On the number of active transmitters in a random multiple-access channel with a uniform spatial distribution

In this paper, a Multiple-access channel is considered in which transmitters randomly located in the squared-shaped area of width D₁. It is assumed the x and y coordinates of each transmitter´s position are Uniform random variables and also the receiver is located at the origin. Hence, the distance between any transmitter-receiver pair and consequently, the path-loss term are random variables. It is asymptotically shown as the total number of transmitters goes to infinity, the maximum number of simultaneously active transmitters (i.e. user capacity) is of the order v (n) = b₁ ln (B₂ n(Pv (n))^3/2α where n is the total number of transmitters and b₁ and b₂ are constants depending on the minimum-rate, the path-loss exponent and the square width. Simulation results shown as the total number of transmitters is large enough, the numerical results converge the theoretical bounds.