Section 6.2

Page 232 #1

Consider a stereo with CD player, FM–AM radio tuner, speakers (dual), and power amplifier (PA) components, as displayed with the reliabilities shown in Figure 6.11. Determine the system’s reliability. What assumptions are required in your model?

Define \(F(t)\) as the cumulative distrubution function of the failure rate,\(f(t)\), of a system or component, where \(t\) refers to time. The reliability \(R(t)\), of a system of component is the complement of \(F(t)\)

\(R(t) = 1-F(t)\)

rs <- c(0.95, 0.98, 0.97, 0.99, 0.99)
names(rs) <- c("PA","CD","Radio","SpeakerA","SpeakerB")

	rs
PA	0.95
CD	0.98
Radio	0.97
SpeakerA	0.99
SpeakerB	0.99

Given:

The diagram above, gives us the assumptions that the following subsystems operate in parallel.

Subsystem A: CD and FM/AM Radio
Subsystem B: Speaker A and Speaker B

In addition, it seems that the parallel subsystems run in serials with each other and the power amplifier.

Therefore, if any of the two subsystems or the Power Amplifier components fail, the system as a whole will fail.

Futhermore, we can assume that the individual components are independent of one another.

Lastly, we can assume that each component and subsystems are running in serials and have independent reliabilities.

Reliability Calculation

Reliability for Parallel Systems

\(R_s(t)\) can be calculated by the union of the reliabilty for the components by using the following formula:

\(\displaystyle R_s(t) = \bigcup_{i=1}^nR_i(t)\)

Calculating the respective reliability for each of the three parallel subsystems

# Subsystem A
r_s1 <- rs["CD"] + rs["Radio"] -  rs["CD"] * rs["Radio"]
names(r_s1) <- NULL

r_s1

## [1] 0.9994

# Subsystem B
r_s2 <- rs["SpeakerA"] + rs["SpeakerB"] -  rs["SpeakerA"] * rs["SpeakerB"]
names(r_s2) <- NULL

r_s2

## [1] 0.9999

Reliability for Systems in Series with Independent Components

\(\displaystyle R_s(t) = \prod_{i=1}^nR_i(t)\)

To calculate the reliability of the system as a whole, we have to assume the Power Amplifier runs in series with the two subsystems.

r_s_t <- rs["PA"] * r_s1 * r_s2

r_s_t

##        PA 
## 0.9493351

Based on the above model, the System has a 95% probability of properly functioning.

Data 609 Assignment #3.1