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Reliability

Probability and Statistics
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Probability Theory
schedule Jul 1, 2022
Last updated
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Reliability can be defined as the probability that a system will run without failure for a given period of time.

A system is usually composed of individual units. We can define an event $U_i$ representing the case where the $i^{th}$ unit is up (i.e. not down / failing). Furthermore, we assume that whether an individual unit is up or down is independent of whether other units are up or down.

\begin{align*} U_i: i^{th} \text{ unit is up} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ U_1, U_2, \ldots , U_n \text{ are independent} \end{align*}

Similarly we can define an event $F_i$ to represent the case where the $i^{th}$ unit is down. Note that $\mathbb{P}(F_i) = 1 - \mathbb{P}(U_i)$.

In the below examples, for convenience we will use the following shorthand notation:

$$P_i = \text{probability that unit }i\text{ is up} = \mathbb{P}(U_i)$$

Examples

Series circuit

Consider the following system:

The calculate the probability the system is up (i.e. there exists a path from left to right where we only pass through "up" units):

\begin{align*} \mathbb{P}(\text{Sytem up}) &= \mathbb{P}(U_1 \cap {U_2} \cap {U_3}) \\ &= \mathbb{P}(U_1)\cdot\mathbb{P}(U_2)\cdot\mathbb{P}(U_3) \\ &= P_1P_2P_3 \end{align*}

Parallel circuit

Next consider the following system:

The calculate the probability the system is up:

\begin{align*} \mathbb{P}(\text{Sytemup})&=\mathbb{P}(U_1\cup{U_2}\cup{U_3})\\ &=1 - \mathbb{P}(F_1\cap {F_2}\cap {F_3})\\ &=1 - \mathbb{P}(F_1)\cdot\mathbb{P}(F_2)\cdot\mathbb{P}(F_3) \\ &=1 - (1-P_1)(1-P_2)(1-P_3) \end{align*}

Here we make use of De Morgan's laws to convert from $\mathbb{P}(U_1\cup{U_2}\cup{U_3})$ to $1 - \mathbb{P}(F_1\cap {F_2}\cap {F_3})$, making use of the fact that $U_i^C = F_i$.

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