A random access protocol for packet-switched, multiple access communication via time slotted busses is investigated. Assuming heavy traffic for all stations, the access probabilities are determined as to allocate a certain portion of the channel capacity to individual stations for achieving prescribed service requirements. In the infinite buffer case with Poisson arrivals the probability generating function of the queue length, and its expectation and variance are determined for a single station. We furthermore calculate the expected delay time of the last arriving packet in a slot, as well as the corresponding variance. Based on these formulas a system of N coupled stations is investigated w.r.t. packet delay and fairness. It turns out that if the bus is able to carry all offered load, then protection of fairness means only deterioration of oneself's performance without improving corresponding parameters for other stations.