【作者】 王浩华；

【导师】 刘次华；

【作者基本信息】 华中科技大学 ， 概率论与数理统计， 2006， 硕士

【摘要】 经典的随机服务系统(也称排队论),起源于二十世纪,最初是丹麦数学家Erlang在利用数学方法研究电话时,发展出来的一套关于随机过程方面的理论,其后四五十年间,特别是二次世界大战以后得到迅猛的发展,成为应用概率论与随机运筹学中最有活力的研究课题之一.它不仅具有较为完备的体系,而且在军事、经济、生产、管理、交通等各个领域都有着广泛的应用.经典的随机服务系统模型,包括M/M/1,M/G/1,GI/G/1,GI/M/N等多种形式,以Kendall, Neuts等为代表的一大批概率和运筹专家对此模型作了深入的研究,并得到了一批令人惊喜的结果.近年来,在经典排队模型的研究基础上,许多学者开始对排队模型进行了各种推广,研究一些更为复杂的排队模型.这种推广主要有以下几个方面:1、对模型中的顾客到达和服务时间做更一般化的假设,如把到达过程假设为马氏更新过程、位相型到达过程等等;2、引入有优先权的顾客到达模型,研究具有优先权的不同类型顾客到达的排队系统;3、引入休假排队系统以及服务器可修排队系统等,通过各种假设研究更一般的模型.本文研究了若干类离散时间排队模型,包括在强占型优先机制下二状态顾客到达的排队、在批量情形下强占型优先机制下二状态顾客批量到达的排队和顾客到达过程依赖于队长的离散时间排队系统.本文第二章至第五章主要运用了概率分析的技巧和嵌入马尔可夫理论进行研究.嵌入马尔可夫链方法是Kendall首次提出,该方法的突出特点是随机点过程不必是马尔可夫过程,只要求在一系列的停时上具有马尔可夫性,扩展了经典排队论中对随机点过程的要求.本文共分五章:第一章引言部分介绍了研究背景、排队论的发展、离散时间排队的有关结果以及本文的主要工作等;第二章介绍了处理排队模型中常用的一些方法其中包括嵌入马尔可夫链、补充变量法和矩阵解析法;第三、四章就离散情形下针对强占优先机制下的两类不同顾客到达系统的排队模型进行了分析得到了各自的队长分布、等待时间和平均忙期等排队指标,并与经典模型进行了对比;第五章对来到过程依赖于队长的离散时间排队系统进行了分析,得到了一些有益的结论.更多还原

【Abstract】 Classical random service system (queuing system), originated from the research of the telephone communication using the mathematic theory by Danish mathematician Erlang at beginning of the 20th century. The theory of random service system has acquired enormous development in past years, and became one of the most vigorous research fields in applying probability and the random operations research. Not only does it have a comparatively complete theoretical system, but also has extensive application in such each field as military, economy, production, management, traffic, etc. Classical random service system, including the forms such as M/M/1, M/G/1,GI/G/1, GI/M/n, has been made deep research by many experts whose majors are the applying probability and the random operations and got many perfect results.In recent years, a lot of scholars began to extend the classical models to the other research fields through various ways and studied some more complicated models on the basis of classical queuing theories. These amplification includes the following several respect mainly: Firstly, it is considered that the general assumption the customers’arrival and the service time of the queue models. For example, some scholars suppose the arrival process of the customers as Markov renew process or PH process; Secondly, some scholars consider the queuing system with the priority classes of customers. At last, the vacation and repairable system is also considered as a part of complicated queuing system.This paper studies various of discrete-time queuing models, including two-state arrival in preemptive priority case, two-state bulk arrival in preemptive case and the discrete-time queue with arrival processes state dependent queue length. Approach used from chapter2 to chapter5 in this paper is mainly the theory of probability analysis and embedded Markov chains which is presented by Kendall. Main feature of embedded Markov chains is that the random point process needn’t be a Markovian process and only acquire Markovian quality in series of stopping time. That extent the require of random point process in the classical queuing model.Our research work include 5 chapters: in chapter1, we provide introduction and preliminary knowledge used in this paper, including research background, development of queuing theory, some results of discrete-time queue and main results of this paper. In chaper2, we introduce some methods on studying queuing models that include embedded Markov chains, supplemental variables method and matrix-geometric method. In chapter3 and chapter4, we discuss two different customers’arrival models in preemptive priority case, and it is given that the distribution of the queue length, the waiting time and busy time in the system. Then, we compare it with the classical models. In chapter5, we consider discrete-time queue with arrival process state dependent queue length and derive some perfect results.更多还原