基于多重工作休假的成批到达离散时间排队的性能分析

ANALYSIS BASED ON DISCRETE TIME BULK INPUT QUEUE WITH MULTIPLE WORKING VACATIONS

研究了一个成批到达的离散时间Geom^([X])/Geom/1多重工作休假排队.首先,建立了模型的二维马尔可夫链,利用矩阵分析的方法,导出了稳态队长复杂的概率母函数.其次,为了展示此模型与经典无休假Geom^([X])/Geom/1排队的联系,给出稳态队长的随机分解结果.尤其重要的是,发现了条件负二项分布的双参数加法定理,利用这些结论,得到了矩母函数序下的稳态等待时间的上下界.进一步,求出了平均队长和平均等待时间的上下界.最后,提出一些数值例子以验证结论.

This paper is concerned with a bulk input Geom^[X]/Geom/l queue with multiple working vacations.Firstly a two-dimensional Markov chain model is established,and with the matrix analysis method,highly complicated PGF of the stationary queue size is derived, from which the stochastic decomposition result for the PGF of the stationary queue size is obtained,which indicates the evident relationship with that of the classical Geom^[X]/Geom/l queue without vacation.The biparameter addition theorems for the conditional negative binomial distribution is established,with which the upper bound and the lower bound of the stationary waiting time is given in the moment generating function order.Furthermore,the mean queue size,the upper bound and the lower bound of the mean waiting time are obtained. Finally,some numerical examples are presented.