Traffic measurements on a ring local area computer network at the Massachusetts Institute of Technology are presented. The analysis of the arrival pattern shows that the arrival processes are neither Poisson nor compound Poisson. An alternative model called "packet train" is proposed. In the train model, the traffic on the network consists of a number of packet streams between various pairs of nodes on the network. Each node-pair stream (or node-pair process, as we call them) consists of a number of trains. Each train consists of a number of packets (or cars) going in either direction (from node A to B or from node B to A). The intercar gap is large (compared to packet transmission time) and random. The intertrain time is even larger. The Poisson and the compound Poisson arrivals are shown to be special cases of the train arrival model. Another important observation is that the packet arrivals exhibit a "source locality." If a packet is seen on the network going from A to B, the probability of the next packet going from A to B or from B to A is very high. Implications of the train arrivals and of source locality on the design of bridges, gateways, and reservation protocols are discussed. A numbet of open problems requiring development of analysis techniques for systems with train arrival processes are also described.