Expectation of arrival times in an interval of a non-homogeneous poisson process

Question

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.

Answer 1

Let $\mathcal{P}_n=\Big\{[t_{k-1},t_k),t_k^*\Big\}_{k=1}^n$ be a sequence of tagged partitions of $[0,t)$ such that $\max_{1\leq k \leq n}\Delta t_k\rightarrow 0$ as $n \rightarrow +\infty$. Take $X_k$ as the number of arrivals on the time interval $[t_{k-1},t_k)$. When $n$ is large we have $$X_k\approx \text{Poisson}\big(\lambda(t_k^*)\Delta t_k\big)$$

Then $S_k=X_1 + \dots + X_k$ counts the number of arrivals on $[0,t)$ and is approximately $\text{Poisson}\Big(\sum_{k=1}^n\lambda(t_k^*)\Delta t_k\Big)$ which clearly tends to $\text{Poisson}\Big(\int_0^t\lambda(x)\mathrm{d}x\Big)$ as $n\rightarrow \infty$.

Take $T_k$ as the arrival time of the $k^{\text{th}}$ arrival. Then for $t>0$ fixed, $$\{T_k<t\}=\bigcup_{j=k}^{\infty} \{N(t)=j\}$$ where $N(t)$ is the number of arrivals on $[0,t)$. This union is disjoint, so that $$P(T_k<t)=\sum_{j=k}^{\infty}P(N(t)=j)=1-e^{-\int_0^t\lambda(x)\mathrm{d}x}\sum_{j=0}^{k-1}\frac{\Big(\int_0^t\lambda(x)\mathrm{d}x\Big)^j}{j!}$$ Taking a derivative and simplifying yields the pdf for the $k^{\text{th}}$ arrival time, namely $ f_{T_k}$: $$f_{T_k}(t)=\lambda(t)e^{-\int_0^t\lambda(x)\mathrm{d}x}\frac{\Big( \int_0^t\lambda(x)\mathrm{d}x \Big)^{k-1}}{(k-1)!}$$