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We present bounds on the maximum channel utilization (with finite average delay) of synchronous multiple access communications protocols serving an infinite population of homogeneous stations. Messages arrive to the system as a series of independent Bernoulli trials in discrete time, with probability p of an arrival at each arrival point (the Poisson limit is explicitly included) and are then randomly distributed among the stations. Pippenger showed that the channel utilization cannot exceed , where and . Using a "helpful genie" argument, we find the exact capacity for all (where we find optimal protocols that obey first-come first-served); for smaller values of p, we present an improved upper bound that decreases monotonically to in the Poisson limit as .