具有多级适应性休假和Min(N,V)-策略的Geo=G=1排队系统分析

Analysis of Geo=G=1 Queueing System with Multiple Adaptive Vacations and Min(N,V)-Policy

本文研究具有多级适应性休假和Min(N,V)-策略的Geo/G/1排队系统,运用更新过程理论,全概率分解技术和z-变换工具,分析在时刻点n^(+)的队长瞬态概率分布的z-变换表达式.此外,基于瞬态分析,使用洛必达法则得到系统在时刻点n+的队长稳态分布的递推表达式,然后给出稳态队长分布的概率母函数以及平均稳态队长的表达式.接着,探讨在不同时刻点n^(-)、n和外部观察点的稳态概率分布.最后,在建立费用结构模型下,由更新报酬过程理论获得系统长期运行单位时间内所产生的成本期望费用的表达式,并通过数值实例讨论最优控制策略N^(*)以及当连续休假次数为正整数J时的二维最优控制策略(N^(*),J^(*)).

This paper treats the Geo=G=1 queueing system with multiple adaptive vacations and Min(N,V)-policy.Employing the renewal process theory,total probability decomposition technique and z-transform,we analyze the transient distribution of the queue size at any epoch n^(+).The z-transform expressions for the transient queue size distribution at any epoch n+are obtained.Moreover,in the analysis of transient analysis,the recursive expressions of the steady-state queue length distribution at any epoch n+are explicitly derived by using L’Hospital rule.Then,the explicit expressions of its probability generating function of the steady-state queue size distribution and the expected queue size are presented.Next,the steady-state distribution of queue length at epochs n-,n and outside observer’s observation epoch are explored.Finally,applying the renewal reward theorem,the explicit expression of the longrun expected cost per unit time is also presented under a given cost structure.And numerical examples are provided to determine the optimal control policy N^(*)for economizing the system cost as well as the combined control policy(N^(*),J^(*))when the number of consecutive vacations is a fixed positive integer J.