基于Min(N,D,V)-策略和单重休假且休假不中断的M/G/1排队系统

M/G/1 Queuing System Based on Min(N,D,V)-Policy and Single Server Vacation Without Interruption

研究在Min(N,D,V)-策略控制下服务员具有单重休假且休假不中断的M/G/1排队系统,其中服务员休假归来如果系统中的顾客数不小于事先给定的正整数阈值N或者系统中的顾客所需的服务时间总和不小于事先给定的另一正数阈值D,服务员就立即开始服务.应用更新过程理论,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,得到了队长的瞬态分布关于时间t的拉普拉斯变换表达式和稳态队长分布的递推表达式.进一步给出了稳态队长的随机分解结构和附加队长的显示表达式.同时,一些特殊情况,如N→∞,或D→∞,或p{V=0}=1等也进行了讨论,并指出了唐应辉和刘名武(2008)的错误.最后,建立费用结构模型,给出了系统长期运行单位时间内的成本期望费用的显示表达式,并通过数值计算例子讨论了使得系统长期运行单位时间内的成本期望费用最小的最优控制策略(N*,D*).

This paper discusses the M/G/1 queuing system for single server vacation without interruption under the control of Min(N,D,V)-policy,in which when the server is transferred on vacation,the server starts service immediately if the number of waiting customers is no less than a given positive integer threshold N or the total service times of waiting customers is no less than a given positive threshold D,whichever occurs first.Using the renewal process theory,the total probability decomposition technique and Laplace transform tool,we discuss the transient queue length distribution and the steady-state queue length distribution.Both the expressions of the Laplace transformation of the transient queue length distribution with respect to time t and the recursive expressions of the steady-state queue length distribution are obtained.Furthermore,the stochastic decomposition structure of the steady-state queue length and the explicit expression of the additional queue length distribution are presented.Meanwhile,some special cases,such as N→∞,or D→∞,or p{V= 0}= 1,and etc.,are discussed.The error in the Tang and Liu(2008) is also pointed.Finally,the explicit expression of the long-run expected cost rate is derived under a given cost structure.And by numerical calculation,we determine the optimal control policy(N*,D*) for minimizing the long-run expected cost rate.