The system is a slotted-aloha system with multiple receivers, and transmissions in a slot do not always collide. The arrival process is assumed to be Poisson. Each transmission (arrival) is then aligned in time slot of length T sec. The frequency of each transmission is selected in random (uniformly distributed) over frequency range F.
A collision is only declared, when two (or more) transmissions in the same timeslot are within Fc Hz (i.e. F1- F2 <= Fc) of each other.
The system is also limited to N receivers per timeslot, i.e. if more the N transmissions are not collided in one timeslot, they will be lost.
The object is find the throughtput of the system in terms of average no. of success transmission per time slot.
My attempt at a solution go like this:
inf throughput S = SUM ( Ps(k) * k ) k
Ps(k) = prob. of k success tx in a timeslot inf = SUM ( P(j) * Ps(k|j) ) j = k
P(j) = prob. of j transmission in T = Pisson distribution
Ps(k|j) = prob. of k success given j transmission in T.
my main problem is with getting the generalized equation for the last expression. Of course, there is also the possibility of not able to sum everything to inf.
However, I got a feeling that the last expression is fairly standard problem. So, please send in any comment, solution or references to published work etc..