I am trying to calculate the variance of the following function: $$F(x) = \mathbb{P}(Y_1(x)-Y_2(x)\le n-1) + G(x)*\mathbb{P}(Y_1(x)-Y_2(x)= n) $$
(this function originates from a paper by Berg and Posner about customer delay in repair systems and this function is used to calculate the probability of a delay of $x$. Hence, $F(x) = \mathbb{P}(\text{delay} \le x)$
$G(x)$ follows an exponential distribution with $\lambda = 1/30$.
The $n$ illustrates the number of spare parts in the system, but is equal to 0 ($n=0$)
$Y_1$ and $Y_2$ are two independent Poisson random variables with the following means: $$\lambda_1(x) = \lambda*\int_x^\infty \bar{G}(u) du $$ $$\lambda_2(x) = \lambda*\int_0^x G (u) du $$
where $$\bar{G}(x)=1-G (x) $$ see this picture for mean calculation of Y1 and Y2 overview. Hence, the $\mathbb{P}(Y_1 - Y_2)$ follows a Skellam distribution (https://en.wikipedia.org/wiki/Skellam_distribution)
I have reproduced this function and plotted the CDF. Visible in this picture
I need to know the variance of the function to compare it to a simple repair system where each customer has to wait for his or her own part. This particular formula assumes that each customer can be serviced with every other person's part and the customer waiting for a part are in a fictional queue which is serviced in a FIFO manner.
I hope the description is clear and someone can help me to calculate the variance of the function.
Kind regards, Noud