Mohan Chaudhry, A. D. Banik, Soumyajit Dev, Sitaram Barik
{"title":"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","authors":"Mohan Chaudhry, A. D. Banik, Soumyajit Dev, Sitaram Barik","doi":"10.1007/s10479-025-06765-8","DOIUrl":null,"url":null,"abstract":"<div><p>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 “<i>b</i>” with a minimum threshold size “<i>a</i>”. 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.</p></div>","PeriodicalId":8215,"journal":{"name":"Annals of Operations Research","volume":"352 1-2","pages":"1 - 24"},"PeriodicalIF":4.5000,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A simple derivation of the waiting-time distribution (in the queue) for the bulk-service \\\\(M/G^{(a,b)}/1\\\\) queueing system\",\"authors\":\"Mohan Chaudhry, A. D. Banik, Soumyajit Dev, Sitaram Barik\",\"doi\":\"10.1007/s10479-025-06765-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>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 “<i>b</i>” with a minimum threshold size “<i>a</i>”. 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.</p></div>\",\"PeriodicalId\":8215,\"journal\":{\"name\":\"Annals of Operations Research\",\"volume\":\"352 1-2\",\"pages\":\"1 - 24\"},\"PeriodicalIF\":4.5000,\"publicationDate\":\"2025-08-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annals of Operations Research\",\"FirstCategoryId\":\"91\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10479-025-06765-8\",\"RegionNum\":3,\"RegionCategory\":\"管理学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"OPERATIONS RESEARCH & MANAGEMENT SCIENCE\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Operations Research","FirstCategoryId":"91","ListUrlMain":"https://link.springer.com/article/10.1007/s10479-025-06765-8","RegionNum":3,"RegionCategory":"管理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"OPERATIONS RESEARCH & MANAGEMENT SCIENCE","Score":null,"Total":0}
A simple derivation of the waiting-time distribution (in the queue) for the bulk-service \(M/G^{(a,b)}/1\) queueing system
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.
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.