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

Banerjee, Sayan; Mukherjee, Debankur

doi:10.1214/19-aap1496

Cited by 12 publications

(16 citation statements)

References 28 publications

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“…Although this approach gives a good handle on the rate of convergence to stationarity, it sheds little light on the form of the stationary distribution of the limiting diffusion process (3.8) itself. In two companion papers Banerjee & Mukherjee [14,15] perform a thorough analysis of the steady state of this diffusion process. Using a classical regenerative process construction of the diffusion process in (3.8), [15,Theorem 2.1] establishes that Q1 (∞) has a Gaussian tail, and the tail exponent is uniformly bounded by constants which do not depend on β, whereas Q2 (∞) has an exponentially decaying tail, and the coefficient in the exponent is linear in β.…”

Section: Diffusion Limit For Jsq Policymentioning

confidence: 99%

“…But the way Q2 (∞) scales as β becomes large, is highly non-trivial. Specifically, it was shown in [14] that there exists β 0 1 and positive constants…”

Section: Diffusion Limit For Jsq Policymentioning

confidence: 99%

“…In this regime as well, Q2 exhibits some surprising behavior. Specifically, it was shown in [14] that there exist positive constants β * , M 1 , and M 2 such that for all…”

Section: Diffusion Limit For Jsq Policymentioning

confidence: 99%

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Scalable load balancing in networked systems: A survey of recent advances

van der Boor,

Borst,

van Leeuwaarden

et al. 2018

Preprint

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The basic load balancing scenario involves a single dispatcher where tasks arrive that must immediately be forwarded to one of N single-server queues. We discuss recent advances on scalable load balancing schemes which provide favorable delay performance when N grows large, and yet only require minimal implementation overhead.Join-the-Shortest-Queue (JSQ) yields vanishing delays as N grows large, as in a centralized queueing arrangement, but involves a prohibitive communication burden. In contrast, power-of-d or JSQ(d) schemes that assign an incoming task to a server with the shortest queue among d servers selected uniformly at random require little communication, but lead to constant delays. In order to examine this fundamental trade-off between delay performance and implementation overhead, we consider JSQ(d(N)) schemes where the diversity parameter d(N) depends on N and investigate what growth rate of d(N) is required to asymptotically match the optimal JSQ performance on fluid and diffusion scale.Stochastic coupling techniques and stochastic-process limits play an instrumental role in establishing the asymptotic optimality. We demonstrate how this methodology carries over to infinite-server settings, finite buffers, multiple dispatchers, servers arranged on graph topologies, and token-based load balancing including the popular Join-the-Idle-Queue (JIQ) scheme. In this way we provide a broad overview of the many recent advances in the field. This survey extends the short review presented at ICM 2018 [121].

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Section: Diffusion Limit For Jsq Policymentioning

confidence: 99%

“…But the way Q2 (∞) scales as β becomes large, is highly non-trivial. Specifically, it was shown in [14] that there exists β 0 1 and positive constants…”

Section: Diffusion Limit For Jsq Policymentioning

confidence: 99%

See 1 more Smart Citation

Scalable load balancing in networked systems: A survey of recent advances

van der Boor,

Borst,

van Leeuwaarden

et al. 2018

Preprint

Self Cite

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show abstract

“…Since then, numerous inert drift systems have been studied [3,4,5]. Moreover, stochastic differential equations somewhat similar in flavor to inert drift systems have recently appeared as diffusion limits of queuing networks like the join-the-shortest-queue discipline [6,7,8]. [9] studied the inert drift model with g > 0, γ = 0.…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

Inert drift system in a viscous fluid: Steady state asymptotics and exponential ergodicity

Banerjee

Brown

2020

Trans. Amer. Math. Soc.

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We analyze a system of stochastic differential equations describing the joint motion of a massive (inert) particle in a viscous fluid in the presence of a gravitational field and a Brownian particle impinging on it from below, which transfers momentum proportional to the local time of collisions. We study the long-time fluctuations of the velocity of the inert particle and the gap between the two particles, and we show convergence in total variation to the stationary distribution is exponentially fast. We also produce matching upper and lower bounds on the tails of the stationary distribution and show how these bounds depend on the system parameters. A renewal structure for the process is established, which is the key technical tool in proving the mentioned results.

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“…This convergence has been extended to steady state by Braverman [6] using a sophisticated generator expansion framework via the Stein's method, enabling the interchange of N → ∞ and t → ∞ limits. Subsequently, the tail and bulk behavior of the stationary distribution of the limiting diffusion have been studied by Banerjee and Mukherjee [3,4], although an explicit characterization of the stationary distribution is yet unknown. Convergence to the above OU process has been extended for a class of power-of-d policies in [24] and for the Join-Idle-Queue policy in [23].…”

Section: Literature Reviewmentioning

confidence: 99%

Many-server asymptotics for Join-the-Shortest Queue in the Super-Halfin-Whitt Scaling Window

Zhao

Banerjee

Mukherjee

2021

Preprint

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The Join-the-Shortest Queue (JSQ) policy is a classical benchmark for the performance of many-server queueing systems due to its strong optimality properties. While the exact analysis of the JSQ policy is an open question to date, even under Markovian assumption on the service requirements, recently, there has been a significant progress in understanding its manyserver asymptotic behavior since the work of Eschenfeldt and Gamarnik (Math. Oper. Res. 43 (2018) 867-886).In this paper, we analyze the many-server limits of the JSQ policy in the super-Halfin-Whitt scaling window when load per server λ N scales with the system size N as lim N→∞ N α (1 − λ N ) = β for α ∈ (1/2, 1) and β > 0. We establish that the centered and scaled total queue length process converges to a certain Bessel process with negative drift and the associated centered and scaled steady-state total queue length, indexed by N, converges to a Gamma(2, β) distribution. Both the transient and steady-state limit laws are universal in the sense that they do not depend on the value of the scaling parameter α, and exhibit fundamentally different qualitative behavior from both the Halfin-Whitt regime (α = 1/2) and the Non-degenerate Slowdown (NDS) regime (α = 1).

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

Cited by 12 publications

References 28 publications

Scalable load balancing in networked systems: A survey of recent advances

Scalable load balancing in networked systems: A survey of recent advances

Inert drift system in a viscous fluid: Steady state asymptotics and exponential ergodicity

Many-server asymptotics for Join-the-Shortest Queue in the Super-Halfin-Whitt Scaling Window

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