Learning Objectives:

• Define reliability.

• Perform reliability computations.

• Explain the purpose of redundancy in a system.

Most products are made up of a number of components. The reliability of each component and the configuration of the system consisting of these components determines the system reliability (i.e., the reliability of the product).

Components connected in Series:

If the components are in series, system performs satisfactorily if all components are fully functional. For example, Christmas lights. \[R = \prod_{i = 1}^{n} P_i\]

Components connected in Parallel:

If the components are in parallel, system performs if any one component remains operational. For example, an airplane with four engines, or a laptop with a power source and a battery. \[R = 1- \prod_{i = 1}^{n}( 1- P_i)\]

Components connected Combination of Series and Parallel:

In such cases, first find out the reliability of series branches and then apply parallel reliability formula.

Solved Example: 99-1-01

A module of a satellite monitoring system has 500 components in series. The reliability of each component is 0.999. Find the reliability of the module.

A. $0.999^{500}$

B. 1 - $0.001^{500}$

C. $500^{0.999}$

D. $0.999^{499}$

In general, if there are n components in series, where the reliability of the $i^{th}$ component is denoted by $R_i$, the system reliability is: \[R_S = R_1 \times R_2 \times R_3 \times....R_n\]

Correct Answer: A

Solved Example: 99-1-02

A system consists of four components in parallel having reliabilities of 0.99, 0.95, 0.98 and 0.97. What is the reliability of the system?

A. 0.99997

B. 0.9999997

C. 0.9997

D. 0.997

\begin{align*} P(\mathrm{failure)} &= P(\mathrm{all\ four\ fail)}\\ &= (1-0.99)(1-0.95)(1-0.98)(1-0.97) \\&= 0.01 \times 0.05 \times 0.02 \times 0.03\\ &= 0.0000003 \end{align*} \begin{align*} P(\mathrm{Success}) &= 1- P(\mathrm{failure})\\ &= 1-0.0000003 \\&= 0.9999997 \end{align*}

Correct Answer: B

Solved Example: 99-1-03

Proper operation of a power plant requires three control panels to work A, B and C. The reliability of these three control panels are respectively 0.75, 0.95, and 0.90. C is a standby for B that comes into operation only if B fails. The overall system reliability is:

A. 0.7125

B. 0.7775

C. 0.74625

D. 0.8425

Plant will work in two cases:
First case: (A AND B) = 0.75 $\times$ 0.95 = 0.7125
Second case:
Let $\bar{B}$ represent probability of failure of B = 1-0.95 =0.05
(A AND $\bar{B}$ AND C) = 0.75 $\times$ 0.05 $\times$ 0.90 = 0.03375
Total probability = 0.7125 + 0.03375 = 0.74625

Correct Answer: C

Solved Example: 99-1-04

A monitoring system has seven components. The reliability of each component is shown in the figure. The system reliability is: (GATE Production and Industrial 2019)

A. 0.174

B. 0.404

C. 0.556

D. 0.872

Now concentrate on parallel part only. \[ R = 1- \prod_{i = 1}^{n}( 1- P_i)\] P(working) = 1 - (0.65)(0.5)(0.4) = 0.87
P(working) = (0.5)(0.87)(0.4) = 0.174

Correct Answer: A