Assume that the starting time $T_0=0$. There are $n$ events that occur sequentially at time $T_1$, $T_2$, …, $T_n$, ($T_k\geqslant T_{k-1}$). Suppose the time intervals $\Delta{T_k}\,(\Delta{T_k} = T_k-T_{k-1})$ are independent exponential random variables with different rate parameters $\lambda_k$, respectively. $(1\leqslant k\leqslant n)$
What is the probability that $k$ events have occurred at time $t$, i.e., $\Pr[N(t)=k]$?
I can calculate the distribution of $T_k$ which is the sum of k independent exponential random variables. But I find it is difficult to formulate $\Pr[N(t)=k]$. Is $N(t)$ a non-homogeneous Poisson process? Besides, I was told this probability has no closed form expression. If that is the case, maybe there is an approximation method?
I feel like this problem has been solved, but I’m having trouble finding sources. Any references to papers/books would be helpful and greatly appreciated.
If the distribution of every $T_k$ is known, one can use the duality $$ [N(t)=k]=[T_k\leqslant t\lt T_{k+1}]=[T_k\leqslant t]\setminus[T_{k+1}\leqslant t]. $$