A simple derivation of the waiting-time distribution (in the queue) for the bulk-service \(M/G^{(a,b)}/1\) queueing system

IF 4.5 3区 管理学 Q1 OPERATIONS RESEARCH & MANAGEMENT SCIENCE
Mohan Chaudhry, A. D. Banik, Soumyajit Dev, Sitaram Barik
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引用次数: 0

Abstract

This paper deals with a Poisson input infinite-buffer single-server queue, where the arrivals occur in singles and the server serves the customers in batches. The server serves customers in batches of maximum size “b” with a minimum threshold size “a”. The service time of each batch follows general distribution (including heavy-tailed distribution) independent of each other as well as of the arrival process. The probability generating function (pgf) of the queue-length distributions at an arbitrary epoch as well as at a post-departure epoch of a batch have been derived using the embedded Markov chain and the argument of the rate-in and rate-out principle. The Laplace-Stieltjes transform (LST) of the actual waiting-time distribution (in the queue) of a random customer has also been derived using functional relation between pgf’s. The proposed analysis is based on the roots of the characteristic equation associated with the LST of the waiting-time distribution (in the queue) of a random customer. Using LSTs, the closed-form expressions for the probability density functions and for an arbitrary number of moments of the waiting-time distributions have been presented. We have also done numerical implementation of this procedure for the case of a bulk service infinite-buffer queueing model, and obtained the probability density function for waiting-time distribution of a random customer in the queue.

大容量服务\(M/G^{(a,b)}/1\)排队系统的等待时间分布(在队列中)的简单推导
本文研究了一个泊松输入无限缓冲单服务器队列,该队列的到货是单个的,而服务端是批量的。服务器以最大大小为“b”的批量为客户提供服务,最小阈值大小为“a”。每批服务时间遵循相互独立的一般分布(包括重尾分布),也与到货过程无关。利用嵌入马尔可夫链和进出率原理的参数,导出了批处理在任意时刻和出发后时刻的队列长度分布的概率生成函数。利用pgf之间的函数关系,导出了随机顾客实际等待时间分布的Laplace-Stieltjes变换(LST)。所提出的分析是基于与随机客户的等待时间分布(在队列中)的LST相关的特征方程的根。利用lst,给出了概率密度函数和任意矩数等待时间分布的封闭表达式。本文还对大容量服务无限缓冲排队模型进行了数值实现,得到了随机顾客在队列中等待时间分布的概率密度函数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Annals of Operations Research
Annals of Operations Research 管理科学-运筹学与管理科学
CiteScore
7.90
自引率
16.70%
发文量
596
审稿时长
8.4 months
期刊介绍: The Annals of Operations Research publishes peer-reviewed original articles dealing with key aspects of operations research, including theory, practice, and computation. The journal publishes full-length research articles, short notes, expositions and surveys, reports on computational studies, and case studies that present new and innovative practical applications. In addition to regular issues, the journal publishes periodic special volumes that focus on defined fields of operations research, ranging from the highly theoretical to the algorithmic and the applied. These volumes have one or more Guest Editors who are responsible for collecting the papers and overseeing the refereeing process.
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