Residual time

Sample evolution of a renewal process with holding times S_i and jump times J_n.

Consider a renewal process ${\displaystyle \{N(t),t\geq 0\}}$, with holding times ${\displaystyle S_{i}}$ and jump times (or renewal epochs) ${\displaystyle J_{i}}$, and ${\displaystyle i\in \mathbb {N} }$. The holding times ${\displaystyle S_{i}}$ are non-negative, independent, identically distributed random variables and the renewal process is defined as ${\displaystyle N(t)=\sup\{n:J_{n}\leq t\}}$. Then, to a given time ${\displaystyle t}$, there corresponds uniquely an ${\displaystyle N(t)}$, such that:

${\displaystyle J_{N(t)}\leq t<J_{N(t)+1}.\,}$

The residual time (or excess time) is given by the time ${\displaystyle Y(t)}$ from ${\displaystyle t}$ to the next renewal epoch.

${\displaystyle Y(t)=J_{N(t)+1}-t.\,}$

Let the cumulative distribution function of the holding times ${\displaystyle S_{i}}$ be ${\displaystyle F(t)=Pr[S_{i}\leq t]}$ and recall that the renewal function of a process is ${\displaystyle m(t)=\mathbb {E} [N(t)]}$. Then, for a given time ${\displaystyle t}$, the cumulative distribution function of ${\displaystyle Y(t)}$ is calculated as:[2]

${\displaystyle \Phi (x,t)=\Pr[Y(t)\leq x]=F(t+x)-\int _{0}^{t}\left[1-F(t+x-y)\right]dm(y)}$

Differentiating with respect to ${\displaystyle x}$, the probability density function can be written as

${\displaystyle \phi (x,t)=f(t+x)+\int _{0}^{t}f(u+x)m'(t-u)du,}$

where we have substituted ${\displaystyle u=t-y.}$ From elementary renewal theory, ${\displaystyle m'(t)\rightarrow 1/\mu }$ as ${\displaystyle t\rightarrow \infty }$, where ${\displaystyle \mu }$ is the mean of the distribution ${\displaystyle F}$. If we consider the limiting distribution as ${\displaystyle t\rightarrow \infty }$, assuming that ${\displaystyle f(t)\rightarrow 0}$ as ${\displaystyle t\rightarrow \infty }$, we have the limiting pdf as

${\displaystyle \phi (x)={\frac {1}{\mu }}\int _{0}^{\infty }f(u+x)du={\frac {1}{\mu }}\int _{x}^{\infty }f(v)dv={\frac {1-F(x)}{\mu }}.}$

Likewise, the cumulative distribution of the residual time is

${\displaystyle \Phi (x)={\frac {1}{\mu }}\int _{0}^{x}[1-F(u)]du.}$

For large ${\displaystyle t}$, the distribution is independent of ${\displaystyle t}$, making it a stationary distribution. An interesting fact is that the limiting distribution of forward recurrence time (or residual time) has the same form as the limiting distribution of the backward recurrence time (or age). This distribution is always J-shaped, with mode at zero.

The first two moments of this limiting distribution ${\displaystyle \Phi }$ are:

${\displaystyle E[Y]={\frac {\mu _{2}}{2\mu }}={\frac {\mu ^{2}+\sigma ^{2}}{2\mu }},}$

${\displaystyle E[Y^{2}]={\frac {\mu _{3}}{3\mu }},}$

where ${\displaystyle \sigma ^{2}}$ is the variance of ${\displaystyle F}$ and ${\displaystyle \mu _{2}}$ and ${\displaystyle \mu _{3}}$ are its second and third moments.

Renewal theory texts usually also define the spent time or the backward recurrence time (or the current lifetime) as ${\displaystyle Z(t)=t-J_{N(t)}}$. Its distribution can be calculated in a similar way to that of the residual time. Likewise, the total life time is the sum of backward recurrence time and forward recurrence time.