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:
Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service.
In systems engineering, dependability is a measure of a system's availability, reliability, and its maintainability, and maintenance support performance, and, in some cases, other characteristics such as durability, safety and security. In software engineering, dependability is the ability to provide services that can defensibly be trusted within a time-period. This may also encompass mechanisms designed to increase and maintain the dependability of a system or software.
Consider a renewal process , with holding times and jump times (or renewal epochs) , and . The holding times are non-negative, independent, identically distributed random variables and the renewal process is defined as . Then, to a given time , there corresponds uniquely an , such that:
The residual time (or excess time) is given by the time from to the next renewal epoch.
Let the cumulative distribution function of the holding times be and recall that the renewal function of a process is . Then, for a given time , the cumulative distribution function of is calculated as: [2]
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable , or just distribution function of , evaluated at , is the probability that will take a value less than or equal to .
Differentiating with respect to , the probability density function can be written as
where we have substituted From elementary renewal theory, as , where is the mean of the distribution . If we consider the limiting distribution as , assuming that as , we have the limiting pdf as
Likewise, the cumulative distribution of the residual time is
For large , the distribution is independent of , 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 are:
where is the variance of and and are its second and third moments.
The fact that (for ) 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 . The average waiting is only when , that is when the renewals are always punctual or deterministic.
When the holding times are exponentially distributed with , the residual times are also exponentially distributed. That is because 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.
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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.
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