On colliding first messages in slotted ALOHA

Considering n nodes performing random access using ALOHA with s slots, we study the probability that there occurs a non-colliding message in the first non-empty slot. If each node transmits with probability p in each slot and the number of slots is sufficiently large, a non-colliding first message occurs with probability Phi = 1-np/2 for large n and small np. If the number of slots is limited, the probability Phi is lower but can be maximized choosing an optimal p. To maximize Phi further, nodes can apply a slot-dependent transmit probability pi with i = 1, ..., s. It is shown that a ldquoslow start strategy,rdquo in which p_i is low for low i and increases with increasing i, is beneficial. Our main contribution is an equation for the p_i values that maximize Phi. We analyze how a higher probability of a non-colliding first message comes at the price of an increased delay of such a message. Besides being of interest for the theory of random access, the results are practically applicable to node selection protocols, such as relay selection in cooperative wireless networks.