An accurate analytical formula for the essential joint probability of framed slotted aloha protocols

Framed slotted aloha (FSA) protocols have been widely used in diverse multiple access communication systems. Assume that the FSA network consists of r users and there are L slots accommodated in a frame. Evaluation of the joint probability for the event that there are exactly S single slots, E empty slots and C collision slots, P[S,E,C], is an important issue to characterize the performance of some FSA-based systems. Consequently, various expressions of P[S,E,C] have been proposed and derived. However, in addition to some approximate approaches, we have found that there are different kinds of mathematical or methodological errors in the existent formulae. In this paper, using the principle of inclusion and exclusion, we develop an accurate analytical formula by which the P[S,E,C] can be precisely evaluated. By using the developed formula, performance evaluation that requires determination of P[S,E,C] for various FSA based systems can be greatly improved. The analysis is validated via simulations.