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Batch sojourn time in the processor sharing queue with geometric batch size
Stochastic Models ( IF 0.7 ) Pub Date : 2021-04-23 , DOI: 10.1080/15326349.2021.1913422
Fabrice Guillemin 1 , Alain Simonian 2 , Ridha Nasri 2 , Veronica Quintuna Rodriguez 1
Affiliation  

Abstract

In this paper, we analyze the sojourn time of an entire batch in a M[X]/M/1 processor sharing queue, where geometrically distributed batches arrive according to a Poisson process and individual jobs require exponentially distributed service times. By conditioning on the number of jobs in the queue and the number of jobs in a tagged batch, we establish recurrence relations for conditional sojourn times, which subsequently allow us to derive a partial differential equation for an associated bivariate generating function. This equation involves an unknown generating function, whose series expansion can be computed by solving an infinite lower triangular linear system. Once this unknown function is determined, we determine the Laplace transform and the mean value of the sojourn time of a batch in the system.



中文翻译:

具有几何批量大小的处理器共享队列中的批量逗留时间

摘要

在本文中,我们分析了整个批次在一个 [X]//1处理器共享队列,其中几何分布的批次根据泊松过程到达,并且单个作业需要呈指数分布的服务时间。通过以队列中的作业数量和标记批次中的作业数量为条件,我们建立了条件逗留时间的递推关系,这随后允许我们推导出相关双变量生成函数的偏微分方程。该方程涉及一个未知的生成函数,其级数展开可以通过求解无限下三角线性系统来计算。一旦确定了这个未知函数,我们就确定了拉普拉斯变换和系统中批次逗留时间的平均值。

更新日期:2021-04-23
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