Consider a scenario with two independent input Poisson processes with rates $P_1 \sim Pois(\lambda_1)$ and $P_2 \sim Pois(\lambda_2)$. These enter a queueing system to form a combined process of rate $\lambda_1 + \lambda_2$ and then incur some additional exponentially-distributed service delay (say with mean $1/\mu$).
If the queue is M/M/1 and $\mu > \lambda_1 + \lambda_2$, the output process should also be Poisson with rate $\lambda_1 + \lambda_2$. If an observer of the output process observes just the samples corresponding to $P_1$ (resp. $P_2$), does she see a Poisson process with rate $\lambda_1$ (resp. $\lambda_2$)? If not, what does this process look like?