In the theory of
renewal process
Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) ...
es, 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
In chronology and periodization, an epoch or reference epoch is an instant in time chosen as the origin of a particular calendar era. The "epoch" serves as a reference point from which time is measured.
The moment of epoch is usually decided by ...
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:
* In
queueing theory
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 ...
, it determines the remaining time, that a newly arriving customer to a non-empty queue has to wait until being served.
* In
wireless networking
A wireless network is a computer network that uses wireless data connections between network nodes.
Wireless networking is a method by which homes, telecommunications networks and business installations avoid the costly process of introducing c ...
, it determines, for example, the remaining lifetime of a wireless link on arrival of a new packet.
* In
dependability
In systems engineering, dependability is a measure of a system's availability, reliability, maintainability, and in some cases, other characteristics such as durability, safety and security. In real-time computing, dependability is the ability to ...
studies, it models the remaining lifetime of a component.
* etc.
Formal definition
![Renewal process](https://upload.wikimedia.org/wikipedia/commons/5/55/Renewal_process.reetep.png)
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.
:
Probability distribution of the residual time
Let the
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
of the holding times
be