God, grant me the serenity to accept the things I cannot change, The courage to change the things I can, And the wisdom to know the difference. (Serenity Prayer, Reinhold Niebuhr)

* Reliability* engineering is an engineering field, that deals with the study of reliability: the ability of a system or component to perform its required functions under stated conditions for a specified period of time (Wikipedia).

Ps = R = e^(-t/MTBF) Where,Ps = R = probability of failure free operation for a time period equal to or greater than t

e = 2.718

t = specified period of failure free operation

MTBF = mean time between failures (the mean TBF distribution)

The probability that failure will occur before time t = (1 - Ps)

- The MTBF is the mean (or average) time between successive failures of a product. This definition assumes that the product in question can be repaired and placed back into operation after each failure. For nonrepairable products, the term mean time to failure (MTTF) is used.
- If the failure rate is constant, the probability that a product will operate without failure for a time equal to or greater than its MTBF is only 37%. This outcome is based on the exponential distribution. (R is equal to .37 when t is equal to MTBF).
- MTBF is not the same as “operating life”, “service life” or other indexes.

An increase in MTBF does not result in a proportional incresase in reliability (the probability of survival).

Px = P1 * P2 * … * Pn, where Pn is the probability of each part's functioning when activated

The formula assumes that (1) the failure of any part causes failure of the system and (2) the reliabilities of the parts are independent of one another. This formula shows the effect of increased complexity of equipment on overall reliability. As the number of parts in a system increases, the system reliability decreases dramatically.

Px = P1 + (1 - P1) * P2, where Pn is the probability of each part's functioning when activated

Px = 1 - [(1 - P1) * (1 - P2) * ... * (1 - Pn)], where Pn is the probability of each part's functioning when activated

Availability has been defined as the probability that a product, when used under given conditions, will perform satisfactorily when called upon. Availability considers the operating time of the product and the time required for repairs. Idle time, during which the product is not needed, is excluded. Availability is calculated as the ratio of operating time to operating time plus downtime.

Total downtime – this period includes active repair (diagnosis and repair time), preventive maintenance time, and logistics time (time spent waiting for personnel, spare parts, etc). When total downtime is used, the resulting ratio is called operational availability (Ao).

Ao = MTBF / (MTBF + MDT)

Active repair time – the resulting ratio is called intrinsic availability (Ai).

Ai = MTBF / (MTBF + MTTR)

Where

MTBF = mean time between failures

MDT = mean downtime

MTTR = mean time to repair