I'm trying to make a small model for the expected life-time of some molecules (I'll edit the question and add info if someone wants to know the context) and I reached the following multiple integral: $$\int_0^T \int_0^{T-t_1}\int_0^{T-t_1-t_2}\ldots \int_0^{T-\sum_{j=1}^{2N-1}t_j} e^{-\sum_{k=1}^{2N}\lambda_k t_k} dt_{2N}\ldots dt_1,$$ where both $T$ and all the $\lambda_k$ are real positive constants. In principle, the integral can be computed explicitely and is straight-forward but, due to the the form of the integration limits, it gets very messy very quickly, and I cannot solve it for general $N.$
So far, apart from trying to compute the integral directly, I have tried to use Laplace/Fourier transform and also tried to change variables, in case I could cast this expression in the form of a chain of convolutions or something similar. But I have gotten nowhere with either approach.
Some ideas?