I can't understand the relation to CLT. When using an RNG you basically sample from the same distribution (thus have only 1 random variable) plus the CLT comes into play when you add (sum, average) different random variables. In the case of the queue, it has nothing to do with the number of cases missed.
Consider a coin labeled +1 on one side and -1 on the other side. A single coin toss is binary distributed with mean of 0.
Toss the coin N times and let S be the sum of the results. The individual tosses are binary distributed, but for larger N, S is approximately Gaussian distributed thanks to the central limit theorem.
