On the Control, Stability, and Waiting Time in a Slotted ALOHA Random-Access System

This concise paper explores some of the boundaries in performance of slotted ALOHA systems by analyzing a simple and almost optimal centrally supervised control. Our control results in a very simple Markov chain model and allows an examination of stability, conditional waiting time distribution of transmitting terminals, and many other system measures. The key to the simplicity is to have a probability of successful packet transmission that is independent of the number of transmitting terminals. Regarding stability, recent papers have shown that a slotted ALOHA system with an infinite population of terminals producing packets at a Poisson rate λ/slot becomes saturated with retransmissions and breaks down. Here we define a stable ALOHA system as one that clears out the blocked packets in a finite time, and has only a finite number of blocked packets in the system, all with arbitrarily low probability. This is a necessary condition for previous definitions of stability to be meaningful. In considering waiting time, we calculate the mean and other moments of the waiting time of a terminal when it enters the system to find ( ) other terminals already there competing for the channel. Under this control, the average time is proportional to . Two things should be pointed out. The first is that the control requires exact knowledge of the number of terminals contending for the channel, and hence is not implementable, except as an approximation. The second is that the analysis takes into account the dynamic comings and goings of terminals.