Residual time

In the theory of renewal processes, a part of the mathematical theory of probability, the residual time or the forward recurrence time is the time between any given time

and the next epoch of the renewal process under consideration.

In the context of random walks, it is also known as overshoot.

Another way to phrase residual time is "how much more time is there to wait?".

The residual time is very important in most of the practical applications of renewal processes: Consider a renewal process

and jump times (or renewal epochs)

are non-negative, independent, identically distributed random variables and the renewal process is defined as

, there corresponds uniquely an

to the next renewal epoch.

Let the cumulative distribution function of the holding times

and recall that the renewal function of a process is

, the cumulative distribution function of

is calculated as:[2] Differentiating with respect to

, the probability density function can be written as where we have substituted

From elementary renewal theory,

, we have the limiting pdf as Likewise, the cumulative distribution of the residual time is For large

, 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

) is also known variously as the waiting time paradox, inspection paradox, or the paradox of renewal theory.

The paradox arises from the fact that the average waiting time until the next renewal, assuming that the reference time point

is uniform randomly selected within the inter-renewal interval, is larger than the average inter-renewal interval

, that is when the renewals are always punctual or deterministic.

, the residual times are also exponentially distributed.

and: This is a known characteristic of the exponential distribution, i.e., its memoryless property.

Intuitively, this means that it does not matter how long it has been since the last renewal epoch, the remaining time is still probabilistically the same as in the beginning of the holding time interval.

Renewal theory texts usually also define the spent time or the backward recurrence time (or the current lifetime) as

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.