Actually I have little experience dealing this problem. I suspect this is related to the superposition of renewal process.

My guess is that all the sequence of inter-arrival times have different distributions, but there will be a limiting distribution. The sequence of distribution can be calculated recursively (like the convolution of \( \text{Uniform}[0,1] \)). E.g. the first one is a mixture of truncated exponential on \( [0, T] \) with point mass on \( T \)