The analysis of the expected successful operation time of slotted ALOHA

It has been well-known for nearly 20 years that the bistable behavior of infinite population slotted ALOHA networks causes the unpleasant effect of eventually reaching an overloaded state, where the number of backlogged stations becomes larger and larger and the useful throughput reduces to zero. The detailed analysis reveals that this statement is true for any average offered load λ>0, regardless of the retransmission probability p. A challenging, and to the best of the authors´ knowledge, not sufficiently solved problem within this context concerns the time until this destabilization occurs. This question is successfully answered based on the fact that the operation of the system may be viewed as a sequence of consecutive busy periods, each starting from backlog 0 and return to backlog 0. It turns out that the whole period of successful operation S consists of a finite sequence of busy periods of finite lengths, which is “terminated” by an infinite busy period (which never returns to backlog 0). Further analysis of this simple renewal process leads to an infinite dimensional system of linear equations, which is shown to have only one meaningful solution. A pair of upper and lower asymptotic bounds for that solution eventually provide the key to the major result, an asymptotic formula for the average number of slots up to the beginning of the infinite busy period, uniformly for p→0 and λ→0