Analysis of tree algorithm for collision resolution †

In multi-user communications the problem is how to serve many senders if one common communication channel is given. The classical solution is a kind of multiplexing, i.e., either time-division multiplexing, or frequency-division multiplexing. For partially active senders, always there are a large number of senders, each which has nothing to send most of the time. In this communication situation the multiplexing is inefficient. One such situation, namely the problem of communicating from remote terminals on various islands of Hawaii via a common radio channel to the main central computer, led to the invention by Abramson (1970) of the first formal random-access algorithm, now commonly called pure ALOHA and to the design of a radio linked computer network, called ALOHANET (cf. Abramson (1985)). The performance of the ALOHA protocol is very poor if the channel occupancy increases beyond a certain level, in fact for Poisson arrivals it is unstable. In this paper we consider the multiple-access collision channel with ternary feedback. An unlimited number of users are allowed to transmit packets of a fixed length whose duration is taken as a time unit and called slot. Stations can begin to transmit packets only at times n ∈ {0, 1, 2, 3, . . . }. A slot is a time interval [n, n + 1). The destination for the packet contents is a single common receiver. All users send their packets through a common channel. Senders of different packets cannot interchange information. Thus it is convenient to suppose that there are infinitely many non cooperating users and that the packet arrivals can be modelled as a Poisson process in time with intensity λ. When two or more users send a packet in the same time slot, these packets “collide” and the packet information is lost, i.e., the receiver cannot determine the packet contents, and retransmission will be necessary. However, all users, also those who were not transmitting can learn—from the ternary feedback just before time instant n+ 1—the story of time slot [n, n+ 1):