Residual lifetime process

The residual lifetime is the time remaining until the next renewal.

For non-arithmetic inter-renewal times with finite positive mean $\mu$, the excess $E(t)=S_{N(t)+1}-t$ converges in distribution to $E_{\infty }$ with

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.

With bounded integer lifetimes, residual lifetime decreases deterministically and resets randomly at zero.

The invariant residual-life mass at i is proportional to the probability that a lifetime exceeds i.

For inter-renewal times with an Erlang distribution of integer shape $r$ and rate $\lambda$, the mean is $r/\lambda$ and survival function is $e^{-\lambda t}{\sum }_{j=0}^{r-1}(\lambda t{)}^j/j!$. The equilibrium first-delay density is survival divided by the mean,This is the equal mixture of Erlang distributions of shapes $1,\dots ,r$. It can be constructed by observing a Poisson clock with uniformly randomized phase modulo $r$, whose every $r$th event is a renewal.