Let's say we have a Poisson process starting at time $0$ with an average arrival rate of $\lambda$ arrivals per hour. The arrival rate doubles immediately after each arrival. Let $X_1$ and $X_2$ be the first and second inter-arrival times. We have to find the expectation and variance of $X_1+X_2$.
Now, I was thinking of modelling this situation as a nonhomogeneous Poisson process with a variable rate $\lambda(t) = \lambda$ for $0 \le t < X_1$ and $\lambda (t) = 2\lambda$ for $X_1 \le t < X_1+X_2$. So far , I know how to deal with variable rates that are deterministic functions of time, which is not the case here as $\lambda (t)$ depends on $X_1$ and that is a random variable.
Maybe, I'm making this unnecessarily complicated. Any ideas on how to proceed would be really appreciated.