An exact closed-form formula of collision probability in diverse multiple access communication systems with frame slotted aloha protocol

For diverse multiple access communication systems based on frame slotted aloha (FSA) protocol, it is important to analyze collision probability for the system performance evaluation. As shown in the literature, for general settings, it is difficult to derive an exact and closed-form solution for collision probability without approximation. Recently, an exact solution based on generic analytical approach (GAA) [31] has been proposed, yet its numerical computation will become difficult when the number of slots is larger than 16. In this paper, we develop an exact closed-form formula (ECFF) for collision probability that can not only overcome the computational deficiency of GAA in the presence of a large number of slots, but also reduce the computation complexity of collision probability. Surprisingly, by introducing a differentiation operator to form a hybrid recursive equation and applying various algebraic properties of Laplace transform and Z transform, the final collision probability can be represented by a compact double summation. Accuracy of the ECFF and comparison with the GAA have been studied by Monte Carlo simulation.