where the dispatcher discovers idle servers as well as methods where idle servers inform the dispatcher about their state.We focus on large-scale systems and our analysis uses the cavity method. e main insight provided is that the performance measures obtained via the cavity method for a load balancing policy with memory reduce to the performance measures for the same policy without memory provided that the arrival rate is properly scaled. us, we can study the performance of load balancers with memory in the same manner as load balancers without memory. In particular this entails closed form solutions for joining the shortest or least loaded queue among d randomly selected queues with memory in case of exponential job sizes.We present simulation results that support our belief that the approximation obtained by the cavity method becomes exact as the number of servers tends to in nity.

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Steady‐state analysis of load balancing with Coxian‐2 distributed service times

Liu

Gong

Ying

2021

Naval Research Logistics

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This paper studies load balancing for many‐server (N servers) systems. Each server has a buffer of size b − 1, and can have at most one job in service and b − 1 jobs in the buffer. The service time of a job follows the Coxian‐2 distribution. We focus on steady‐state performance of load balancing policies in the heavy traffic regime such that the normalized load of system is λ = 1 − N−α for 0 < α < 0.5. We identify a set of policies that achieve asymptotic zero waiting. The set of policies include several classical policies such as join‐the‐shortest‐queue (JSQ), join‐the‐idle‐queue (JIQ), idle‐one‐first (I1F) and power‐of‐d‐choices (Po d) with d = O(Nα log N). The proof of the main result is based on Stein's method and state space collapse. A key technical contribution of this paper is the iterative state space collapse approach that leads to a simple generator approximation when applying Stein's method.

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Steady-State Analysis of the Join-the-Shortest-Queue Model in the Halfin–Whitt Regime

Cited by 43 publications

References 45 publications

Join-the-Shortest Queue diffusion limit in Halfin–Whitt regime: Sensitivity on the heavy-traffic parameter

Join-the-Shortest Queue diffusion limit in Halfin–Whitt regime: Sensitivity on the heavy-traffic parameter

Performance Analysis of Workload Dependent Load Balancing Policies

Steady‐state analysis of load balancing with Coxian‐2 distributed service times

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