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The Kardar-Parisi-Zhang (KPZ) equation is a stochastic PDE describing various objects in statistical mechanics such as random interface growth, directed polymers, interacting particle systems. We study large deviations of the KPZ equation, both in the short time and long time regime. We prove the first short time large deviations for the KPZ equation and detects a Gaussian - 5/2 power law crossover in the lower tail rate function. In the long-time regime, we study the upper tail large deviations of the KPZ equation starting from a wide range of initial data and explore how the rate function depends on the initial data. The KPZ equation plays a role as the weak scaling limit of various models in the KPZ universality class. We show the stochastic higher spin six vertex model, a class of models which sit on top of the KPZ integrable systems, converges weakly to the KPZ equation under certain scaling. This extends the weak universality of the KPZ equation. On the other hand, we show that under a different scaling, the stochastic higher spin six vertex model converges to a hyperbolic stochastic PDE called stochastic telegraph equation. One key tool behind the proof of these two stochastic PDE limits is a property called Markov duality.

Since the emergence of the Asynchronous Transfer Mode ( A TM ) concept, various switching architectures have been proposed. The multistage interconnection networks have been proposed for the switching architecture under the A TM environment. In this thesis, we propose a new model for the performance analysis of an A TM switching fabric based on single-buffered Banyan network. In this model, we use a three-state, i.e., "empty", "new" and "blocked" Markov chain model to describe the behavior of the buffer within a switching element. In addition to traditional statistical analysis including throughput and delay, we also examine the delay variation. Performance results show that the proposed model is more accurate in describing the switch behavior under uniform traffic environment in comparison with the "two-state" Markov chain model developed by Jenq, et. al.[4] [6] . Based on the "three-state" model, we study a packet priority scheme which gives the blocked packet higher priority to be routed forward during contention. It is found that the standard deviation of the network delay is reduced by about 30%.