A Class Of Queuing Models Are Often Back To Back Speed Of The Markov Chain

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) whentâ†'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 whentâ†'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 whentâ†'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.