 Residual lifetime process

The residual lifetime process of a renewal process is the time remaining until the next renewal:The time since the last renewal is $A_t=t-S_{N_t}$, and $A_t+E_t$ is the length of the renewal interval containing a fixed time. Between renewal epochs, $E_t$ decreases at speed one. On reaching zero it resets to a freshly drawn interarrival time. For strictly positive iid interarrivals, this deterministic motion and independent resetting make $E_t$ a Markov process, even when the count $N_t$ alone is not Markov.

Its initial and equilibrium laws can differ. An ordinary process started at a renewal has $E_0={\xi }_1$. A stationary observation instead has the equilibrium residual-life distribution, whose survival function is the integrated interarrival tail divided by its mean. For deterministic interarrivals of length $c$, the initial residual is $c$, but the equilibrium residual is uniform on $(0,c)$. For exponential interarrivals the two laws agree, reflecting the memoryless property.