Queueing Models With Feedback In Network Traffic

The current world is a universal communicative one which depends mostly on the development of network technology and the network technology is also developing rapidly. The ATM Technique has brought a revolution in exchanging manner for the LAN and Internet network. The large communications based on the ATM technique play the important role in the Board-band, integrated and intellectualized network, which brings a challenge for the questions of how to improve the quality of service and how to reduce the bursty in network traffic. Furthermore, it is also a grate chance for applying the queueing theorems to the research field of practical network.Once the transmission's error or data packets' loss occurred, the terminal user might bring forward new service requirements, which exists widely in network traffic. The queueing models with feedback are investigated in the different perspective since being brought forward by Takacs in 1963.By constructing the QBD process, we studied the classical queueing models with feedback in detail. A new strategy is put forward. Comparing with the traditional strategy, we find that the feedback priority strategy can improve the quality of service in network traffic. Moreover. A tandem queue with feedback is discussed. Some new interesting perspectives are discovered applying the perturbed Chebyshelv polynomials. The density evolution method is used to handle the self-similar queueing models. The analysis to the tail behavior of queueing length will guide the design of buffer capacity in network traffic. The main contents and results of this paper can be described as follows.(1) Overviews of the main results, methods and developments of queuing theory. In the first part of the paper, we introduce some elementary theorems and resultsin queuing theory. Different approaches to set up queueing models are also briefly mentioned. For the convenience of the other parts some notations widely accepted in classic queueing theory are given there. The developments of network and ATM technology provide a variety background for the queuing theory, in which new phenomena appear and new methodology is required to solve these problem. We present the outline the studying methods widely used and results obtained in Self-similarity network traffic. Furthermore, the background and history of the queuing models with feedback in classic and self-similar queueing models are provided.(2) The feedback priority strategy is put forward, which can improve the quality of service by changing protocol of network.We analyze queuing models with feedback, such as M/M/1 , M/PH/1 . and MAP/PH /1, by constructing their corresponding QBD process. Performances for these models under different queuing discipline are calculated. Comparing thecalculating results we find that the feedback priority strategy is better than the traditional strategy. The result is useful for the drawing of network protocols in ATM.(3) A general condition for the separating theorem of nonlinear matrix equation is given.The rate matrix is important in QBD process. It determines the decay rate for stationary distribution. In general cases, it is difficult to give an explicit expression for the rate matrix. By analyzing the queuing models such as M/M/1, M/PH/1, and MAP/PH/1, queuing model with feedback, we deduce a general condition for the existence of a explicit expression for matrix R, i.e. if the matrix C can be dissolved into the production of a column and row vectors then we can obtain the explicit solution. Furthermore the explicit solution is also possible for the stationary distribution.(4) A simple solution to derive the decreasing rate of the stationary queueing length distribution is given for the M/PH/1 queuing model with feedback.In order to obtain the decreasing rate of the stationary distribution of the QBD process with finite phases, the usual way is to acquire the rate matrix R firstly, and then its spectral radius sp(R). Based on the known parameters, we can directly calculate the decreasing rate of t...