# Availability

In telecommunications and reliability theory, the term availability has the following meanings:

1. The degree to which a system, subsystem, or equipment is operable and in a committable state at the start of a mission, when the mission is called for at an unknown, i.e., a random, time. Simply put, availability is the proportion of time a system is in a functioning condition.

Note 1: The conditions determining operability and committability must be specified.

Note 2: Expressed mathematically, availability is 1 minus the unavailability.

2. The ratio of (a) the total time a functional unit is capable of being used during a given interval to (b) the length of the interval.

Note 1: An example of availability is 100/168 if the unit is capable of being used for 100 hours in a week.

Note 2: Typical availability objectives are specified either in decimal fractions, such as 0.9998, or sometimes in a logarithmic unit called nines, which corresponds roughly to a number of nines following the decimal point, such as "five nines" for 0.99999 reliability.

## Representation

The most simple representation for availability is as a ratio of the expected value of the uptime of a system to the aggregate of the expected values of up and down time, or $A = \frac{E[\mathrm{Uptime}]}{E[\mathrm{Uptime}]+E[\mathrm{Downtime}]}$

If we define the status function $X(t)$ as $X(t)= \begin{cases} 1, & \mbox{sys functions at time } t\\ 0, & \mbox{otherwise} \end{cases}$

therefore, the availability is represented by $A(t)=\Pr[X(t)=1].$ $E[X(t)]=X.\Pr[X(t)=1] \quad t > 0.$

Average availability must be defined on an interval of the real line. If we consider an arbitrary constant $c$, then average availability is represented as $A_c = \frac{1}{c}\int_0^c A(t)\,dt,\quad c > 0.$

Limiting (or steady-state) availability is represented by $A = \lim_{t \rightarrow \infty} A(t).$

Limiting average availability is also defined on an interval $(0,c]$ as, $A_{\infty}=\lim_{c \rightarrow \infty} A_c = \lim_{c \rightarrow \infty}\frac{1}{c}\int_0^c A(t)\,dt,\quad c > 0.$

## Literature

Availability is well established in the literature of stochastic modeling and optimal maintenance. Barlow and Proschan  define availability of a repairable system as "the probability that the system is operating at a specified time t." While Blanchard  gives a qualitative definition of availability as "a measure of the degree of a system which is in the operable and committable state at the start of mission when the mission is called for at an unknown random point in time." This definition comes from the MIL-STD-721. Lie, Hwang, and Tillman  developed a complete survey along with a systematic classification of availability.

Availability measures are classified by either the time interval of interest or the mechanisms for the system downtime. If the time interval of interest is the primary concern, we consider instantaneous, limiting, average, and limiting average availability. The aforementioned definitions are developed in Barlow and Proschan , Lie, Hwang, and Tillman , and Nachlas . The second primary classification for availilability is contingent on the various mechanisms for downtime such as the inherent availability, achieved availability, and operational availability. (Blanchard , Lie, Hwang, and Tillman ). Mi  gives some comparison results of availability considering inherent availability.

Availability considered in maintenance modeling can be found in Barlow and Proschan  for replacement models, Fawzi and Hawkes  for an R-out-of-N system with spares and repairs, Fawzi and Hawkes  for a series system with replacement and repair, Iyer  for imperfect repair models, Murdock  for age replacement preventive maintenance models, Nachlas [1998, 1989] for preventive maintenance models, and Wang and Pham  for imperfect maintenance models.