On the distributions of infinite server queues with batch arrivals

Daw, Andrew; Pender, Jamol

doi:10.1007/s11134-019-09603-4

Cited by 17 publications

(11 citation statements)

References 33 publications

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“…Next, the authors extend this result to the M B /G/∞ case, again where each arriving batch consists of n customers, by reinterpreting the queueing system as a collection of n sub-queues. The authors then continue in Daw and Pender [4] by showing that the same results still apply when the batch sizes are random. Our first primary goal in this work is to illustrate how these observations carry over to the M Bt t /G t /∞ system, where we show that the marginal distributions of both the queue-length process, as well as the departure process of this queueing system are equal in distribution to an infinite sum of independent, scaled Poisson random variables.…”

Section: Introduction and Model Descriptionmentioning

confidence: 87%

“…Our first primary goal in this work is to illustrate how these observations carry over to the M Bt t /G t /∞ system, where we show that the marginal distributions of both the queue-length process, as well as the departure process of this queueing system are equal in distribution to an infinite sum of independent, scaled Poisson random variables. While this could also be carried out using the sub-queue construction featured in Daw and Pender [4], we choose to instead use point process reasoning to find these marginal distributions. Next, we build on this point process approach even further, by illustrating how it can be used to calculate the finite-dimensional distributions of both the queue-length process and the departure process of the M Bt t /G t /∞ system.…”

Section: Introduction and Model Descriptionmentioning

confidence: 99%

“…The research literature contains a large body of work addressing batch/bulk queueing systems: to the best of our knowledge, the first study featuring queues with batch arrivals is that of Miller Jr [21]. Since then, many other papers have been written that feature a study of queues with batch arrivals that operate under various different conditions: see for example Foster [10], Shanbhag [28], Brown and Ross [1], Holman et al [13], Fakinos [9], Chatterjee and Mukherjee [2], Lucantoni [18], Takagi and Takahashi [29], Economou and Fakinos [7], Masuyama and Takine [19], Liu and Templeton [16], Lee et al [15], Daw and Pender [4]. Later work has expanded the concept to a variety of related models, including priority queues and queues with server vacations.…”

Section: Introduction and Model Descriptionmentioning

confidence: 99%

“…This paper contributes to the literature on infinite-server queues with batch arrivals in multiple ways. First, it was recently discovered in Daw and Pender [4] that the stationary distribution of a M B /G/∞ queueing system is equal in distribution to an infinite convolution of scaled Poisson random variables, where the means of these random variables are expressed in terms of the order statistics associated with the amounts of work found in an arriving batch. This discovery was made by first realizing that, for the case where all batch sizes are deterministic-more particularly, of size n for some integer n ≥ 1-the moment generating function (MGF) of the M B /M/∞ queue can be rewritten in a manner that allows for the MGF to be inverted analytically.…”

Section: Introduction and Model Descriptionmentioning

confidence: 99%

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Non-Stationary Queues with Batch Arrivals

Daw¹,

Fralix²,

Pender³

2020

Preprint

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Motivated by applications that involve setting proper staffing levels for multi-server queueing systems with batch arrivals, we present a thorough study of the queue-length process {Q(t); t ≥ 0}, departure process {D(t); t ≥ 0}, and the workload process {W (t); t ≥ 0} associated with the M Bt t /G t /∞ queueing system, where arrivals occur in batches, with the batch size distribution varying with time. Notably, we first show that both Q(t) and D(t) are equal in distribution to an infinite sum of independent, scaled Poisson random variables. When the batch size distribution has finite support, this sum becomes finite as well. We then derive the finite-dimensional distributions of both the queue-length process and the departure process, and we use these results to show that these finite-dimensional distributions converge weakly under a certain scaling regime, where the finite-dimensional distributions of the queue-length process converge weakly to a shot-noise process driven by a non-homogeneous Poisson process. Next, we derive an expression for the joint Laplace-Stieltjes transform of W (t), Q(t), and D(t), and we show that these three random variables, under the same scaling regime, converge weakly, where the limit associated with the workload process corresponds to another Poisson-driven shot-noise process.

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Section: Introduction and Model Descriptionmentioning

confidence: 87%

Section: Introduction and Model Descriptionmentioning

confidence: 99%

Section: Introduction and Model Descriptionmentioning

confidence: 99%

Section: Introduction and Model Descriptionmentioning

confidence: 99%

See 2 more Smart Citations

Non-Stationary Queues with Batch Arrivals

Daw¹,

Fralix²,

Pender³

2020

Preprint

Self Cite

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“…In [13], a multistage batch arrival queueing system is considered. In [14], a method for analyzing a queueing system with a random number of batch arrivals is proposed. The proposed approaches for analysis with batch arrivals for a number of fundamental reasons cannot be used in assessing the effectiveness of the service system of urban public transport passengers.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

Development of a model of the service system of batch arrivals in the passengers flow of public transport

Раскін

Сіра

Palant

et al. 2019

EEJET

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Запропоновано математичну модель системи масового обслуговування потоку пасажирiв мiського громадського транспорту. Отримана модель вiдрiзняється вiд канонiчних моделей теорiї масового обслуговування урахуванням принципових особливостей реальних систем. По-перше, процес обслуговування розбитий на наступнi один за одним рiзнi сеанси обслуговування. По-друге, заявки на входi i на виходi системи є груповими. По-третє, iнтенсивностi вхiдного потоку в рiзних сеансах обслуговування рiзнi. По-четверте, закони розподiлу числа вимог в групових заявках для рiзних сеансiв рiзнi. По-п'яте, закони розподiлу числа заявок в групах на входi системи i виходi з неї також рiзнi. Розроблено критерiй ефективностi системи обслуговування. Критерiй заснований на розрахунку розподiлу ймовiрностей станiв системи обслуговування на входi системи i аналогiчного розподiлу на виходi системи. Цi розподiлу визначаються незалежно для кожного сеансу обслуговування, на якi розбивається весь цикл обслуговування. Чисельне значення критерiю задається вiдношенням середнього числа вiдмов в обслуговуваннi до середнього числа вимог в сумарному входить потоцi за весь цикл обслуговування. Його можна використовувати для оцiнки ефективностi системи обслуговування на будь-якому обраному часовому iнтервалi протягом доби, т. я. значення запропонованого критерiю залежить вiд довжини iнтервалу мiж сеансами, яка визначається числом транспортних засобiв на маршрутi. Отриманi моделi адекватно вiдображають процес функцiонування системи, що дає можливiсть прогнозувати безлiч рiзних ситуацiй i оцiнювати наслiдки пропонованих рiшень. При цьому виникає можливiсть прогнозування транспортної забезпеченостi населення та визначення кiлькiсних значень ефективностi функцiонування системи мiського громадського пасажирського транспорту Ключовi слова: система масового обслуговування, мiський громадський транспорт, розподiл вимог в груповiй заявцi, розподiл числа вiдмов UDC 519.

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Open networks of infinite server queues with non-homogeneous multivariate batch Poisson arrivals

Mehra,

Taylor

2023

Queueing Syst

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In this paper, we consider the occupancy distribution for an open network of infinite server queues with multivariate batch arrivals following a non-homogeneous Poisson process, and general service time distributions. We derive a probability generating function for the transient occupancy distribution of the network and prove that it is necessary and sufficient for ergodicity that the expected occupancy time for each batch be finite. Further, we recover recurrence relations for the transient probability mass function formulated in terms of a distribution obtained by compounding the batch size with a multinomial distribution.

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On the distributions of infinite server queues with batch arrivals

Cited by 17 publications

References 33 publications

Non-Stationary Queues with Batch Arrivals

Non-Stationary Queues with Batch Arrivals

Development of a model of the service system of batch arrivals in the passengers flow of public transport

Open networks of infinite server queues with non-homogeneous multivariate batch Poisson arrivals

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