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Chapter 8 – System Reliability

Chapter 8 – System Reliability. Motivation. How do you know how long your design is going to last? Is there any way we can predict how long it will work? Why do Reliability Engineers get paid so much?. 8.1 Probability Review. Definitions Random Experiment Event or Outcomes Event Space.

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Chapter 8 – System Reliability

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  1. Chapter 8 – System Reliability

  2. Motivation • How do you know how long your design is going to last? • Is there any way we can predict how long it will work? • Why do Reliability Engineers get paid so much?

  3. 8.1 Probability Review Definitions • Random Experiment • Event or Outcomes • Event Space

  4. Axioms of Probability • What is an axiom? • 2 of the 3 axioms

  5. Probability Density Functions • What is a random variable (RV)? • What is a PDF? • Math Definition

  6. Common PDFs Normal Density

  7. Common PDFs Uniform Density

  8. Data for lifelengths of batteries (in hundreds of hours) 0.406 2.343 0.538 5.088 5.587 2.563 0.023 3.334 3.491 1.267 0.685 1.401 0.234 1.458 0.517 0.511 0.225 2.325 2.921 1.702 4.778 1.507 4.025 1.064 3.246 2.782 1.514 0.333 1.624 2.634 1.725 0.294 3.323 0.774 2.330 6.426 3.214 7.514 0.334 1.849 8.223 2.230 2.920 0.761 1.064 0.836 3.810 0.968 4.490 0.186

  9. Common PDFs Exponential Density

  10. Example: Based on the data for lifelengths of batteries • previously given, the random variable X representing • the lifelength has associated with it an exponential • density function with  = 0.5. • Find the probability that the lifelength of a particular • battery is less than 200 or greater than 400 hours. • (b) Find the probability that a battery lasts more than • 300 hours given that it has already been in use for • more than 200 hours.

  11. Square Roots of the lifelengths of batteries 0.637 0.828 2.186 1.313 2.868 1.531 1.184 1.228 0.542 1.493 0.733 0.484 2.006 1.823 1.709 2.256 1.207 1.032 0.880 0.872 2.364 0.719 1.802 1.526 1.032 1.601 0.715 1.668 2.535 0.914 0.152 0.474 1.230 1.793 1.952 1.826 1.525 0.577 2.741 0.984 1.868 1.709 1.274 0.578 2.119 1.126 1.305 1.623 1.360 0.431 WeiBull Distribution

  12. Example: The length of service time during which a certain type of thermistors produces resistances within its • specifications has been observed to follow a Weibull • Distribution with  = 1/50 and  = 2 (measurements in • thousand of hours). • Find the probability that one of these thermistors, to be installed in a system today, will function properly for over 10,000 hours. • (b) Find the expected lifelength for the thermistor of this • type.

  13. 8.2 Reliability Prediction • Reliability (defn) • Failure Rate

  14. The Bathtub Curve

  15. Derivations • See the book for derivation of R(t). • If the failure rate is constant, then R(t) = ?

  16. Example: Consider a transistor with a constant failure rate of •  = 1/106 hours. • What is the probability that the resistor will be operable in 5 years? • (b) Determine the MTTF and the reliability at the MTTF.

  17. Series Systems • Def’n (Series System) = • We model this as

  18. Parallel Systems Definition: Redundancy Definition: Parallel System

  19. Parallel System Model

  20. Example: Redundant Array of Independent disks (RAID) In a RAID, multiple hard drives are used to store the same data, thus achieving redundancy and increased reliability. One or more of the disks in the system can fail and the data can still be recovered. However, if all disks fail, then the data is lost. Assume that each disk drive has a failure rate of  = 10 failures/106 hrs. How many disks must the system have to achieve a reliability of 98% in 10 years?

  21. Combination Systems

  22. Quiz: Redundant Array of Independent disks (RAID) Your company intends to design, manufacture, and market a new RAID for network servers. The system must be able to store a total of 500 GB of user data and must have a reliability of at least 95% in 10 years. In order to develop the RAID system, 20-GB drives will be designed and utilized. To meet the requirement, you have decided to use a bank of 25 disks (25x20 GB = 500 GB) and utilize a system redundancy of 4 (each of the 25 disks has a redundancy of 4). What must the reliability of the 20 GB drive be in 10 years in order to meet the overall system reliability requirement?

  23. Failure Rate Estimates • What factors influence the failure rate?

  24. Failure rate estimates • Low Frequency FET, Appendix C. • How would you find each of these?

  25. Example 8.4

  26. Physical Model

  27. Resistive Model

  28. Physical Model with Heat Sink

  29. Resistive Model

  30. Power Derating Curve

  31. 8.3 System Reliability • So far, we have only looked at a single device. • We are interested in collection of devices into a system! • For example

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