在Min(N,V)-策略下有Bernoulli中断休假和启动故障的M/(G_(1),G_(2))/1可修排队  

An M/(G_(1), G_(2))/1 Repairable Queue with Bernoulli Interrupting Vacations and Starting Failures Under Min(N, V)-Policy

作　　者：何亚兴 唐应辉 HE Yaxing;TANG Yinghui(School of Mathematical Sciences,Sichuan Normal University,Chengdu 610068)

机构地区：[1]四川师范大学数学科学学院,成都610068

出　　处：《系统科学与数学》2024年第6期1744-1764,共21页Journal of Systems Science and Mathematical Sciences

基　　金：国家自然科学基金项目(71571127);四川师范大学学科建设专项项目(XKZX2021-04);河西学院校长基金创新团队项目(CXTD2022013)资助课题。

摘　　要：文章研究了在Min(N,V)-策略控制下有Bernoulli中断休假和启动故障且提供两阶段服务的M/(G_(1),G_(2))/1可修排队系统,其中每当系统变空时服务员就去休假,如果在休假中到达系统的顾客数达到N个时,服务员以概率p (0≤p≤1)中断休假,以概率1-p不中断休假,而且服务台在其忙期和闲期均可能发生故障.运用更新过程理论、全概率分解技术和拉普拉斯变换,讨论了系统在任意时刻队长的瞬态和稳态分布,获得了队长瞬态分布关于时间t的拉普拉斯变换表达式,以及稳态队长分布的递推表达式,同时求出了系统一些重要排队性能指标.最后,从经济效益角度出发讨论了系统的费用优化问题,并应用更新报酬定理得到了系统长期单位时间内期望费用的显示表达式,然后在(没有)平均等待时间约束下,及服务时间和修理时间服从PH分布时,通过数值实例讨论了使得系统期望费用最小的一维最优控制策略N*,以及二维最优控制策略(N*,T*),并讨论了参数p对最优控制策略的影响.This paper examines an M/(G_(1),G_(2))/1 repairable queuing system with Bernoulli interrupting vacations,starting failures and providing two phases of services under Min(N,V)-policy control,where the server goes on vacation once the system becomes empty,and if the number of customers arriving at the system during the vacation reaches N,the server interrupts the vacation with probability p(0≤p≤1),or continue the vacation with probability 1-p,and the service station may fail during its busy and idle periods.The transient and stationary distributions of the queue size at any time are discussed by using the renewal process theory,the total probability decomposition technique and Laplace transform.We obtain Laplace transform expressions of transient queue size distribution with respect to t,and the explicit expressions for the probability generating function of the stationary queue size distribution.In addition,some other important performance measures are also discussed.Finally,the system cost optimization problem is discussed from an economic perspective.And employing the renewal reward theory,the explicit expression for the long-run expected cost per unit time is given.Then,when the service time and the repair time obey the PH distributions,the one-dimensional optimal control policy N*and the two-dimensional optimal control policy(N*,T*)with(without)expected waiting time constraint to minimize the expected system cost are investigated through numerical examples,and the effect of the parameter p on the optimal control policies is discussed.

关 键 词：Min(N V)-策略 启动故障 队长分布 最优控制策略 

分 类 号：O226[理学—运筹学与控制论]