Yes, one way that the poisson process is defined is as a counting process with inter arrival times that are iid exponentials. The poisson process is the first

Yes, one way that the poisson process is defined is as a counting process with inter arrival times that are iid exponentials. The poisson process is the first (continuous time) point process discussed in many textbooks because the counting increments are identically distributed, since the rate parameter is a constant. You can think that in the case of modeling waiting times for subway trains, a better model is with the rate parameter that is dependent on time, since the inter arrival times are smaller during rush hour than when the subway is closed. You can read about this in Bertsekas, and more thoroughly in Parzen or Cox's books on stochastic processes.