Suppose that a bus has 30 places. A way to model in how long the bus is full where people come independently is to use Poisson process. But in this type of model, we only consider people that come "one by one". Is there a similar way to model it where we don't consider people that come one by one, but rather by small group ? i.e. at a time we can have a group of 3 people, then after a group of 5 people, then after just one people... where the number of people in a group is independent of each other group ?
So it's a sort of poisson process, but instead have $\mathbb P\{N_{t+h}-N_t>1\}=o(h)$ it would be rather $\mathbb P\{N_{t+h}-N_t=k\}=$ I don't know what...
One thing you could look at is a compound Poisson process. Essentially, you have a Poisson process that determines when the jumps happen, but instead of being one, the size of the jumps are drawn independently from some distribution. Here you could choose a size distribution that is natural number-valued... perhaps a geometric distribution?