### Abstract

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 language | English |
---|---|

Pages (from-to) | 821-864 |

Journal | Mathematics of Operations Research |

Volume | 44 |

Issue number | 3 |

Early online date | May 2019 |

DOIs | |

Publication status | Published - Aug 2019 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### Cite this

*Mathematics of Operations Research*,

*44*(3), 821-864. https://doi.org/10.1287/moor.2018.0947

}

*Mathematics of Operations Research*, vol. 44, no. 3, pp. 821-864. https://doi.org/10.1287/moor.2018.0947

**Heavy-traffic analysis through uniform acceleration of queues with diminishing populations.** / Gianmarco, Bet; van der Hofstad, Remco; van Leeuwaarden, Johan.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Heavy-traffic analysis through uniform acceleration of queues with diminishing populations

AU - Gianmarco, Bet

AU - van der Hofstad, Remco

AU - van Leeuwaarden, Johan

PY - 2019/8

Y1 - 2019/8

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.

KW - queueing theory

KW - transitory queueing systems

KW - uniform acceleration technique

KW - heavy-traffic approcimations

KW - asymptotic analysis

KW - continuous-mapping approach

KW - martingale functional CLT

U2 - 10.1287/moor.2018.0947

DO - 10.1287/moor.2018.0947

M3 - Article

VL - 44

SP - 821

EP - 864

JO - Mathematics of Operations Research

JF - Mathematics of Operations Research

SN - 0364-765X

IS - 3

ER -