From the in-homogeneous Poisson process $\lambda(t)=10-5cos(2\pi t)dt $, we know $\lambda(t)$ is the arrival rate at time $t$ and $\int_{0}^{t}10-5cos(2\pi t)$ is the average arrivals during time $0$ to $t$. I'm wondering how can we calculate the average waiting time need (from time $a$ to time $b$) to have $x$ (a specific number) arrivals? I've tried to do the following:
For example, $x=3,a=0$.
$\int_{0}^{b}10-5cos(2\pi t)dt=3$. What I get $b-a=b-0=b$ from the equation is supposed to be greater than the actual waiting time because normally the $x^{th}$ arrival time is before time $b$. I know this method is wrong, but how should we do?