I am trying to make a model for a specific stochastic process, however I'm not sure how to tackle this problem.
PROCESS. Discrete time process.
At each time step we have a random number of unit arrivals with known probabilities. The probability that we have $n$ arrivals is given by known $a_n$ and $\sum_{n=0}^\infty a_n = 1$. For example $a_n$ could follow the Poisson distribution.
Right before arrivals we also have possibility of departures. At each time step, each unit has a probability $p$ of departing.
However we have a maximum capacity, say $n_{\text{max}}$, and there is no queueing system, so the exceeding arrivals just disappear.
So my question is the following:
What is the easiest way to model this stochastic process?
Particularly, I'm interested in the number of units at each time step (just after departures and arrivals).
I've been looking at various processes such as Markov Chains, poisson processes, however since I'm not that well-versed in stochastic processes I'm not sure what is the easiest approach.