Context: I'm taking a stochastic processes class right now and we got a bit into queueing theory. In all the queue's we've considered, arrivals follow a poisson process. This seems unrealistic in some of the cases I want to consider, like arrivals in a store.
If you consider a fixed interval, say 10 minutes, I would suspect arrivals to a store to be poisson distributed at any interval we choose to sample. But! I also suspect the mean of those distributions to be vastly different at 9AM and say 12PM.
It seems to me that the mean of arrivals is likely to change depending on the time of the day, but in the very short term, it seems like arrivals do follow a poisson distribution.
Clearly this is a violation of the property that:
If X is a poisson process, X(t) - X(s) has a poisson distribution with parameter lambda(t-s) for all s,t such that 0 < s < t
If I supposed that the mean of arrivals followed some continuous function, could I find a pdf by breaking down the interval [0,t] into smaller and smaller partitions? By say taking the midpoint of each interval, evaluating the mean at that point, and finding P (x_1 + x_2 +...... = k) where x_1, x_2 etc are the poisson distributions for each of the time intervals i've broken down?
Thank you all!