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Dept. of Electrical & Computer engineering Duke University

Probability and Statistics with Reliability, Queuing and Computer Science Applications: Chapter 4 on Expected Value and Higher Moments. Dept. of Electrical & Computer engineering Duke University Email: bbm@ee.duke.edu , kst@ee.duke.edu. Expected (Mean, Average) Value.

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Dept. of Electrical & Computer engineering Duke University

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  1. Probability and Statistics with Reliability, Queuing and Computer Science Applications: Chapter 4 onExpected Value and Higher Moments Dept. of Electrical & Computer engineering Duke University Email: bbm@ee.duke.edu, kst@ee.duke.edu

  2. Expected (Mean, Average) Value • There are several ways to abstract the information in the CDF into a single number: median, mode, mean. • Mean: • E(X) may also be computed using distribution function Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  3. Higher Moments • RV’s X and Y (=Φ(X)). Then, • Φ(X) = Xk, k=1,2,3,.., E[Xk]: kthmoment • k=1 Mean; k=2: Variance (Measures degree of variability) • Example: Exp(λ)  E[X]= 1/ λ; σ2 = 1/λ2 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  4. Bernoulli Random Variable • For a fixed t, X(t) is a random variable. The family of random variables {X(t), t 0} is a stochastic process. • Random variable X(t) is the indicator or Bernoulli random variable so that: • Probability mass function: • Mean E[X(t)]: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  5. Y(t) is binomial with parameters n,p Binomial Random Variable (cont.) Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  6. Poisson Distribution • Probability mass function (pmf) (or discrete density function): • Mean E[N(t)] : Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  7. Exponential Distribution • Distribution Function: • Density Function: • Reliability: • Failure Rate: failure rate is age-independent (constant) • MTTF: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  8. Exponential Distribution • Distribution Function: • Density Function: • Reliability: • Failure Rate (CFR): • Failure rate is age-independent (constant) • Mean Time to Failure: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  9. Failure Rate: IFR for DFR for MTTF: Shape parameter  and scale parameter  Weibull Distribution (cont.) Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  10. Using Equations of the underlying Semi-Markov Process (Continued) • Time to the next diagnostic is uniformly distributed over (0,T) Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  11. Using Equations of the underlying Semi-Markov Process (Continued) Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  12. E[ ] of a function of mutliple RV’s • If Z=X+Y, then • E[X+Y] = E[X]+E[Y] (X, Y need not be independent) • If Z=XY, then • E[XY] = E[X]E[Y] (if X, Y are mutually independent) Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  13. Variance: function of Mutliple RV’s • Var[X+Y]=Var[X]+Var[Y] (If X, Y independent) • Cov[X,Y] E{[X-E[X]][Y-E[Y]]} • Cov[X,Y] = 0 and (If X, Y independent) • Cross Cov[ ] terms may appear if not independent. • (Cross) Correlation Co-efficient: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  14. Moment Generating Function (MGF) • For dealing with complex function of rv’s. • Use transforms (similar z-transform for pmf) • If X is a non-negative continuous rv, then, • If X is a non-negative discrete rv, then, Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  15. MGF (contd.) • Complex no. domain characteristics fn. transform is • If X is Gaussian N(μ, σ), then, Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  16. MGF Properties • If Y=aX+b (translation & scaling),then, • Uniqueness property Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  17. MGF Properties • For the LST: • For the z-transform case: • For the characteristic function, Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  18. MFG of Common Distributions • Read sec. 4.5.1 pp.217-227 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  19. MTTF Computation • R(t) = P(X > t), X: Lifetime of a component • Expected life time or MTTF is • In general, kthmoment is, • Series of components, (each has lifetime Exp(λi) • Overall lifetime distribution: Exp( ), and MTTF = Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  20. Series system (Continued) • Other versions of Equation (2) Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  21. Series SystemMTTF (contd.) • RV Xi : ith comp’s life time (arbitrary distribution) • Case of least common denominator. To prove above Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  22. Homework 2: • For a 2-component parallel redundant system with EXP( ) behavior, write down expressions for: • Rp(t) • MTTFp • Further assuming EXP(µ) behavior and independent repair, write down expressions for: • Ap(t) • Ap • downtime Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  23. Homework 3: • For a 2-component parallel redundant system with EXP( ) and EXP( ) behavior, write down expressions for: • Rp(t) • MTTFp • Assuming independent repair at rates µ1and µ2, write down expressions for: • Ap(t) • Ap • downtime Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  24. TMR (Continued) • Assuming that the reliability of a single component is given by, we get: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  25. TMR (Continued) • In the following figure, we have plotted RTMR(t) vs t as well as R(t) vs t. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  26. Homework 5: • specialize the bridge reliability formula to the case where Ri(t) = • find Rbridge(t) and MTTF for the bridge Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  27. MTTF Computation (contd.) • Parallel system: life time of ith component is rv Xi • X = max(X1, X2, ..,Xn) • If all Xi’s are EXP(λ), then, • As n increases, MTTF also increases as does the Var. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  28. Standby Redundancy • A system with 1 component and (n-1) cold spares. • Life time, • If all Xi’s same,  Erlang distribution. • Read secs. 4.6.4 and 4.6.5 on TMR and k-out of-n. • Sec. 4.7 - Inequalities and Limit theorems Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  29. EXP() EXP() Cold standby Lifetime of Active EXP() X Y Lifetime of Spare EXP() Total lifetime 2-Stage Erlang Assumptions: Detection & Switching perfect; spare does not fail Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  30. EXP(+) EXP() Warm standby • With Warm spare, we have: • Active unit time-to-failure: EXP() • Spare unit time-to-failure: EXP() • 2-stage hypoexponential distribution Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  31. Warm standby derivation • First event to occur is that either the active or the spare will fail. Time to this event is min{EXP(),EXP()} which is EXP( + ). • Then due to the memoryless property of the exponential, remaining time is still EXP(). • Hence system lifetime has a two-stage hypoexponential distribution with parameters 1 =  + and 2 =  . Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  32. EXP(2) EXP() Hot standby • With hot spare, we have: • Active unit time-to-failure: EXP() • Spare unit time-to-failure: EXP() • 2-stage hypoexponential Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  33. The WFS Example File Server Computer Network Workstation 1 Workstation 2 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  34. RBD for the WFS Example Workstation 1 File Server Workstation 2 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  35. Rw(t): workstation reliability Rf (t): file-server reliability System reliability R(t) is given by: Note: applies to any time-to-failure distributions RBD for the WFS Example (cont.) Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  36. RBD for the WFS Example (cont.) • Assuming exponentially distributed times to failure: • failure rate of workstation • failure rate of file-server • The system mean time to failure (MTTF) is given by: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  37. Comparison Between Exponential and Weibull Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  38. Homework 2: • For a 2-component parallel redundant system with EXP( ) behavior, write down expressions for: • Rp(t) • MTTFp Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  39. Solution 2: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  40. Homework 3 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  41. Homework 3: • For a 2-component parallel redundant system with EXP( ) and EXP( ) behavior, write down expressions for: • Rp(t) • MTTFp Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  42. Solution 3: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  43. Homework 4: • Specialize formula (3) to the case where: • Derive expressions for system reliability and system meantime to failure. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  44. Homework 4 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  45. Control channels-Voice channels Example: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  46. Homework 5 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  47. Homework 5: • specialize the bridge reliability formula to the case where Ri(t) = • find Rbridge(t) and MTTF for the bridge Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  48. Bridge: conditioning C1 C2 C3 fails S T C1 C2 C4 C5 C3 S T C3 is working C4 C5 C1 C2 S T Factor (condition) on C3 C4 C5 Non-series-parallel block diagram Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  49. C1 C2 S T C4 C5 Bridge: Rbridge(t) When C3 is working Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

  50. C1 C2 S T C4 C5 Bridge: Rbridge(t) When C3 fails Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

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