# mean and variance in backlog process

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?