First Passage Times to Congested States of Many-Server Systems in the Halfin–Whitt Regime

Fralix, Brian; Knessl, Charles; Leeuwaarden, Johan S. H. van

doi:10.1080/15326349.2014.900385

Cited by 6 publications

(7 citation statements)

References 12 publications

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“…In the limit m → ∞, with the scaling in (2.104) and (3.25), the transform of the first passage distribution Q n (θ) for the M/M/m model has the limit P(x, θ) where We have previously obtained these results in [5], by directly solving the parabolic PDE satisfied by the diffusion approximation. Since 2D 1−θ (−β)+βD −θ (−β) = −2D ′ −θ (−β), Corollary 6 agrees with Theorems 1 and 2 in [5]. Now, we can also consider the Halfin-Whitt limit for the first passage distribution in the M/M/m + M model (with a fixed η > 0), and then Theorem 3 reduces to the following.…”

Section: First Passage Timesmentioning

confidence: 98%

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Transient analysis of the Erlang A model

Knessl

Leeuwaarden

2015

Math Meth Oper Res

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We consider the Erlang A model, or queue, with Poisson arrivals, exponential service times, and m parallel servers, and the property that waiting customers abandon the queue after an exponential time. The queue length process is in this case a birth–death process, for which we obtain explicit expressions for the Laplace transforms of the time-dependent distribution and the first passage time. These two transient characteristics were generally presumed to be intractable. Solving for the Laplace transforms involves using Green’s functions and contour integrals related to hypergeometric functions. Our results are specialized to the queue, the M / M / m queue, and the M / M / m / m loss model. We also obtain some corresponding results for diffusion approximations to these models.

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Section: First Passage Timesmentioning

confidence: 98%

“…(2014), by directly solving the parabolic PDE satisfied by the diffusion approximation. Since , Corollary 6 agrees with Theorems 1 and 2 in Fralix et al.…”

Section: First Passage Timesmentioning

confidence: 99%

Transient analysis of the Erlang A model

Knessl

Leeuwaarden

2015

Math Meth Oper Res

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“…The limiting diffusion process (D(t)) t≥0 in Theorem 2.2 is a combination of a negative-drift Brownian motion in the upper half plane and an Ornstein-Uhlenbeck process in the lower half plane. We refer to this hybrid diffusion process as the Halfin-Whitt diffusion [143,42,27]. Studying this diffusion process provides valuable information for the systems performance.…”

Section: )mentioning

confidence: 99%

“…An alternative derivation of this spectral gap was presented in by [142,143], along with expressions for the Laplace transform over time, and the large-time asymptotics for the time-dependent density. First passage times to large levels corresponding to highly congested states were obtained in [105,42]. For obvious reasons, the QED regime is also referred to as the Halfin-Whitt regime, and both these names are used interchangeably in the literature.…”

Section: )mentioning

confidence: 99%

Economies-of-Scale in Many-Server Queueing Systems: Tutorial and Partial Review of the QED Halfin--Whitt Heavy-Traffic Regime

Leeuwaarden¹,

Mathijsen²,

Zwart³

2019

SIAM Rev.

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Resource sharing systems describe situations in which users compete for service from scarce resources. Examples include check-in lines at airports, waiting rooms in hospitals or queues in contact centers, data buffers in wireless networks, and delayed service in cloud data centers. These are all situations with jobs (clients, patients, tasks) and servers (agents, beds, processors) that have large capacity levels, ranging from the order of tens (checkouts) to thousands (processors). This survey investigates how to design such systems to exploit resource pooling and economies-of-scale. In particular, we review the mathematics behind the Quality-and-Efficiency Driven (QED) regime, which lets the system operate close to full utilization, while the number of servers grows simultaneously large and delays remain manageable. We also discuss emerging research directions related to load balancing, overdispersion and model uncertainty.arXiv:1706.05397v1 [math.PR]

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“…Now consider an asymptotic regime where the number of servers grows large, and additionally assume that N − λ(N) √ N → β as N → ∞ for some positive coefficient β > 0, i.e., the load per server approaches unity as 1 − β/ √ N. In terms of the aggregate traffic load and total service capacity, this scaling corresponds to the socalled Halfin-Whitt heavy-traffic regime which was introduced in the seminal paper [12] and has been extensively studied since. The set-up in [12], as well as the numerous model extensions in the literature (see [8,9,10,12,23,24,25], and the references therein), primarily considered a single centralized queue and server pool (M/M/N), rather than a scenario with parallel queues. Eschenfeldt and Gamarnik [7] initiated the study of the scaling behavior for parallel-server systems in the Halfin-Whitt heavy-traffic regime.…”

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confidence: 99%

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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First Passage Times to Congested States of Many-Server Systems in the Halfin–Whitt Regime

Cited by 6 publications

References 12 publications

Transient analysis of the Erlang A model

Transient analysis of the Erlang A model

Economies-of-Scale in Many-Server Queueing Systems: Tutorial and Partial Review of the QED Halfin--Whitt Heavy-Traffic Regime

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

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