Age and Residual Life in a Renewal Process

Locating the Renewal Interval

At a time \(t\), the age of a renewal process records how long it has been since the most recent renewal. Its residual life records how long remains until the next renewal. Both quantities are determined by locating the interval between consecutive renewal epochs that contains \(t\). The endpoint convention matters: if \(t\) is itself a renewal time, the age is zero, and the next renewal is the one after \(t\).

Let \((\Omega,\mathcal{F},\mathbb{P})\) be a probability space, and let \(X_1,X_2,\ldots\) be independent, identically distributed, strictly positive and finite random variables on that space. These are the interarrival times. Define the renewal epochs by \(S_0=0\) and \(S_n=\sum_{k=1}^n X_k\) for \(n\ge1\). Assume \(S_n\to\infty\) almost surely, so only finitely many renewals occur in any bounded time interval. The definitions below are pathwise on the event of probability one where the assumptions hold.

The maximum defining \(N(t)\) exists: \(S_0=0\le t\), and \(S_n\to\infty\) means there are only finitely many indices with \(S_n\le t\). Since the interarrival times are strictly positive, the epochs are strictly increasing. Consequently there is exactly one interval \([S_n,S_{n+1})\) containing \(t\). No distributional calculation is needed to find age and residual life on a specified sample path.

Proof. The set \(I_t=\{n\ge0:S_n\le t\}\) is nonempty because it contains 0. It is finite because \(S_n\to\infty\), so it has a maximum \(n\). By the definition of maximum, \(S_n\le t\), while \(S_{n+1}>t\); otherwise \(n+1\) would also belong to \(I_t\). Strict increase of the epochs means no other interval can contain \(t\), proving uniqueness. The maximum is \(N(t)\), and substituting it into the definitions gives the formulas for \(A(t)\) and \(R(t)\). The inequalities follow from \(S_n\le t<S_{n+1}\). Finally, \(S_{n+1}-S_n=X_{n+1}\), so \(A(t)+R(t)=(t-S_n)+(S_{n+1}-t)=X_{n+1}\). \(\square\)

The sum \(A(t)+R(t)\) is the length of the interval currently containing \(t\), not a new interarrival time formed by adding two unrelated quantities. At a renewal time \(t=S_n\), the interval containing \(t\) is the one beginning there: \(N(t)=n\), \(A(t)=0\), and \(R(t)=X_{n+1}\). At a time strictly between renewals, both age and residual life are positive.

How Age and Residual Life Change Between Renewals

The epoch formulas give a direct description of the paths of these two quantities. Within one renewal interval, age increases at unit rate, while residual life decreases at unit rate. The next result states this precisely and includes the endpoint behavior.

Proof. If \(0\le u<X_{n+1}\), then \(S_n\le S_n+u<S_n+X_{n+1}=S_{n+1}\). By the unique bracketing property, \(N(S_n+u)=n\). Therefore \(A(S_n+u)=S_n+u-S_n=u\), and \(R(S_n+u)=S_{n+1}-(S_n+u)=X_{n+1}-u\). At the endpoint \(S_{n+1}\), the interval beginning there is selected because the bracketing convention includes its left endpoint. Thus \(N(S_{n+1})=n+1\), \(A(S_{n+1})=0\), and \(R(S_{n+1})=S_{n+2}-S_{n+1}=X_{n+2}\). \(\square\)

In particular, as time approaches \(S_{n+1}\) from below, the age approaches \(X_{n+1}\) and the residual life approaches 0. At \(S_{n+1}\), age resets to 0 and residual life becomes the next full interarrival time \(X_{n+2}\). This reset is why the equality convention in \(S_n\le t<S_{n+1}\) cannot be replaced casually by a different endpoint convention.

Endpoint Conventions and Common Errors

The convention \(S_n\le t<S_{n+1}\) assigns every renewal time to the interval that starts at that time. In particular, \(A(S_n)=0\) for every \(n\ge0\), while \(R(S_n)=X_{n+1}\). The residual life is strictly positive under the assumption \(X_{n+1}>0\), including at a renewal epoch. If zero interarrival times were allowed, epochs could coincide and the stated uniqueness and positivity properties would require additional conventions; that case is excluded here.

A frequent error is to use the interval that ends at an exact renewal time. For the periodic example, an incorrect calculation at \(t=6\) might use the interval \([4,6]\), yielding age \(6-4=2\) and residual life \(6-6=0\). That calculation treats the right endpoint as part of the old interval, contrary to the defining strict inequality \(t<S_{n+1}\). The correct interval is \([6,8)\), giving age 0 and residual life 2.

A second error is an off-by-one mistake in the interarrival index. The interval from \(S_n\) to \(S_{n+1}\) has length \(X_{n+1}\), not \(X_n\), because \(S_{n+1}-S_n=X_{n+1}\). Once the bracketing index \(n=N(t)\) is known, the formulas \(A(t)=t-S_n\) and \(R(t)=S_{n+1}-t\) avoid this ambiguity.

These definitions are pathwise: they describe the elapsed and remaining portions of the particular renewal interval containing the observation time. They do not by themselves specify a limiting distribution or an equilibrium law. Such distributional questions require additional probabilistic arguments; the starting point for each is the correct identification of the containing interval and its endpoints.