This dissertation considers an important issue in high-speed networks - the understanding of the interaction between the network and the bursty traffic it handles. The first part of the dissertation analyzes the leaky bucket (LB), a proposed congestion control scheme for ATM networks. There are many preliminary results in the literature on the LB. Most of them, however, are numerical ones under some Markovian assumptions. In this dissertation, it is shown that under very mild assumptions, the LB is burst reducing. More importantly, this burst reduction property obeys some monotonicity properties in all the parameters of the LB. This property is then used to the design of the LB. Bounds and approximations for some other performance measures of the LB are also derived to facilitate the design. The mathematical tool developed in this part can also be used to study a large class of non-stationary stochastic processes.

In the second part of the dissertation, some general results on the calculus of burstiness are obtained for renewal processes. These results are useful in understanding the effects on the burstiness of the traffic in some systems through multiplexing, splitting, rate control, and scheduling. The burstiness is characterized by both the peakedness functional of the traffic and the squared coefficient of variation of its asymptotic version. As examples, problems in network scheduling and in the design of multiplexer with rate control mechanism are discussed.