## 11.1.4 Nonhomogeneous Poisson Processes

*nonhomogeneous Poisson process*. Such a process has all the properties of a Poisson process, except for the fact that its rate is a function of time, i.e., $\lambda=\lambda(t)$.

__Nonhomogeneous Poisson Process__

Let $\lambda(t):[0,\infty) \mapsto [0,\infty)$ be an integrable function. The counting process $\{N(t), t \in [0, \infty)\}$ is called a

**nonhomogeneous Poisson process**with

**rate**$\lambda(t)$ if all the following conditions hold.

- $N(0)=0$;
- $N(t)$ has
__independent__increments; - for any $t \in [0,\infty)$, we have \begin{align*} &P(N(t+\Delta)-N(t)=0) =1-\lambda(t) \Delta+ o(\Delta),\\ &P(N(t+\Delta)-N(t)=1)=\lambda(t) \Delta+o(\Delta),\\ &P(N(t+\Delta)-N(t) \geq 2)=o(\Delta). \end{align*}

\begin{align*}
N(t+s)-N(t) \; \sim \; Poisson \left( \int_t^{t+s} \lambda (\alpha) d \alpha\right).
\end{align*}