1. Dept. of Electrical & Computer engineering
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
1/16/2019

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
In case, the summation or the integration does is not absolutely convergent, then E(X) does not exist.
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]: kth moment
k=1 Mean; k=2: Variance (Measures degree of variability)
Example: Exp(λ)  E[X]= 1/ λ; σ2 = 1/λ2
shape of the pdf (or pmf) for small and large variance values.
σ is commonly referred to as the ‘standard deviation’
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. Binomial Random Variable (cont.)
Y(t) is binomial with parameters n,p
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. Weibull Distribution (cont.)
Failure Rate:
IFR for DFR for
MTTF:
Shape parameter  and scale parameter 
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,
M[θ] is not guaranteed to exist. But for most distributions of our interest, it does exist.
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
Summation in one domain  convolution in the other domain.
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 pp
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, kth moment is,
Series of components, (each has lifetime Exp(λi)
Overall lifetime distribution: Exp( ), and MTTF =
The last equality follows from by integrating by parts,
int_0^∞ t R’(t) = -t R(t)|0 to ∞ + Int_0^∞ R(t)
-t R(t) 0 as t ∞ since R(t)  0 faster than t  ∞. Hence, the first term disappears.
Note that the MTTF of a series system is much smaller than the MTTF of an individual component. Failure of any component implies failure of the overall system.
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 System MTTF (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 µ1 and 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.
These are notes.
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 and on TMR and k-out of-n.
Sec Inequalities and Limit theorems
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

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

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

33. The WFS Example File Server Computer Network Workstation 1
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. RBD for the WFS Example (cont.)
Rw(t): workstation reliability
Rf (t): file-server reliability
System reliability R(t) is given by:
Note: applies to any time-to-failure distributions
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 Non-series-parallel block diagram C1 C2 C3 fails
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. Bridge: Rbridge(t) When C3 is working C1 C4 C2 C5 S T
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

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

51. Bridge: Rbridge(t)
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

52. Bridge: MTTF
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

53. Homework 7
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

54. Homework 7:
Derive & compare reliability expressions for Cold, Warm and Hot standby cases.
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

55. Cold spare:
EXP()
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

56. Warm spare: EXP(+ ) EXP()
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

57. Hot spare: EXP(2) EXP()
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

58. Comparison graph:
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

59. Homework 8
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

60. Homework 8:
For the 2-component system with non-shared repair, use a reliability block diagram to derive the formula for instantaneous and steady-state availability.
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

61. Solution 8:
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

62. TMR and TMR/simplex as hypoexponentials
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University