This thesis investigates the worst possible behaviour of a source if the traffic emanating from it is constrained to pass through a selection of leaky buckets. Various criteria for judging the worst ease arc considered, including the average queue length when the traffic is passed through an infinite buffer served at a constant rate and the rate of loss when the traffic is passed through a finite buffer, again served at a constant rate. Both of these criteria may be used when the source traffic is buffered alone or in combination with other traffic. Another functional considered is the effective bandwidth function which governs the asymptotic loss rate when the number of sources becomes infinite. This functional turns out to be the most tractable and we concentrate our attention on it. In all cases considered it is found that the functional to be maximised is convex in the space of traffic processes. This leads to the use of convex optimisation methods to characterise the worst case traffic process. In addition, Optimal Control Theory is used to show that the worst case traffic exhibits periodic behaviour.