Nonhomogeneous Poisson Process

Definition

A Nonhomogeneous Poisson Process is a counting process $\{N(t),t\ge 0\}$ with intensity function $\lambda (t)$, $t\ge 0$, such that:

1. $N(0)=0$
2. The process has independent increments
3. $P(N(t+h)-N(t)=1)=\lambda (t)h+o(h)$
4. $P(N(t+h)-N(t)\ge 2)=o(h)$

Then $N(t)\sim \text{Po}(m(t))$ where the mean value function is: $m(t)={\int }_0^t\lambda (s)ds$

Interpretation

Unlike the standard Poisson process where the rate $\lambda$ is constant, here the rate varies with time. Think of customer arrivals at a store — more during lunch rush, fewer at 3 AM. The expected count is the area under the rate curve.

Key Difference from Standard Poisson Process

The nonhomogeneous process does not have stationary increments — the distribution depends on when the interval starts, not just its length.

Related

• Poisson Process
• Counting Process
• Compound Poisson Process