In all the examples of ALOHA I've seen, the Poisson distribution is used. Theoretically, how could the throughput be calculated if a Binomial distribution was used instead?
For example, in the case that the offered load follows a Binomial distribution with probability $p$, I know that the probability of $k$ attempts within $t$ time slots is as follows:
$$\Pr\big[k \text{ attempts within $t$ time slots}\big] = \binom{t}{k}p^k(1-p)^{t-k}\,.$$
I know that the throughput, $S$ is equivalent to $G$ (average offered load) multiplied by the probability of success:
$$S = G\cdot\Pr\big[\text{successful}\big]\,,$$
but what would this be in the case that a binomial distribution is used?
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