Sigman 1 Non - stationary Poisson processes and Compound ( batch ) Pois - son processes

Assuming that a Poisson process has a fixed and constant rate λ over all time limits its applicability. (This is known as a time-stationary or time-homogenous Poisson process, or just simply a stationary Poisson process.) For example, during rush hours, the arrivals/departures of vehicles into/out of Manhattan is at a higher rate than at (say) 2:00AM. To accommodate this, we can allow the rate λ = λ(t) to be a deterministic function of time t. For example, consider time in hours and suppose λ(t) = 100 per hour except during the time interval (morning rush hour) (8, 9) when λ(t) = 200, that is λ(t) = 200, t ∈ (8, 9), λ(t) = 100, t / ∈ (8, 9). In such a case, for a given rate function λ(s), the expected number of arrivals by time t is thus given by m(t) def = E(N(t)) = ∫ t