Problem 4: The Guidance System Of A Ship Is Controlled By A

Problem 4s 5the Guidance System Of A Ship Is Controlled By A Computer

Problem 4s 5the Guidance System Of A Ship Is Controlled By A Computer

Problem 4S-5 The guidance system of a ship is controlled by a computer that has three major modules. In order for the computer to function properly, all three modules must function. Two of the modules have reliabilities of .91, and the other has a reliability of .96.

a. What is the reliability of the computer? (Round your answer to 4 decimal places.)

b. A backup computer identical to the one being used will be installed to improve overall reliability. Assuming the new computer automatically functions if the main one fails, determine the resulting reliability. (Round your intermediate calculations and final answers to 4 decimal places.)

c. If the backup computer must be activated by a switch in the event that the first computer fails, and the switch has a reliability of .97, what is the overall reliability of the system? (Both the switch and the backup computer must function in order for the backup to take over.) (Round your intermediate calculations and final answers to 4 decimal places.)

Paper For Above instruction

The reliability of a guidance system in a ship, which is controlled by a computer comprising three major modules, is a crucial factor in ensuring safe and efficient navigation. Given that two modules have reliabilities of 0.91 and the third has a reliability of 0.96, it is essential to calculate the probability that the entire system functions correctly. Furthermore, understanding the impact of backup systems and switching mechanisms on overall system reliability is vital for designing robust navigational controls.

Part A: Reliability of the Primary System

The primary system's reliability depends on all three modules functioning properly. Since the modules must all function for the system to be operational, the combined reliability under independent operation is the product of individual reliabilities:

Reliability of the system = R1 × R2 × R3

Given R1 = 0.91, R2 = 0.91, and R3 = 0.96, the calculation is:

Reliability = 0.91 × 0.91 × 0.96 = 0.7961776

Rounding to four decimal places, the reliability of the primary computer system is approximately 0.7962.

Part B: Reliability with Automatic Backup System

If a backup identical computer is installed and automatically functions if the primary fails, the combined reliability increases because the system can still operate successfully if either the primary or backup system functions. The probability that both systems fail simultaneously is the product of their failure probabilities:

Failure probability of primary system = 1 - 0.7962 ≈ 0.2038

Failure probability of backup system = same as primary, 0.2038

Therefore, the probability that both systems fail is:

0.2038 × 0.2038 ≈ 0.0415

The overall system reliability with the backup is thus:

1 - 0.0415 = 0.9585

After rounding to four decimal places, the reliability becomes 0.9585.

Part C: Reliability with Switch-Activated Backup

When the backup system is activated manually via a switch with reliability 0.97, both the switch and the backup computer must function for the backup to take over successfully. The combined reliability of the backup system in this case is the product of the backup computer reliability and switch reliability:

Backup system reliability = 0.96 (backup computer) × 0.97 (switch) = 0.9312

The total system reliability in this case depends on whether the primary system fails or not:

  • If the primary system functions, the system reliability is 0.7962.
  • If the primary fails (probability 0.2038), then the backup, and the switch must both function, with probability 0.9312.

Thus, the overall system reliability is:

Reliability = Primary system reliability + (failure probability of primary × backup system reliability)

= 0.7962 + (0.2038 × 0.9312) = 0.7962 + 0.1901 ≈ 0.9863

Rounding to four decimal places, the overall reliability with switch-activated backup is approximately 0.9863.

References

  • Carlson, B. (2018). Reliability Engineering and Risk Analysis. Springer.
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  • Mehran, K. (2019). System Reliability Modeling and Analysis. CRC Press.
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