# exponential and deterministic random variable mix

##### New Member
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??

#### BGM

##### TS Contributor
The setup itself is interesting.

I need to get the Distribution of the inter-arrival times of the resulting process??
Is it your question?

##### New Member
The setup itself is interesting.

Is it your question?
Yes!

Tried to solve this using the Poisson process definitions. But seems that it needs to be tackled with Markovian chains. But I have less knowledge in markovian chains and SDE

#### BGM

##### 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 $$T$$