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We have a discrete process $(N_k)_{k\in\mathbb{N}}$ defined as follows. At each time $k$ an amount of Poisson$(\nu)$ packets come in, but there is also a probability $1/e$ that exactly $1$ packet leaves.

Example: $N_1=3$

At $k=2$ exactly one of these three leaves. It could also have happened that nothing was left, with probability $1-1/e$, but let it do happen this time.

We run Poisson$(\nu)$ and the outcome is $4$.

So $N_2= $3-1+4=6$.

My question is: how to calculate the mean and variance of this process? We assume $\nu<1/e$, so the process is stable.

I know it is doable if we know the probability distribution $p=(p_1,p_2,...)$, but calculating $p$ is a lot of work, so maybe there is an easier way?

We can also see this as an $M/G/1$ queue with $G\sim$ geometric$(1/e)$, but then we will just get an approximation, because the Poisson-process here is continuous, not discrete.

Is there anyone with a nice idea?

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