Martingale Method Of Queuing Networks With Service Interruptions

Since the late 1960s, the modern martingale theory and theory of stochastic analysis has flourished. Among these the modern martingale theory has been applied to study the theory of stochastic point process and unique superiority has been shown in application. In recent years, as one of the powerful research tools, it disciplines in queuing theory, stochastic control and survival analysis etc. Therefore using the martingale theory to study stochastic process queuing theory has important theoretical significance and practical value.The simplest understanding of martingale methods in the application of queuing theory is finding out martingale, the special type of stochastic process from all queuing system then using the good properties of martingale to do deep discussion. Based on studying martingale method of queuing network this paper gets main conclusion as following:(1) Comprehensive exposition on the basic theory of space and provide continuous mapping theorem from this theory. Based on one-dimensional reflection map multidimensional reflection map definitions can be obtained according to using special network model. Meanwhile existence of the reflection map is shown and the topological various continuous results are listed.(2) Apply some of the modern martingale methods and the conclusions to the queuing theory, which includes the nonlinear filtering, martingale description, decomposition theorem for nonnegative submartingales convergence theorem, local square integral martingale and martingale representation etc. It is also shows how to connected the martingale and counting process in the queuing models and make the martingale method to research stochastic process of queuing network more direct.(3) This paper studies deeply two kinds of network queue model of stochastic fluid network model and stochastic fluid network model with service interruptions. The Main contents are combining martingale method and two kinds of network, analyzing the heavy traffic limits of queuing network using some important martingale method and martingale theory. First, using the definition of martingale clear that the queuing network of potential buffer-content process is a martingale. Then through the tectonic of queuing network of input process which can satisfy the decomposition theorem for nonnegative submartingales or meet the variation and covariation process. Finally using martingale convergence theorem combined with composite mapping and continuous mapping theorem obtains we want, and the result shows the feasibility and validity of the martingale method for queuing network.