Characterizing superposition of two renewal processes

This is a follow-up question of "When superposition of two renewal processes is another renewal process?".

How can we characterize the superposition of two renewal processes?

The superposition of two independent renewal processes is not generally a renewal process, but it can be described within a larger class of processes called the Markov-renewal processes. How can we characterize the superposition of them according to the parameters of each of the renewal processes?

1 Answer

In the case where the two component renewal processes have the same interrenewal distribution $F$, the interarrival distribution of the superimposed process has density $$g(x) = -\frac1\mu\frac{\mathsf d}{\mathsf dx}\left[\bar F(x)\int_x^\infty\bar F(u)\ \mathsf d u\right],$$ as derived in this answer. Here $\bar F=1-F$ is the survival function of the component interrenewal times and $\mu$ is the mean component interrenewal time. We can further simplify the above to $$\frac1\mu\left(\bar F(x)^2 + f(x)\int_x^\infty \bar F(u)\ \mathsf d u \right).$$ For example, if the component renewal processes have $U(0,1)$ interrenewal times, I computed the density of the interarrival times in the superimposed process to be $$g(x) = 3(1-x)^2\mathsf 1_{(0,1)}(x).$$

• thanks a lot for your that. How about the general case where inter-renewal distributions are not identical? – Susan_Math123 Nov 24 '15 at 20:22