Diffusion Approximation Of G-networks In Heavy Traffic By Martingale Method

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.