导　　师： 赵敏智

学科专业： 070103

授予学位： 硕士

作　　者： ;

机构地区： 浙江大学

摘　　要： 设{Xn}n≥0为一Markov链,转移概率为pij=aj-/(i-1/)+1{j≥/(i-1/)+},/(?/)j,i≥0本文主要讨论这类排队模型常返Markov链的首回速度,通过对首回时的α阶矩有限性的讨论给出零常返、正常返的进一步的分类.本文分三章. 第一章为引言.介绍了Markov链的历史背景、回顾了常返,暂留等一些基本概念、给出了本文所研究的这类排队论的模型和相关结论、给出相关符号的释义和本文的主要结果. 第二章主要讨论{Xn}n≥0为零常返时的首回速度.用F/(t/),G/(t/)分别表示首回时的母函数和{an/)n≥0的母函数.首先我们用G/(t/)估计t趋于1时1-F/(t/)的渐近行为.然后给出首回时α阶矩有限的充要条件.最后我们给出下面两个推论：若Gn/(1/)<∞,则首回时的α阶矩有限当且仅当a<1//2；若t→1时,1-G'/(t/)～/(1-t/)β,首回时的α阶矩有限当且仅当α<1//β+1.这说明了当t→1时,1-G'/(t/)趋于。的速度越快,则首次返回的速度就越慢. 第三章主要讨论了{Xn}n≥0为正常返时的首回速度.首先我们给出F/(k/)<∞当且仅当G/(k/)<∞,然后由此我们证明了首回时的整数阶矩有限当且仅当分布{an}n≥0的整数阶矩有限,最后我们给出更一般的结果：首回时的任意α>0阶矩的有限性与分布{an}n≥0的α阶矩的有限性是一致的. Let {Xn}n≥0 be a Markov chain with transition probability pij= aj-/(i-1/)+1{j≥/(i-1/)},/(?/)j,i≥0. We will research the first returning speed of recurrent Markov Chains of the queuing model as{Xn}n≥0, and give a further classification of null recurrent and positive recurrent by the finiteness of a order moment. This dissertation consists of three chapters. Chapter One is the preface. In this chapter, we will review the history of Markov chains and some basic concepts such as recurrent and transient. Then we will introduce the background of the queuing model and the main results about it. At last, we will give the main results of this dissertation. In Chapter Two, we will research the first returning speed of{Xn}n≥0 when it is null recurrent. Let F/(t/) and G/(t/) be the generating function of the first returning time and distribution of {an}n≥0 respectively. First, we estimate the asymptotic behavior of 1-F/(t/) whent→1.Then we give a necessary and sufficient condition of whether the a order moment of the first returning time is finite. At last we give two inferences: ifG'/(1/)<∞, then the a order moment of the first returning time is finite if and only ifα<1//2. If whent→1,1-G'/(t/)-/(1-t/)β, then the a order moment of the first returning time is finite if and only ifα<1///(β+1/). That is to say whent→1, the faster 1-G /(t/) tends to 0, the slower the first returning is. In Chapter Three, we will research the first returning speed of{Xn}n≥0 when it is positive recurrent. First we get the conclusion that F/(k/)< oo if and only if G/(k/)<∞, and then we prove that the integer-order moment of the first returning time is finite if and only if the integer-order moment of distribution of {an}n≥0 is finite, at last we get a more general conclusion that for any a> 0, the finiteness of the a order moment of the first returning time is consistent with the finiteness of theαorder moment of distribution of {an}n≥0.

关 键 词： 常返 暂留 模型 首回时

领　　域： [理学] [理学]