摘要
随着1991年Gelenb首次提出“负顾客”的概念,带有正顾客和“负顾客”的G-网络得到了广大学者的关注。近年来,人们从不同的角度对G-网也做出了大量的研究,其主要应用在神经网络系统、遗传学管理系统、电脑安全系统等。1939年,法国概率学家Levy首先提出了鞅的概念,此后美国概率学家Doob将鞅的概念进行了推广,提出了下鞅的概念,并对它做了很多系统性的研究。Burkholder D.L.,Meyer P.等在此基础之上,又进行了一系列的研究工作,从而就形成了近代鞅理论。随着鞅理论在概率论和随机过程中地位的日益突出,鞅方法已经成为研究随机过程和排队论的强有力工具。在本篇文章中主要运用鞅方法来研究高负荷条件下G-网队长过程的扩散逼近。我们首先给出鞅的有关知识,并在此基础之上应用非负下鞅的Doob-Meyer分解定理构造出两类特殊的G-网模型:二阶串联系统和二阶并联系统队长过程的鞅分解表示,进而给出其刻画过程的鞅分解表示,然后再运用概率测度收敛和随机过程极限的有关知识来研究其扩散逼近过程。并且将其结论推广到一般G-网模型,给出一般G-网模型中队长过程在高负荷条件下的扩散逼近过程,表明了鞅方法在研究G-网模型时的可行性和有效性。在本篇论文的最后一章也将给出二阶串联系统队长过程的数值模拟仿真,并画出其图像,验证了二阶串联系统的高负荷极限定理。
The concept of 'negative customer' was introduced for the first time in 1991, from then on, the G-networks with positive customers and 'negative customers' became the focus of the scholars. In recent years, G-networks have been extensively studied from different directions, including neural network modeling, computer networks, genetic regulatory systems, and so on. In 1939, the probability of scientist Levy gave the concept of martingale. Then Doob developed that concept, and introduced the concept of sub-martingale. Based on that, Burkholder D.L. and Meyer P. made a series of studies, contributing to the formation of the modern martingale theory. With the increasingly prominent status of martingale theory in probability and stochastic processes, martingale method has become a powerful tool in stochastic processes and queuing theory. In this paper, we study in the diffusion approximation of state-dependent G-networks under heavy traffic. At first, we we'd like to give the associated knowledge of martingale. And using Doob-Meyer decomposition for nonnegative sub-martingales, we construct the martingale decomposition of two types of networks, including two queues in tandem and two-layer network. Then we will present the martingale representation for the scaled queuelength processes. On that basis, we want to study the diffusion approximation processes of queuelength processes in the two models, using the knowledge in convergence of probability measures and stochastic-process limits. What's more, we will present the heavy-traffic limits of queuelength process, which applies the conclusions of the two special models. According to that, we show the feasibility and validity of the martingale method in queuing network. At the last chapter of this paper, we will simulate the queuelength process in the model of two queues in tandem and draw the images, which verifies the heavy traffic limit theorem of two queues in tandem.
引文
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