QUASI-BIRTH-DEATH PROCESSES AND THEIR USE IN THE MODELING AND ANALYSIS OF COMPUTER NETWORKS (MATRIX-GEOMETRIC, RECURSIVE FORM SOLUTIONS, CSMA/CD)
Abstract
Quasi-birth-death (QBD) processes are often very natural models of the activity in computer communication networks. For example, they provide a very good model of the behavior of carrier sense multiple access (CSMA/CD) networks. The success of the QBD model of CSMA/CD has led to further studies of CSMA/CD and, in turn, to a new study of QBD processes themselves. In the first part of this thesis, a QBD model of a CSMA/CD network is used to find various waiting time characteristics of the channel. First the caudal characteristic curve and throughput delay curves are computed. Then, the distribution of the delay suffered by a packet in a CSMA/CD network, the variance of that delay, and the rate of decay of the tail of the delay distribution are found for both the case in which users are served in random order and the first come first served service order case. These results are the tools that are needed to design CSMA/CD networks which must carry real time control information or speech. Next, a study of QBD process in general is presented. A state splitting technique is developed which leads to a closed form recursive solution for the steady state probability distribution of all QBD processes. A matrix W(i) is found in closed form for all QBD processes such that (pi)(i) = (pi)(i - 1) W(i). The relationship between this recursive solution and matrix-geometric solutions is also presented. The existence of such recursive equations should allow more complete analysis of networks modeled as QBD processes. It should also lead to more theoretical studies of QBD processes themselves. Finally, the state splitting technique is applied to a two dimensional QBD process. This allows the stationary probabilities of the non-boundary states of the QBD process to be written in terms of just the boundary states. This is accomplished using a recursive technique, which is an alternative to the product form solution method. From there, it should be possible to recursively compute the stationary probabilities of all of the states in the process.
Degree
Ph.D.
Subject Area
Electrical engineering
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