Tel Aviv University School of Mathematical Science Asymptotic Methods for Queueing Systems and Networks with Application to Telecommunications

This dissertation is concerned with the study of non-Markovian queueing systems and networks, with applications to communication networks. The main contribution of the dissertation consists in deriving results for non-Markovian networks that have been obtained so far only for Markovian queueing networks. We study large closed client/server communication networks and losses in single-server queueing systems, with an application to communication networks of loss queues. We apply stochastic calculus and the theory of martingales to the case when one of the client stations is a bottleneck in the limit, where the total number of tasks in the server increases to infinity. The main results of this study are (i) an explicit expression for the interrelation between the limiting non-stationary distributions in non-bottleneck client stations; thus when one distribution is found in a simulation, the others can be computed; (ii) derivation of diffusion and fluid approximations for the non-Markovian queue length in the bottleneck client station. For the loss networks considered, we find an asymptotic expression for the loss probability and other performance measures, as buffer capacity increases to infinity. We also find the changes in the loss probability when redundant packets are added to the messages. The application of martingale methods for the study of the asymptotic behavior of non-Markovian queueing systems seems to be new.