Continuous approximation of $ M_t/M_t/ 1 $ distributions with application to production

IF 1 Q3 Engineering

Journal of Computational Dynamics Pub Date : 2018-07-18 DOI:10.3934/jcd.2020010

D. Armbruster, Simone Gottlich, S. Knapp

引用次数: 1

Abstract

A single queueing system with time-dependent exponentially distributed arrival processes and exponential machine processes (Kendall notation $M_t/M_t/1$) is analyzed. Modeling the time evolution for the discrete queue-length distribution by a continuous drift-diffusion process a Smoluchowski equation on the half space is derived approximating the forward Kolmogorov equations. The approximate model is analyzed and validated, showing excellent agreement for the probabilities of all queue lengths and for all queuing utilizations, including ones that are very small and some that are significantly larger than one. Having an excellent approximation for the probability of an empty queue generates an approximation of the expected outflow of the queueing system. Comparisons to several well-established approximation from the literature show significant improvements in several numerical examples.

查看原文本刊更多论文

$ M_t/M_t/ 1 $分布的连续逼近及其在生产中的应用

分析了一类具有时间依赖指数分布到达过程和指数机过程(Kendall符号$M_t/M_t/1$)的单排队系统。用连续漂移扩散过程对离散队列长度分布的时间演化进行建模，导出了一个半空间上近似于正向Kolmogorov方程的Smoluchowski方程。对近似模型进行了分析和验证，显示出所有队列长度和所有队列利用率的概率非常一致，包括非常小的队列利用率和明显大于1的队列利用率。如果对空队列的概率有一个很好的近似值，就可以生成排队系统预期流出量的近似值。与文献中几个已建立的近似相比较，在几个数值例子中显示出显著的改进。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Journal of Computational Dynamics Engineering-Computational Mechanics

CiteScore

2.30

自引率

10.00%

发文量

期刊介绍： JCD is focused on the intersection of computation with deterministic and stochastic dynamics. The mission of the journal is to publish papers that explore new computational methods for analyzing dynamic problems or use novel dynamical methods to improve computation. The subject matter of JCD includes both fundamental mathematical contributions and applications to problems from science and engineering. A non-exhaustive list of topics includes * Computation of phase-space structures and bifurcations * Multi-time-scale methods * Structure-preserving integration * Nonlinear and stochastic model reduction * Set-valued numerical techniques * Network and distributed dynamics JCD includes both original research and survey papers that give a detailed and illuminating treatment of an important area of current interest. The editorial board of JCD consists of world-leading researchers from mathematics, engineering, and science, all of whom are experts in both computational methods and the theory of dynamical systems.