Editing Availability (section)

==Representation==
The simplest representation of '''availability''' (''A'') is a ratio of the expected value of the uptime of a system to the aggregate of the expected values of up and down time (that results in the "total amount of time" ''C'' of the observation window)

: <math>A = \frac{E[\mathrm{uptime}]}{E[\mathrm{uptime}]+E[\mathrm{downtime}]} = \frac{E[\mathrm{uptime}]}{C}</math>

Another equation for '''availability''' (''A'') is a ratio of the Mean Time To Failure (MTTF) and Mean Time Between Failure (MTBF), or

: <math>A = \frac{MTTF}{MTTF + MTTR} = \frac{MTTF}{MTBF}</math>

If we define the status function <math>X(t)</math> as

: <math>X(t)=
  \begin{cases}
   1, & \text{sys functions at time } t\\
   0, &  \text{maintenance}
  \end{cases}
</math>

therefore, the availability ''A''(''t'') at time ''t''&nbsp;>&nbsp;0 is represented by

: <math> A(t)=\Pr[X(t)=1]=E[X(t)]. \, </math>

Average availability must be defined on an interval of the real line. If we consider an arbitrary constant <math>c>0</math>, then average availability is represented as

: <math> A_c = \frac{1}{c} \int_0^c A(t)\,dt.
</math>

Limiting (or steady-state) availability is represented by<ref>Elsayed, E., ''Reliability Engineering'', Addison Wesley, Reading, MA,1996</ref>
: <math> A = \lim_{c \rightarrow \infty} A_c. </math>

Limiting average availability is also defined on an interval <math>[0,c]</math> as,

: <math> A_\infty =\lim_{c \rightarrow \infty} A_c = \lim_{c \rightarrow \infty}\frac{1}{c} \int_0^c A(t)\,dt,\quad c > 0.
</math>

Availability is the probability that an item will be in an operable and committable state at the start of a mission when the mission is called for at a random time, and is generally defined as uptime divided by total time (uptime plus downtime).

=== Series vs Parallel components ===
[[File:Series vs parallel components.png|alt=series vs parallel components|thumb|397x397px|series vs parallel components]]



Let's say a series component is composed of components A, B and C. Then following formula applies: 

Availability of series component = (availability of component A) x (availability of component B) x (availability of component C) <ref name=":0">{{Cite book |title=System Sustainment: Acquisition And Engineering Processes For The Sustainment Of Critical And Legacy Systems |year=2022 |isbn=9789811256868 |last1=Sandborn |first1=Peter |last2=Lucyshyn |first2=William |publisher=World Scientific }}</ref><ref name=":1">{{Cite book |title=Reliability and Availability Engineering: Modeling, Analysis, and Applications |year=2017 |isbn=978-1107099500 |last1=Trivedi |first1=Kishor S. |last2=Bobbio |first2=Andrea |publisher=Cambridge University Press }}</ref>

Therefore, combined availability of multiple components in a series is always lower than the availability of individual components. 

On the other hand, following formula applies to parallel components:

Availability of parallel components = 1 - (1 - availability of component A) X (1 - availability of component B) X (1 - availability of component C) <ref name=":0" /><ref name=":1" />
[[File:System availability chart.png|alt=10 hosts, each having 50% availability. But if they are used in parallel and fail independently, they can provide high availability.|thumb|10 hosts, each having 50% availability. But if they are used in parallel and fail independently, they can provide high availability.]]
In corollary, if you have N parallel components each having X availability, then:

Availability of parallel components = 1 - (1 - X)^ N <ref name=":1" />

Using parallel components can exponentially increase the availability of overall system. <ref name=":0" />  For example if each of your hosts has only 50% availability, by using 10 of hosts in parallel, you can achieve 99.9023% availability. <ref name=":1" />

Note that redundancy doesn’t always lead to higher availability. In fact, redundancy increases complexity which in turn reduces availability. According to Marc Brooker, to take advantage of redundancy, ensure that:<ref>{{Cite book |title=Understanding Distributed Systems, Second Edition: What every developer should know about large distributed applications |isbn=978-1838430214 |last1=Vitillo |first1=Roberto |date=23 February 2022 |publisher=Roberto Vitillo }}</ref>

# You achieve a net-positive improvement in the overall availability of your system
# Your redundant components fail independently
# Your system can reliably detect healthy redundant components
# Your system can reliably scale out and scale-in redundant components.

=== Methods and techniques to model availability ===

[[Reliability block diagram|Reliability Block Diagrams]] or [[Fault Tree Analysis]] are developed to calculate availability of a system or a functional failure condition within a system including many factors like:

* Reliability models
* Maintainability models
* Maintenance concepts
* Redundancy
* Common cause failure
* Diagnostics
* Level of repair
* Repair status
* Dormant failures
* Test coverage
* Active operational times / missions / sub system states
* Logistical aspects like; spare part (stocking) levels at different depots, transport times, repair times at different repair lines, manpower availability and more.
* Uncertainty in parameters

Furthermore, these methods are capable to identify the most critical items and failure modes or events that impact availability.

=== Definitions within systems engineering ===
'''Availability, inherent (A<sub>i</sub>)''' <ref>{{cite web|title=Inherent Availability (AI) |url=https://dap.dau.mil/glossary/Pages/2045.aspx |work=Glossary of Defense Acquisition Acronyms and Terms |publisher=Department of Defense |access-date=10 April 2014 |url-status=dead |archive-url=https://web.archive.org/web/20140413164657/https://dap.dau.mil/glossary/Pages/2045.aspx |archive-date=13 April 2014 }}</ref>
The probability that an item will operate satisfactorily at a given point in time when used under stated conditions in an ideal support environment. It excludes logistics time, waiting or administrative downtime, and preventive maintenance downtime. It includes corrective maintenance downtime. 
Inherent availability is generally derived from analysis of an engineering design:
# The impact of a repairable-element (refurbishing/remanufacture isn't repair, but rather replacement) on the availability of the system, in which it operates, equals [[mean time between failures]] MTBF/(MTBF+ [[mean time to repair]] MTTR).
# The impact of a one-off/non-repairable element (could be refurbished/remanufactured) on the availability of the system, in which it operates, equals the [[mean time to failure]] (MTTF)/(MTTF + the [[mean time to repair]] MTTR).
 
It is based on quantities under control of the designer.

'''Availability, achieved (Aa)''' <ref>{{cite web|title=Achieved Availability (AI) |url=https://dap.dau.mil/glossary/Pages/1380.aspx |work=Glossary of Defense Acquisition Acronyms and Terms |publisher=Department of Defense |access-date=10 April 2014 |url-status=dead |archive-url=https://web.archive.org/web/20140413164705/https://dap.dau.mil/glossary/Pages/1380.aspx |archive-date=13 April 2014 }}</ref> 
The probability that an item will operate satisfactorily at a given 
point in time when used under stated conditions in an ideal support environment (i.e., that personnel, tools, spares, etc. are instantaneously available). It excludes logistics time and waiting or administrative downtime. 
It includes active preventive and corrective maintenance downtime.

'''Availability, operational (Ao)''' <ref>{{cite web|title=Operational Availability (AI) |url=https://dap.dau.mil/glossary/Pages/Archived/1476.aspx |work=Glossary of Defense Acquisition Acronyms and Terms |publisher=Department of Defense |access-date=10 April 2014 |url-status=dead |archive-url=https://web.archive.org/web/20130312154509/https://dap.dau.mil/glossary/Pages/Archived/1476.aspx |archive-date=12 March 2013 }}</ref>
The probability that an item will operate satisfactorily at a given point in time when used in an actual or realistic operating and support environment. It includes logistics time, ready time, and waiting or administrative downtime, and both preventive and corrective maintenance downtime. This value is equal to the mean time between failure ([[Mean time between failures|MTBF]]) divided by the mean time between failure plus the mean downtime (MDT). This measure extends the definition of availability to elements controlled by the logisticians and mission planners such as quantity and proximity of spares, tools and manpower to the hardware item.

Refer to [[Systems engineering]] for more details

=== Basic example===
If we are using equipment which has a [[mean time to failure]] (MTTF) of 81.5 years and [[mean time to repair]] (MTTR) of 1 hour:

: MTTF in hours = {{math|1=81.5&nbsp;×&nbsp;365&nbsp;×&nbsp;24&nbsp;=&nbsp;713940}} (This is a reliability parameter and often has a high level of uncertainty!)

: Inherent availability (Ai) {{math|1= = 713940 / (713940+1) = 713940 / 713941 = 99.999860% }}

: Inherent unavailability {{math|1= = 1 / 713940 = 0.000140%}}

Outage due to equipment in hours per year = 1/rate = 1/MTTF = 0.01235 hours per year.
<!-- article does not discuss 1+0
That was the case if we are using 1+0 link.. if we are using 1+1 we would use below formula to calculate availability

A= Under root  of MTBFa/MTBFa+20ms * MTBFb/MTBFb+20ms
-->