Unavailability, in mathematical terms, is the

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

that an item will not operate correctly at a given time and under specified conditions. It opposes

availability In reliability engineering, the term availability has the following meanings: * The degree to which a system, subsystem or equipment is in a specified operable and committable state at the start of a mission, when the mission is called for at a ...

Numerical value A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...

s associated with the

calculation A calculation is a deliberate mathematical process that transforms one or more inputs into one or more outputs or ''results''. The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm, to th ...

of availability are often awkward, consisting of a series of 9s before reaching any significant numerical information (e.g. 0.9999999654). For this reason, it is more convenient to use the complement measure of availability, namely, unavailability. Expressed mathematically, unavailability is 1 minus the

. Therefore, a

system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...

with availability 0.9999999654 is more concisely described as having an unavailability of 3.46 × 10⁻⁸.

Calculations using unavailability

Often fault trees and

reliability block diagram A reliability block diagram (RBD) is a diagrammatic method for showing how component reliability contributes to the success or failure of a redundant. RBD is also known as a dependence diagram (DD). An RBD is drawn as a series of blocks conn ...

s will use unavailability of the various components in the calculation of the top level failure rates through AND gates or parallel redundant components.

Repairable Model

Unavailability (Q), using the repairable model, may be expressed mathematically by the equation: :

Q = \frac

where MTTR is the mean time to repair, and MTBF is the mean time between failures of a repairable system. Alternatively, this can be written as: :

Q = \frac

where λ is the failure rate and μ is the repair rate. When μ >> λ, the preceding formula is often approximated to: :

Q \approx \frac

Non-Repairable Model

For the non-repairable model of unavailability (Q), the unreliability function (often F(t) the CDF of the

exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...

) is used to approximate the worst-case-unavailability. If the rate of failure is constant the

Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...

and

describe this rate. The unreliability function approximating the worst case unavailability is as follows: :Q = 1 - e^-λt Where t is the time at risk.

Telecommunications

telecommunication Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than that fe ...

, an unavailability is an expression of the degree to which a

, subsystem, or equipment is not operable and not in a committable state at the start of a mission, when the mission is called for at an unknown, i.e. random,

time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...

. The conditions determining operability and committability must be specified.

Railway

In the railway industry, the railway normally keeps operating for 24 hours a day 7 days per week all year round making the idea of mission time meaningless. Both the repairable model and non-repairable model are known to be used in railway. The repairable model is used for total system availability or unavailability and the non-repairable model is used for system safety. Safe down time is the time between when a wrong side failure happens and when it is detected and mitigated.CENELEC EN 50129:2018

Aerospace

The aerospace industry often uses a mission time equal to the expected flight time given that certain pre-flight tests are performed.

Space

The mission time for space systems may be as long as a satellite or system in orbit. Space systems are exceedingly difficult to repair making mission time a consideration when evaluating unavailability.

References

{{Reflist Systems engineering