有限容量多服务员排队系统全忙期分布函数的一种直观计算途径  

An Intuitive Numerical Approach to Compute the Full Busy Period Distribution of a Finite Capacity Multi-server Queue

作　　者：余玅妙 唐建芳 YU Miaomiao;TANG Jianfang(School of Mathematical Science,Sichuan Normal University,Chengdu 610066;School of Mathematics and Statistics,Sichuan University of Science and Engineering,Zigong 643000)

机构地区：[1]四川师范大学数学科学学院,成都610066 [2]四川轻化工大学数学与统计学院,自贡643000

出　　处：《工程数学学报》2023年第6期991-1000,共10页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(71571127);四川师范大学学科建设专项基金(XKZX2021-04).

摘　　要：排队系统忙期长度是衡量系统运行效率的重要性能指标,它在与排队系统相关的优化问题建模中有着广泛而重要的应用。受数学方法的制约,一段时间以来多服务员有限缓冲空间排队系统的忙期概率分布函数未得到充分的重视与研究。依托马氏多服务员有限缓冲空间排队系统的子全忙期分布及与其相关联的有限缓冲空间排队系统全忙期分布之间的重要关系,给出了寻求马氏多服务员有限容量排队系统服务员全忙期分布函数Laplace-Stieltjes变换的一个迭代计算公式。同时,为了计算该全忙期分布函数在任意给定时刻处的函数值,利用有理函数部分分式分解方法和极点处的留数计算规则对上述Laplace-Stieltjes变换进行了从s域到时间域的反演,并给出了基于变换特征根的半解析化全忙期分布函数反演公式。使用Mathematica软件编程,在数值算例中通过与现有文献结果的对比,证实了所提出分析方法的可行性与简洁性。这为实际工程应用提供了一种较为直观且操作性较强的多服务台有限缓冲空间排队系统全忙期分布函数的计算方法。The length of the busy period for a multi-server queueing system is a key performance measure.It has important and wide applications in optimization problems related to queueing systems.Limited by mathematical methods,the busy period probability distribution function of multi-server finite-buffer queueing systems has not been sufficiently investigated for a long time.Using an importantly quantitative relationship between the sub-full busy period distribution of a multi-server finite-buffer Markovian queue and the full-busy period distribution of its corresponding finite waiting space queue,an elegant recursive formula for finding the Laplace-Stieltjes transform of the full busy period distribution of a multi-server Markovian queue with finite capacity is presented.Meanwhile,to calculate the full-busy period distribution function for any given time,the above Laplace-Stieltjes transform in the s-domain can be inverted into the time-domain by using the partial fractional decomposition method of rational functions and the rules for calculating residues.A semi-analytic inversion formula for the full busy period distribution function based on the characteristic roots of transform is also presented.Programming with Mathematica and comparing results with the exiting literature in the numerical examples validate the feasibility and simplicity of the proposed approach.This work provides a more intuitive and easy-to-implement way to determine the full-busy distribution function of a multi-server queueing system with finite buffer space.

关 键 词：忙期 多服务员排队 拉普拉斯–斯蒂尔切斯变换 特征根 留数 

分 类 号：O226[理学—运筹学与控制论]