{"title":"Group service system with three queues and load balancing","authors":"M. P. Savelov","doi":"10.1515/dma-2022-0019","DOIUrl":null,"url":null,"abstract":"Abstract A group service system for three queues is considered. At each time t = 1, 2, . . ., with some probability, a customer enters the system, selects randomly two queues, and goes to the shorter one. At each moment such that there is at least one customer in each queue, each queue performs instantly the service of one customer. By means of Lyapunov functions, a criterion for the ergodicity of the Markov chain corresponding to this queuing system is established. The limiting joint distribution of queue lengths is found, and the connection with the problem of balanced allocations of particles into cells is described. In the corresponding problem of balanced allocation of particles, the limiting distribution of the range is found, i. e. the difference between the maximal and minimal numbers of particles in cells.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/dma-2022-0019","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract A group service system for three queues is considered. At each time t = 1, 2, . . ., with some probability, a customer enters the system, selects randomly two queues, and goes to the shorter one. At each moment such that there is at least one customer in each queue, each queue performs instantly the service of one customer. By means of Lyapunov functions, a criterion for the ergodicity of the Markov chain corresponding to this queuing system is established. The limiting joint distribution of queue lengths is found, and the connection with the problem of balanced allocations of particles into cells is described. In the corresponding problem of balanced allocation of particles, the limiting distribution of the range is found, i. e. the difference between the maximal and minimal numbers of particles in cells.
期刊介绍:
The aim of this journal is to provide the latest information on the development of discrete mathematics in the former USSR to a world-wide readership. The journal will contain papers from the Russian-language journal Diskretnaya Matematika, the only journal of the Russian Academy of Sciences devoted to this field of mathematics. Discrete Mathematics and Applications will cover various subjects in the fields such as combinatorial analysis, graph theory, functional systems theory, cryptology, coding, probabilistic problems of discrete mathematics, algorithms and their complexity, combinatorial and computational problems of number theory and of algebra.