exponential and deterministic random variable mix

Assume that there is a Poisson random process which has a rate of λ that produce samples which has an exponentially distributed inter-arrival times, t (i.e distribution function given by λe−λt. Then there is another deterministic process which produces samples at a rate of 1 sample in every T time interval (i.e the sample-inter arrival times of this process are not random and you get 1 sample in every T time). Now assume both process start together and we measure the inter arrival times of the samples of resulting process irrespective of which process they are generated from)

I need to get the Distribution of the inter-arrival times of the resulting process??


TS Contributor
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 \( T \)