The process can be analyzed using the method of probability generating function.
2.
We say that the discrete random variable Y satisfying probability generating function characterization
3.
We define that any discrete random variable Y satisfying probability generating function characterization
4.
Probability generating functions obey all the rules of power series with non-negative coefficients.
5.
N are independent " and " identically distributed with common probability generating function " G"
6.
The probability generating function is an example of a generating function of a sequence : see also formal power series.
7.
A stationary distribution for the number of customers in the queue and mean queue length can be computed using probability generating functions.
8.
Then from Wiener L�vy theorem, we show that G ( z ) have the probability generating function of discrete pseudo compound Poisson distribution.
9.
That is, if " X " and " Y " have identical probability generating functions, then they have identical distributions.
10.
So the radius of convergence of any probability generating function must be at least 1, by Abel's theorem for power series with non-negative coefficients.
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