Mathematical Analysis of Stochastic Networks

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0406191Ramanan The main objective of this project is to extend the general theory of deterministic and stochastic constrained processes that have discontinuous dynamics and to apply it to gain better insight into the performance and control of stochastic networks. The project has four components. The first part studies reflected diffusions that arise as approximations to multi-class stochastic networks with cooperative servers (i.e. servers that use all the excess capacity allocated to one class in order to serve other classes). The estimation of functionals of these diffusions is complicated by the fact that these diffusions often fail to be semimartingales. The current project describes new techniques for the pathwise analysis of these diffusions. The second part of the research aims to first establish general sufficient conditions for uniqueness of differential equations that have discontinuous drifts and to then use these conditions to characterize fluid limits of queueing networks that exhibit discontinuous transition mechanisms in the interior of their domains. The third part is devoted to the analysis of phase transitions and Gibbs measures associated with a class of stochastic loss networks that appear as models of multicasting in telecommunication networks. The last part focuses on the design of optimal controls for a class of non-stationary queueing networks. Over the last decade the increasing complexity of telecommunication, computer and manufacturing systems has emphasized the need for the systematic development of tools for their analysis, design and control. Since an exact analysis of these stochastic systems is often infeasible one resorts to an asymptotic analysis that is appropriate for the performance measure of interest. Three asymptotic regimes of interest are the fluid regime, which captures the mean behavior of the system, the diffusion regime, which describes stochastic variations around the mean and the large deviations regime, which analyzes probabilities of events that are rare, but of crucial importance to the operation of the network (such as, for example, data loss in a computer network). This project will develop tools that facilitate the asymptotic analysis (in each of the three regimes) of important classes of stochastic networks that arise naturally in applications. In particular, the tools developed are expected to be useful for the estimation of important network performance measures such as stability and buffer overflow probabilities and also for the design of optimal controls.