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 Analysis

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).

The Exponential Formula for Reliability

When the failure rate is constant, and distributed exponentially, the probability of survival (or reliability) at least as long as time t:
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 Meaning of Mean Time between Failures

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

The Relationship Between Part and System Reliability

Reliability of a System of Independent Events

It is often assumed that system reliability (i.e., the probability of survival Ps) is the product of the individual reliabilities of the n parts within the system:
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.

Reliability of a Component with a Single Backup

If two events are independent and success is defined as the probability that at least one of the events will occur, the probability of a component of two parts with a single backup is:
Px = P1 + (1 - P1) * P2, where Pn is the probability of each part's functioning when activated

Reliability of a Component with Multiple Backups

If multiple events are independent and success is defined as the probability that at least one of the events will occur, the probability of a component of n parts where each serve as a backup is:
Px = 1 - [(1 - P1) * (1 - P2) * ... * (1 - Pn)], where Pn is the probability of each part's functioning when activated

Availability

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