Application Of Markov Skeleton Processes To GI/G/1 Queueing Systems

Queueing theory is a greatly important kind of application stochastic processes. The M/G/1 and GI/G/1 queueing systems are the most typical and the important queueing modes among all the queueing systems. In this dissertation, we first introduce the basic knowledge of Markov skeleton processes, the present condition of queueing theory, and the Markovization of queueing system and the criteria for several types of ergodicity. In the past, the research on queueing system had mostly focused on stability and distribution, that is, the common ergodicity. While recently, the Institute of Probability and Statistics in Central South University has undergone a systematic research on the instantaneous distribution and all kinds of ergodicity. Based on the research, this dissertation has studied the M/G/1 and GI/GI/1 queueing systems, and drew the following conclusions: Firstly, we present the equation which satisfies the transient distribution of the lengthof (L(T),θ1(t),θ2(t) for GI/G/1 queueing systems, and proves that the length(L(t),θ1(t),θ2(t)) for GI/G/1 queueing systems satisfy two types of equation, and aretheir minimal nonnegative solutions. Furthermore, we give out not only the expression of Laplace transformation, but also the expression of Laplace transformation of busy-idle period, as well as the equation that satisfies the waitingtime of (W(t),θ(t)).Secondly, for M/G/1 queueing system, we state the necessary and sufficient conditions of Harris ergodicity, 1-ergodicity, geometric ergodicity of the length of(L(t),θ(t)), and prove the length of (L(t),θ(t)) is not uniform ergodicity. Lastly, we put forward the limited distribution of the length (L(t), θ1 (t), θ2 (t)) and the waiting time of (W(t),θ(t)) of GI/G/1 queueing system, in the mean time, we obtain the conditions of several kind of ergodicity of the length (L(t),θ1(t),θ2(t))and the waiting time (W(t),0(t)).