在双水平(p,N_(1),N_(2))-策略下随机启动时间M/G/1排队的性能与约束优化

Performance and Constraint Optimization of M/G/1 Queue with Randomized Startup Time and the Bi-Level (p,N_(1),N_(2))-Policy

文章考虑在双水平(p,N_(1),N_(2))-策略控制下有随机启动时间的M/G/1排队系统,其中双水平(p,N_(1),N_(2))-策略是指当系统中的顾客数达到一个事先给定的低阈值N_(1)(≥1)时,服务员以概率p(0≤p≤1)启动系统,以概率1-p不启动系统直到系统中的顾客数累计达到另一个事先设定的较高阈值N_(2)(N_(2)≥N_(1))个时才启动系统,同时在一段随机长度的启动时间完成后服务员才开始为顾客服务.运用更新过程理论、全概率分解分析方法和拉普拉斯变换工具,研究了系统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,同时求出了系统其他一些重要排队性能指标.进一步,通过数值实例说明了稳态队长分布表达式在系统容量设计中的重要应用.最后,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,并在平均等待时间约束下当服务时间和启动时间服从PH分布时,通过数值实例讨论了使得系统期望费用最小的最优双水平控制策略(N_(1)^(*),N_(2)^(*)),以及参数p对系统期望费用和顾客平均等待时间的影响.

This paper considers an M/G/1 queue with randomized startup time under the control of the bi-level(p,N_(1),N_(2))-policy,where the bi-level(p,N_(1),N_(2))-polic means that if the number of customs in the system is equal to a given low intege threshold value Ni(≥1),the server starts the system with probability p(0≤p≤1)or still left with probability 1-p.When the number of customs in the system reaches a higher integer threshold value N_(2)(N_(2)≥N_(1)),the server starts the system immediately.Meanwhile,it takes a random length of time to start the system.When the system startup is completed,the server begins service immediately.Using the renewal process theory,the total probability decomposition technique and the Laplace transformation,we discuss the transient and steady-state distributions of the queue size.Both the expressions of the Laplace transform of the transient queue-length distribution with respect to time t and the recursive expressions of the steady-state queue length distribution are obtained.Meanwhile,some other important queueing performance indicators of the system are derived.Furthermore,we illustrate the important application of the recursive expressions of the steady-state queue length distribution in the system capacity design by a numerical example.Finally,employing the renewal reward theorem,the recursive expression of the long-run expected cost per unit time of the system is obtained.Moreover,under assuming that the service time and set-up time obey PH distributions and the limit of the average waiting time of customer,numerical examples are presented to discuss the bi-level optimal control strategy(N_(1)^(*),N_(2)^(*)) for minimizing the long-run expected cost per unit time of the system as well as the influence of parameter p on the long-run expected cost per unit time of the system and expected waiting time of customer.