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 formula_1 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:
Formal definition.
Consider a renewal process formula_2, with "holding times" formula_3 and "jump times" (or renewal epochs) formula_4, and formula_5. The holding times formula_3 are non-negative, independent, identically distributed random variables and the renewal process is defined as formula_7. Then, to a given time formula_1, there corresponds uniquely an formula_9, such that:
The residual time (or excess time) is given by the time formula_11 from formula_1 to the next renewal epoch.
Probability distribution of the residual time.
Let the cumulative distribution function of the holding times formula_3 be formula_15 and recall that the renewal function of a process is formula_16. Then, for a given time formula_1, the cumulative distribution function of formula_11 is calculated as:
Differentiating with respect to formula_20, the probability density function can be written as 
where we have substituted formula_22 From elementary renewal theory, formula_23 as formula_24, where formula_25 is the mean of the distribution formula_26. If we consider the limiting distribution as formula_24, assuming that formula_28 as formula_24, we have the limiting pdf as
Likewise, the cumulative distribution of the residual time is 
For large formula_1, the distribution is independent of formula_1, 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 formula_34 are:
where formula_37 is the variance of formula_26 and formula_39 and formula_40 are its second and third moments. 
Waiting time paradox.
The fact that formula_41 (for formula_42) 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 formula_1 is uniform randomly selected within the inter-renewal interval, is larger than the average inter-renewal interval formula_44. The average waiting is formula_45 only when formula_46, that is when the renewals are always punctual or deterministic.
Special case: Markovian holding times.
When the holding times formula_3 are exponentially distributed with formula_48, the residual times are also exponentially distributed. That is because formula_49 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.
Related notions.
Renewal theory texts usually also define the spent time or the backward recurrence time (or the current lifetime) as formula_51. 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.