Consider a counting process $\{N(t), t\geq 0\}$ where the time distribution between any two consecutive events, say $k$ and $k+1$ has a Poisson rate $\lambda(k)$, which is an explicit function of $k$. I would like to know whether the number of events that happen in a time interval, say $[0, T]$, has a particular distribution. In general, what we can say about the number of events in time interval $[0, T]$.



non-homogeneous counting process
$\newcommand\la\lambda$Let $\la\_k:=\la(k)$. Let $P\_{\la\_0,\la\_1,\dots}(t,m)$ denote the probability that $N(t)=m$ given the rates $\la\_0,\la\_1,\dots$. Then, conditioning on the time of the first event, we get the recurrence
$$P\_{\la\_0,\la\_1,\dots}(t,m)
=\int\_0^t\la\_0\,ds\,e^{-\la\_0 s}P\_{\la\_1,\la\_2,\dots}(t-s,m-1)$$
for all real $t>0$ and all natural $m$, with $P\_{\la\_0,\la\_1,\dots}(t,0)=e^{-\la\_0 t}$.


One can now check by induction on $m$ that the solution of this recurrence is given by
$$P\_{\la\_0,\la\_1,\dots}(t,m)=(-1)^m \Big(\prod\_{j=0}^{m-1}\la\_j\Big)
\sum\_{j=0}^m \Big(e^{-\la\_j t}\Big/\prod\_{i\in[[m]]\setminus\{j\}}(\la\_j-\la\_i)\Big)$$
if the $\la\_j$'s are all distinct, where $[[m]]:=\{0,1,\dots,m\}$.


