{"title":"Transient GI/MSP/1/N Queue.","authors":"Andrzej Chydzinski","doi":"10.3390/e26090807","DOIUrl":null,"url":null,"abstract":"<p><p>A non-zero correlation between service times can be encountered in many real queueing systems. An attractive model for correlated service times is the Markovian service process, because it offers powerful fitting capabilities combined with analytical tractability. In this paper, a transient study of the queue length in a model with MSP services and a general distribution of interarrival times is performed. In particular, two theorems are proven: one on the queue length distribution at a particular time <i>t</i>, where <i>t</i> can be arbitrarily small or large, and another on the mean queue length at <i>t</i>. In addition to the theorems, multiple numerical examples are provided. They illustrate the development over time of the mean queue length and the standard deviation, along with the complete distribution, depending on the service correlation strength, initial system conditions, and the interarrival time variance.</p>","PeriodicalId":11694,"journal":{"name":"Entropy","volume":"26 9","pages":""},"PeriodicalIF":2.1000,"publicationDate":"2024-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11431701/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Entropy","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.3390/e26090807","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
A non-zero correlation between service times can be encountered in many real queueing systems. An attractive model for correlated service times is the Markovian service process, because it offers powerful fitting capabilities combined with analytical tractability. In this paper, a transient study of the queue length in a model with MSP services and a general distribution of interarrival times is performed. In particular, two theorems are proven: one on the queue length distribution at a particular time t, where t can be arbitrarily small or large, and another on the mean queue length at t. In addition to the theorems, multiple numerical examples are provided. They illustrate the development over time of the mean queue length and the standard deviation, along with the complete distribution, depending on the service correlation strength, initial system conditions, and the interarrival time variance.
在许多实际排队系统中，服务时间之间可能存在非零相关性。马尔可夫服务过程是一个很有吸引力的相关服务时间模型，因为它具有强大的拟合能力和可分析性。本文对具有马尔可夫服务过程和一般到达时间分布的模型中的队列长度进行了瞬态研究。本文特别证明了两个定理：一个是特定时间 t（t 可以任意小或大）的队列长度分布定理，另一个是 t 时的平均队列长度定理。它们说明了队列平均长度和标准偏差以及完整分布随时间的变化，这取决于服务相关强度、初始系统条件和到达时间方差。
期刊介绍：
Entropy (ISSN 1099-4300), an international and interdisciplinary journal of entropy and information studies, publishes reviews, regular research papers and short notes. Our aim is to encourage scientists to publish as much as possible their theoretical and experimental details. There is no restriction on the length of the papers. If there are computation and the experiment, the details must be provided so that the results can be reproduced.
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