The task of controlling the access to a shared communication medium is addressed. More specifically, the problem of finding the control strategy which maximizes the throughput is considered for the infinite population multiple access collision channel. A new method for the efficient design and analysis of access control strategies is presented. This method is based on the recursive computation of the expected number of attempted and successful transmissions during the collision resolution phase of these strategies. It is shown that the throughput of any splitting algorithm can be improved if, instead of transmitting a set which contains exactly one packet with high probability by itself, this set is transmitted together with a subset of another collision set. This observation implies that splitting algorithms are not optimum. Besides establishing a new lower bound for the capacity of the channel, the new strategy is theoretically important since it represents the first algorithm to transmit combinations of sets both of which are not Poisson distributed and therefore lends support to the conjecture that the optimum control strategy involves arbitrarily complex combinations of sets. Motivated by the difficulties of finding the optimum strategy directly, an upper bound for the capacity is computed as the capacity of a channel for which additional information beyond the broadcast feedback is provided. The additional information removes the uncertainty about the number of packets in a collision set and possible dependencies between sets which had access to the channel at the same time. For the channel with side information, the capacity can actually be computed as the throughput of a strategy which achieves an upper bound for the capacity
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