A Poisson point process consists of a sequence of points $0\leq t_1\leq t_2<\cdots$ where $t_i = t_{i-1} + X_i$ where $X_i$ is an exponentially distributed random variable with some rate parameter $\lambda$.
This process has commonly been used to model the spike distribution of neurons. Given a decay parameter $\tau$ the following function, $z(t)$, which depends on the Poisson point process is commonly used in neuroscience models. It is usually referred to as a synaptic trace. We can describe it using an ODE as follows: \begin{align} \frac{dz(t)}{dt} = -\tau z(t) + \delta(t\in T) \end{align} where $T=\{t_i\}$ denotes the set of points of the Poisson process. So in other words $z(t)$ decays exponentially except that at every point in the Poisson point process one adds $1$ to $z(t)$.
Usually we can assume that $\tau > 1/\lambda$, i.e. we do not expect the trace to decay by much before the next point. Now, my question is the following. Given that $z(0) = 0$ and given a threshold $\psi<\mathbb{E}[z(t)]$ can we get a good upper bound on the time $t'$ when with probability $1-o(1)$ we have that $z(t') > \psi$?
If I fix $t'$ I can compute exactly the probability that $z(t')>\psi$ but this requires with my approach to condition on the timing of all the points in the Poisson point process up to time $t'$ and their count. Is there some easier approach than this which gives me a somewhat tight formal statement?