Analytically Explicit Results for the Distribution of the Number of Customers Served during a Busy Period for Special Cases of the M/G/1 Queue

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY Accounts of Chemical Research Pub Date : 2019-08-27 DOI:10.1155/2019/7398658

M. Chaudhry, V. Goswami

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引用次数: 2

Abstract

This paper presents analytically explicit results for the distribution of the number of customers served during a busy period for special cases of the M/G/1 queues when initiated with m customers. The functional equation for the Laplace transform of the number of customers served during a busy period is widely known, but several researchers state that, in general, it is not easy to invert it except for some simple cases such as M/M/1 and M/D/1 queues. Using the Lagrange inversion theorem, we give an elegant solution to this equation. We obtain the distribution of the number of customers served during a busy period for various service-time distributions such as exponential, deterministic, Erlang-k, gamma, chi-square, inverse Gaussian, generalized Erlang, matrix exponential, hyperexponential, uniform, Coxian, phase-type, Markov-modulated Poisson process, and interrupted Poisson process. Further, we also provide computational results using our method. The derivations are very fast and robust due to the lucidity of the expressions.

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M/G/1队列特殊情况下繁忙时段服务人数分布的解析显式结果

本文给出了M/G/1队列在有M个顾客时的特殊情况下繁忙时段服务顾客数分布的解析式结果。在繁忙时段服务的顾客数量的拉普拉斯变换的函数方程是众所周知的，但一些研究人员指出，一般来说，除了一些简单的情况，如M/M/1和M/D/1队列，它不容易反演。利用拉格朗日反演定理，给出了该方程的一个优美解。我们得到了各种服务时间分布，如指数分布、确定性分布、Erlang-k分布、gamma分布、卡方分布、逆高斯分布、广义Erlang分布、矩阵指数分布、超指数分布、均匀分布、Coxian分布、相型分布、马尔可夫调制泊松过程分布和中断泊松过程分布。此外，我们还使用我们的方法给出了计算结果。由于表达式的清晰性，推导非常快速和健壮。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.