双阈值(m,N)-策略不中断单重休假M/G/1排队分析  

Analysis of M/G/1 Queue with Bi-level Threshold(m,N)-Policy and Uninterrupted Single Vacation

作　　者：高文萍 唐应辉 Wen Ping GAO;Ying Hui TANG(School of Mathematical Sciences,Sichuan Normal University,Chengdu 610068,P.R.China)

机构地区：[1]四川师范大学数学科学学院,成都610068

出　　处：《数学学报（中文版）》2023年第2期363-388,共26页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(71571127);四川师范大学学科建设专项项目(xkzx2021-04)。

摘　　要：研究具有启动时间、双阈值(m,N)-策略和服务员单重休假且休假不中断的M/G/1排队系统,其中当服务员休假转来时,如果系统中的顾客数不小于一个事先给定的正整数阈值m(m≥1),服务员就立即启动系统.系统启动完成后,如果系统中的顾客数不小于另一个事先给定的正整数阈值N(N≥m),服务员就立即开始服务直到系统再次变空.假定服务员的休假时间和系统的启动时间均为一般分布,使用更新过程理论、全概率分解技术和拉普拉斯变换工具,讨论了系统队长的瞬态分布和稳态分布,获得了系统队长的瞬态分布关于时间t的拉普拉斯变换表达式,进一步通过直接计算获得了系统队长的稳态分布的递推表达式,同时给出了稳态队长的随机分解结构和附加队长分布的显示表达式.最后,在建立系统费用模型的基础上,应用更新报酬过程理论导出了系统长期单位时间内期望费用的显示表达式,并通过数值实例确定了使得系统在长期单位时间内期望费用最小的最优控制策略(m^(*),N^(*)).This paper studies the M/G/1 queueing system with startup time,bi-level threshold(m,N)-policy and single server vacation without interruption.In this system,when the server is transferred on vacation,the server starts the system immediately if the number of waiting customers is no less than a given positive integer threshold m(m≥1),and when the system startup is complete,the server begins service immediately if the number of waiting customers is no less than another given positive integer threshold N(N≥m).Assume that the server’s vacation time and the startup time of the system follow general distributions,both the transient queue-length distribution and the steady-state queue-length distribution of the system are discussed by using the renewal process theory,the total probability decomposition technique and Laplace transform tool.The expressions of the Laplace transformation of the transient queue-length distribution with respect to time t are obtained.Furthermore,the recursive expressions of the steady-state queue-length distribution are derived by a direct calculation.Meanwhile,the stochastic decomposition structure of the steady-state queue-length and the explicit expression of the additional queue-length distribution are presented.Finally,the explicit expression of the long-run expected cost per unit time is derived under a given cost model.And the numerical example is given to determine the optimal control policy(m^(*),N^(*))for minimizing the long-run expected cost per unit time.

关 键 词：M/G/1排队 双阈值(m N)-策略 单重休假且休假不中断 

分 类 号：O226[理学—运筹学与控制论]