Exponential thinning of Poisson Process

Question

Let $(N_t)_{t\ge0}$ be a Poisson Process of parameter $\lambda>0$ whose arrival times are $(T_k)_{k\ge1}$. Consider a thinning of $(N_t)_{t\ge0}$ such that we remove an arrival time $T_k=t$ with probability $\pi(t)=1-e^{-at}$ where $a>0$ independantly of other arrival times. Determine the probability that at least one arrival time is observed in $[1,\infty)$.

If we let $A$ be the event "at least one arrival time is observed in $[1,\infty)$", $A_k$ be the event "the arrival time $T_k$ is removed" and $k_0=\min\{k\ge 1, T_k\ge 1\}$, then we have : $$\mathbb P(A)=1-\mathbb P(A^c)=1-\mathbb P\left(\bigcap_{k=k_0}^\infty A_k\right)=1-\prod_{k=k_0}^\infty\mathbb P(A_k)=1-\prod_{k=k_0}^\infty\pi(T_k)$$ where the $3$rd equality is obtained thanks to the independance of the $(A_k)$. The issue now is that the right-hand side is fairly difficult to compute and I'm not quite sure where to go from there. I also tried calculating $\mathbb P(A)$ directly by using subsets of $[\![k_0,\infty[\![$ but again to no avail. Any help would be greatly appreciated.

Answer 1

Here is how I approached this problem.

Let $[x,y)$ be an interval and $\{[t_{i-1},t_i),s_i\}_{i=1}^n$ a tagged partition of $[x,y)$.

If $n$ is very large, the arrival process in $[t_{i-1},t_i)$ splits $-$ those arrivals which survive and those that don't. The counting process for surviving arrivals is approximately $\text{Poisson}\left(\lambda \Delta t_i e^{-as_i}\right)$ where $\Delta t_{i}=t_i-t_{i-1}$.

This means the surviving arrivals on $[x,y)$ is approximately $\text{Poisson}\left(\sum_{i=1}^n \lambda \Delta t_i e^{-as_i}\right)$ which approaches $\text{Poisson}\left(\int_x^y\lambda e^{-at}\mathrm{d}t\right)$ as $n$ approaches $\infty$.

Now take $x=1$ and $y=\infty$. Can you finish?

Answer 2

Your question is a special case of a sampled Poisson process or non-homogeneous Poisson process:

Suppose we have a Poisson process in which there are $k$ possible types of events, and suppose that the probability that an event is of type $i$ is time-dependent, say $P_i(t)$. We then have the following result:

Proposition. If $N_i(t)$ represents the number of type-$i$ events occurring by time $t$, then $N_i(t)$, $i = 1,\ldots,k$, are independent Poisson random variables having means $$ E\big(N_i(t)\big) = \lambda \int_0^t P_i(s) \,ds \text{.} $$

I found this claim as Proposition 5.3 of Introduction to Probability Models by Sheldon M. Ross, though I'm sure that there are freely accessible proofs on the Web as well.

Applied to your question, the number of observed events after time $t = 1$ is a Poisson random variable with expected value $\lambda e^{-a} / a$. That should help you get the probability you're looking for.