The problem of optimizing the efficiency of a digital communication network through the use of feedback control is considered. An important goal is to reduce the effects of traffic uncertainties on network efficiency, especially in cases where statistical models of the traffic uncertainties are not available. Optimization is based on the notion of 'asymptotic efficiency', which characterizes the efficiency of high capacity networks and is introduced in this paper. This part of the paper concentrates on the optimal selection of the data admitted into the network.