Heavy-traffic analysis through uniform acceleration of queues with diminishing populations

Bet Gianmarco, Remco van der Hofstad, Johan van Leeuwaarden

Research output: Contribution to journalArticleScientificpeer-review

Abstract

We consider a single-server queue that serves a finite population of n customers
that will enter the queue (require service) only once, also known as the Δ(i)/G/1 queue. This paper presents a method for analyzing heavy-traffic behavior by using uniform acceleration, which simultaneously lets n and the service rate grow large, while the initial resource utilization approaches one. A key feature of the model is that, as time progresses, more customers have joined the queue, and fewer customers can potentially join. This diminishing population gives rise to a class of reflected stochastic processes that vanish over time
and hence do not have a stationary distribution. We establish that, when the arrival times are exponentially distributed, by suitably rescaling space and time, the queue-length process converges to a Brownian motion with a negative quadratic drift, a stochastic-process limit that captures the effect of the diminishing population. When the arrival times are generally distributed, our techniques provide information on the typical queue length and the first busy period.
Original languageEnglish
Pages (from-to)821-864
JournalMathematics of Operations Research
Volume44
Issue number3
Early online dateMay 2019
DOIs
Publication statusPublished - Aug 2019
Externally publishedYes

Fingerprint

Traffic Analysis
Heavy Traffic
Diminishing
Random processes
Queue
Arrival Time
Queue Length
Stochastic Processes
Brownian movement
Customers
Single Server Queue
Busy Period
Servers
Finite Population
Rescaling
Stationary Distribution
Join
Brownian motion
Vanish
Converge

Keywords

  • queueing theory
  • transitory queueing systems
  • uniform acceleration technique
  • heavy-traffic approcimations
  • asymptotic analysis
  • continuous-mapping approach
  • martingale functional CLT

Cite this

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title = "Heavy-traffic analysis through uniform acceleration of queues with diminishing populations",
abstract = "We consider a single-server queue that serves a finite population of n customersthat will enter the queue (require service) only once, also known as the Δ(i)/G/1 queue. This paper presents a method for analyzing heavy-traffic behavior by using uniform acceleration, which simultaneously lets n and the service rate grow large, while the initial resource utilization approaches one. A key feature of the model is that, as time progresses, more customers have joined the queue, and fewer customers can potentially join. This diminishing population gives rise to a class of reflected stochastic processes that vanish over timeand hence do not have a stationary distribution. We establish that, when the arrival times are exponentially distributed, by suitably rescaling space and time, the queue-length process converges to a Brownian motion with a negative quadratic drift, a stochastic-process limit that captures the effect of the diminishing population. When the arrival times are generally distributed, our techniques provide information on the typical queue length and the first busy period.",
keywords = "queueing theory, transitory queueing systems, uniform acceleration technique, heavy-traffic approcimations, asymptotic analysis, continuous-mapping approach, martingale functional CLT",
author = "Bet Gianmarco and {van der Hofstad}, Remco and {van Leeuwaarden}, Johan",
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Heavy-traffic analysis through uniform acceleration of queues with diminishing populations. / Gianmarco, Bet; van der Hofstad, Remco; van Leeuwaarden, Johan.

In: Mathematics of Operations Research, Vol. 44, No. 3, 08.2019, p. 821-864.

Research output: Contribution to journalArticleScientificpeer-review

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AU - van der Hofstad, Remco

AU - van Leeuwaarden, Johan

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N2 - We consider a single-server queue that serves a finite population of n customersthat will enter the queue (require service) only once, also known as the Δ(i)/G/1 queue. This paper presents a method for analyzing heavy-traffic behavior by using uniform acceleration, which simultaneously lets n and the service rate grow large, while the initial resource utilization approaches one. A key feature of the model is that, as time progresses, more customers have joined the queue, and fewer customers can potentially join. This diminishing population gives rise to a class of reflected stochastic processes that vanish over timeand hence do not have a stationary distribution. We establish that, when the arrival times are exponentially distributed, by suitably rescaling space and time, the queue-length process converges to a Brownian motion with a negative quadratic drift, a stochastic-process limit that captures the effect of the diminishing population. When the arrival times are generally distributed, our techniques provide information on the typical queue length and the first busy period.

AB - We consider a single-server queue that serves a finite population of n customersthat will enter the queue (require service) only once, also known as the Δ(i)/G/1 queue. This paper presents a method for analyzing heavy-traffic behavior by using uniform acceleration, which simultaneously lets n and the service rate grow large, while the initial resource utilization approaches one. A key feature of the model is that, as time progresses, more customers have joined the queue, and fewer customers can potentially join. This diminishing population gives rise to a class of reflected stochastic processes that vanish over timeand hence do not have a stationary distribution. We establish that, when the arrival times are exponentially distributed, by suitably rescaling space and time, the queue-length process converges to a Brownian motion with a negative quadratic drift, a stochastic-process limit that captures the effect of the diminishing population. When the arrival times are generally distributed, our techniques provide information on the typical queue length and the first busy period.

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KW - transitory queueing systems

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KW - asymptotic analysis

KW - continuous-mapping approach

KW - martingale functional CLT

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