How can I find the expectation of arrival times on an interval $[0,t]$ of a non-homogeneous poisson process.
For example if customers arrive to a shop following a non-homogeneous poisson process with a rate function $\lambda(t)$, how can I find the expected arrival time of customers in the time interval $[0,t]$.
That is, given that some customers have arrived at the shop between time $0$ and time $t$, what is the average time of their arrivals?
I know that with homogeneous poisson processes, the arrival times on an interval are uniformly distributed due to Campbell's theorem. However, I can't find anything similar for non-homogeneous poisson processes.