Chapter 15, Problem 25P

(a)

Summary Introduction

Interpretation:

Theten year reliability of D. T.’s current satellite design.

Concept Introduction:

An overall consistency of a measure is known asreliability. If the measure produces similar results under consistent conditions then it is called a high reliability.

Explanation of Solution

Given information:

D. T. is designing a new workhorse satellite to eventually replace all the orbital satellite it manages in its telecommunication network. The design of this satellite consists of five interdependent subsystems, as indicated here:

  

Figure $\left(1\right)$

The satellite Damon currently has in place use subsystems that are $99\%$ reliable over a $10$ year period, but D. T. is annoyed to have already lost use of two of these older satellites. Some managers argue that the new generation of satellite should feature two computers and two solar arrays, a few argue that subsystem known to have been lost in D.T.’s failed satellites. A few argue that the antennae array and transmitter/repeater subsystem represent obsolete satellite design that invites problem: both these systems should be designed as one integrated system built by one contractor.

D. T. has in place satellite, each of which has five sub system. The subsystems are antenna array, solar array, computer, transmitter/repeater and positioning system. The subsystems has a reliability of $99\%$ each over a $10$ year period.

Calculate the reliability of the current satellite design,

Reliability for the $10$ years of the antenna array is,

  $R_1=0.99$

Reliability for the $10$ years of the solar array is,

  $R_2=0.99$

Reliability for the $10$ years of the computer is,

  $R_3=0.99$

Reliability for the $10$ years of the positioning system is,

  $R_4=0.99$

Reliability for the $10$ years of the transmitter/repeater is,

  $R_5=0.99$

The reliability for the $10$ years of the current satellite design $R_s$ is the combined reliability of all the five sub system.

  $R_s=R_1\times R_2\times R_3\times R_4\times R_5$

Substitute the value,

  $\begin{array}{c}{R}_s=0.99\times 0.99\times 0.99\times 0.99\times 0.99\\ =0.95099\end{array}$

Rounding off the $10$ years reliability of the D. T.’s current satellite design is

  $R_s=95.1\%$ .

(b)

Summary Introduction

Interpretation:

The reliability of the computer sub system, if a redundant computer were added.

Concept Introduction:

An overall consistency of a measure is known asreliability. If the measure produces similar results under consistent conditions then it is called a high reliability.

Explanation of Solution

Given information:

D. T. is designing a new workhorse satellite to eventually replace all the orbital satellite it manages in its telecommunication network. The design of this satellite consists of five interdependent subsystems, as indicated here;

  

Figure $\left(1\right)$

The satellite Damon currently has in place use subsystems that are $99\%$ reliable over a $10$ year period, but D. T. is annoyed to have already lost use of two of these older satellites. Some managers argue that the new generation of satellite should feature two computers and two solar arrays, a few argue that subsystem known to have been lost in D’s failed satellites. A few argue that the antennae array and transmitter/repeater subsystem represent obsolete satellite design that invites problem: both these systems should be designed as one integrated system built by one contractor.

The redundant computer has a reliability of,

  $R_{SR}=0.99$

The reliability of the computer system $R_{CS}$ is worked out, and given the computer system has a reliability of $R_3=0.99$ is backed by a redundant computer of the same reliability,

  $R_{SR}=0.99$

Calculate the combined reliability $R_{CS}$ as shown in figure $\left(2\right)$ .

  

Figure $\left(2\right)$

The reliability of $R_{CS}$ is the probability that the system will not fail.

Calculate the combined probability,

  $R_{CS}=1-\left[\left(1-R_3\right)\times \left(1-R_{SR}\right)\right]$

Substitute the value,

  $\begin{array}{c}{R}_{CS}=1-\left[\left(1-0.99\right)\times \left(1-0.99\right)\right]\\ =1-\left[\left(0.01\right)\times \left(0.01\right)\right]\\ =1-0.0001\\ =0.9999\end{array}$

Therefore, the reliability of the computer system will be $99.99\%$ .

(c)

Summary Introduction

Interpretation:

The $10$ year reliability of the satellite, if the computer and soar array subsystem were duplicated as proposed.

Concept Introduction:

An overall consistency of a measure is known asreliability. If the measure produces similar results under consistent conditions then it is called a high reliability.

Explanation of Solution

Given information:

D. T. is designing a new workhorse satellite to eventually replace all the orbital satellite it manages in its telecommunication network. The design of this satellite consists of five interdependent subsystems, as indicated here;

  

Figure $\left(1\right)$

The satellite Demon currently has in place use subsystems that are $99\%$ reliable over a $10$ year period, but D. T. is annoyed to have already lost use of two of these older satellites. Some managers argue that the new generation of satellite should feature two computers and two solar arrays, a few argue that subsystem known to have been lost in D’s failed satellites. A few argue that the antennae array and transmitter/repeater subsystem represent obsolete satellite design that invites problem: both these systems should be designed as one integrated system built by one contractor.

If the solar array and the computer system are duplicated, the duplication raises the reliability from $0.99$ to $0.9999$ ,

Reliability for the $10$ years of the antenna array is,

  $R_1=0.99$

Reliability for the $10$ years of the solar array is,

  $R_2=0.9999$

Reliability for the $10$ years of the computer is,

  $R_3=0.9999$

Reliability for the $10$ years of the positioning system is,

  $R_4=0.99$

Reliability for the $10$ years of the transmitter/repeater is,

  $R_5=0.99$

The reliability for the $10$ years of the current satellite design $R_{CS}$ is the combined reliability of all the five sub system.

  $R_{CS}=R_1\times R_2\times R_3\times R_4\times R_5$

Substitute the value,

  $\begin{array}{c}{R}_{CS}=0.99\times 0.9999\times 0.9999\times 0.99\times 0.99\\ =0.9701\end{array}$

Rounding off the $10$ years reliability of the D. T.’s current satellite design is

  $R_{CS}=97.01\%$ .

(d)

Summary Introduction

Interpretation:

The resulting reliability of the satellite.

Concept Introduction:

An overall consistency of a measure is known asreliability. If the measure produces similar results under consistent conditions then it is called a high reliability.

Explanation of Solution

Given information:

D. T. is designing a new workhorse satellite to eventually replace all the orbital satellite it manages in its telecommunication network. The design of this satellite consists of five interdependent subsystems, as indicated here;

  

Figure $\left(1\right)$

The satellite Damon currently has in place use subsystems that are $99\%$ reliable over a $10$ year period, but D. T. is annoyed to have already lost use of two of these older satellites. Some managers argue that the new generation of satellite should feature two computers and two solar arrays, a few argue that subsystem known to have been lost in D’s failed satellites. A few argue that the antennae array and transmitter/repeater subsystem represent obsolete satellite design that invites problem: both these systems should be designed as one integrated system built by one contractor.

If no redundant elements were added, but the antennae array and transmitter/repeater were successfully redesigned into a single subsystem with a $10$ year reliability of $99\%$ .

When the antenna array and transmitter/repeater are redesigned as a single system with a $10$ year reliability of $99\%$

Calculate the resulting reliability,

Reliability for the $10$ years of the antenna array and the transmitter/repeater is,

  $R_1=0.99$

Reliability for the $10$ years of the solar array is,

  $R_2=0.99$

Reliability for the $10$ years of the computer is,

  $R_3=0.99$

Reliability for the $10$ years of the positioning system is,

  $R_4=0.99$

The reliability for the $10$ years of the current satellite design $R_S$ is the combined reliability of all the four sub system.

  $R_S=R_1\times R_2\times R_3\times R_4$

Substitute the value,

  $\begin{array}{c}{R}_{CS}=0.99\times 0.99\times 0.99\times 0.99\\ =0.9606\end{array}$

Rounding off the $10$ years reliability of the D. T.’s current satellite design is

  $R_S=96.06\%$ .