Assume we have a renewal process $P_1$ with inter-renewal distribution $p_1(x)$ and rate $\lambda_1 = \lim_{t \to \infty} \frac{N_1(t)}{t}$, where $N_1(t)$ is the total number of renewals of $P_1$ during $[0,t]$, and we have another renewal process $P_2$, with similar parameters $p_2(x)$, $\lambda_2$, and $N_2(t)$.
Now, how can we find the distribution of the superposition $P_1 + P_2$ for each of the following cases:
$P_2$ is Poisson.
$p_2(x)$ is scaled version of $p_1(x)$, (where the scaling changes the mean from $\lambda_1$ to $\lambda_2$.
For the first case, if $P_1$ is also Poisson, it is really simple since the outcome is also Poisson. But here, $P_1$ is general, not only Poisson.
For the second case, I am just thinking as righting the outcome as a general Markov Renewal Process and and finding the moments of that, which is hard! Any better idea?
Please see following related questions: Characterizing superposition of two renewal processes and When Superposition of Two Renewal Processes is another Renewal Process?.
