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

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.

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

• Thanks for the answer but I'm not sure I understand. Does this not just tell you that the number of arrivals in [0,t] is poisson distributed with its parameter being the integral of the rate function between 0 and t. How do I use this to know the distribution of the times of these arrivals as opposed to the distribution of the number of arrivals? Commented Nov 10, 2021 at 20:15
• My bad. I read your question too quickly. I only illustrated the distribution for the number of arrivals on $[0,t)$ Let me get back to your on the distribution of the arrival times.
– user801306
Commented Nov 10, 2021 at 20:23
• no worries, thank you very much Commented Nov 10, 2021 at 20:29
• I updated my answer to illustrate the pdf of the $k^{\text{th}}$ arrival. Note taking $\lambda(t)$ to be constant yields the appropriate Erlang distribution.
– user801306
Commented Nov 10, 2021 at 21:29
• Thanks, so to find the expectation, I would take the integral of this pdf multiplied by t and then after that plug in my value of t? Commented Nov 11, 2021 at 8:44