Exponential first interarrival in a renewal superposition containing a Poisson process
= Exponential first interarrival in a renewal superposition containing a Poisson process
If an independent superposition of a Poisson process and a non-arithmetic finite-mean renewal process is itself a renewal process, then its first interarrival is exponential. Equating the limiting excess law computed as a renewal excess with the minimum of the Poisson and second-process excesses yields an integral equation whose survival-function solutions are exponential.