Reliability

Reliability is defined as the ability of an item to perform as required in a stated operating context and for a stated period of time.

Reliability could be measured in various way, e.g., by the

Availability
Expected number of failure per unit time
Mean time to failure
Probability that the item survives a given time period, i.e., survival probability

Availability

Availability is defined as the ability of an item (under combined aspects of its reliability, maintainability, and maintenance support) to perform its required function at a stated instant of time or over a stated period of time

To quantify availability we often use the following formula:

$ \color{blue}{A =\frac{\mathrm{MTTF}}{\mathrm{MTTF}+\mathrm{MDT}} } $ where

$ \color{blue}{\mathrm{MTTF}=} $ Mean Time To Failure
$ \color{blue}{\mathrm{MDT}=} $ Mean Down Time

Availability could be interpreted as the portion of time the item is functioning or the probability that the item is functioning at an arbitrary point of time.

Reliability of series and parallel structures

A series structure is a conceptual understanding of a system where all items in the system need to function in order for the system to function. A parallel structure is a conceptual understanding of a system where it is required that at least one item is functioning in order to ensure that the system is functioning.

If availability is used as a metric for reliability, and items in a series structure are stochastically independent, then the system reliability is given by:

$ \color{blue}{A =\prod_i A_i } $ (series structure, product of availabilities)

and for a parallel structure we similarly have:

$ \color{blue}{A =1-\prod_i \left( 1- A_i \right)} $ (parallel structure, co-product of availabilities)

For a parallel structure of two components we have:

$ \color{blue}{A =A_1 + A_2 - A_1 A_2} $ (parallel structure of two components)