1. Expected Value and MTTF Calculation
In the Name of the Most High
Expected Value and MTTF Calculation
Behzad Akbari
Spring 2009
Tarbiat Modares University
These slides are based on the slides of Prof. K.S. Trivedi (Duke University)
4/21/2019

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.

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’

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:
Mean E[X(t)]:

5. Binomial Random Variable (cont.)
Y(t) is binomial with parameters n,p

6. Poisson Distribution
Probability mass function (pmf) (or discrete density function):
Mean E[N(t)] :

7. Exponential Distribution
Distribution Function:
Density Function:
Reliability:
Failure Rate:
failure rate is age-independent (constant)
MTTF:

8. Weibull Distribution (cont.)
Failure Rate:
IFR for DFR for
MTTF:
Shape parameter  and scale parameter 

9. 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)

10. 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:

11. 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.

12. MGF Properties If Y=aX+b (translation & scaling), then,
Uniqueness property
Summation in one domain  convolution in the other domain.

13. MGF Properties
For the LST:
For the z-transform case:

14. 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.

15. Series system (Continued)
Other versions of Equation (2)

16. Series System MTTF (contd.)
RV Xi : ith comp’s life time (arbitrary distribution)
Case of least common denominator. To prove above

17. TMR (Continued)
Assuming that the reliability of a single component is given by,
we get:

18. TMR (Continued)

19. 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.

20. 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.

21. 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

22. 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()

23. 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 =  .

24. 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()

25. The WFS Example File Server Computer Network Workstation 1

26. RBD for the WFS Example
Workstation 1
File Server
Workstation 2

27. 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

28. 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:

29. Homework 1: For a 2-component parallel redundant system
with EXP( ) behavior, write down expressions for:
Rp(t)
MTTFp

30. Solution 1:

31. Homework 2: For a 2-component parallel redundant system
with EXP( ) and EXP( ) behavior, write down expressions for:
Rp(t)
MTTFp

32. Solution 2:

33. Homework 3: specialize the bridge reliability formula to the case
where
Ri(t) =
find Rbridge(t) and MTTF for the bridge

34. 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

35. Bridge: Rbridge(t)
When C3 is working C1 C4 C2 C5 S T

36. Bridge: Rbridge(t) C1 C5 C2 C4 S T
When C3 fails

37. Bridge: Rbridge(t)

38. Bridge: MTTF

39. Homework 4:
Derive & compare reliability expressions for Cold, Warm and Hot standby cases.

40. Cold spare:
EXP()

41. Warm spare:
EXP(+ )
EXP()

42. Hot spare:
EXP(2)
EXP()

43. Comparison graph:

44. TMR and TMR/simplex as hypoexponentials

45. Conditional Probability and Expectation
4/21/2019

46. Conditional pmf Conditional probability:
Above works if x is a discrete rv.
For discrete rv’s X and Y, conditional pmf is,
Above relationship also implies,
Hence we have another version of the theorem of total probability
If x and y are mutually independent, then, p(y|x) = p(y).

47. Independence, Conditional Distribution
Conditional distribution function
Using conditional pmf,

48. Example n total jobs, k are passed on to server A Two servers k jobs
p: prob. that the next job
goes to server A p A
Poisson ( λ)
Job stream
Bernoulli trial
n jobs
1-p B
n total jobs, k are passed on to server A
pY(k) = [(pk/k!)e-λ \sum_{k-n}^\infty \frac{\lambda^k (\lambda (1-p))^{n-k}}{(n-k)!}
= (\lamda p)^k / k! \sum_{n=k}^\infty \frac{(\lambda (1-p))^{n-k}}{n-k!}
substituting m= n-k gives the result.

49. Conditional pdf For continuous rv’s X and Y, conditional pdf is, Also,
Independent X, Y 
Marginal pdf (cont. version of the TTP),
Conditional distribution function

50. Conditional Reliability
Software system after having incurred (i-1) faults,
Ri(t) = P(Ti > t) (Ti : inter-failure times)
Ti : independent exponentially distributed Exp(λi).
λi : Failure rate itself may be random, then
Conditional reliability:
λi:: random  that with fault occurrence, followed by its removal, failure rate λi changes (in a random manner denoted by the random variable Λi ). If ψi = a1+a2i, then what happens to E[Λi]and R_i(t) ? E[Λi] = 1/ψi = i.e. decreasing with i and there is a likelihood (but no guarantee) that R_i(t) improves with time.

51. Conditional Moments Conditional Expectation is E[Y|X=x] or E[Y|x]
E[Y|x]: a.k.a regression function
For the discrete case,
In general, then,
The mathematical function used to describe the deterministic variation in the response variable is sometimes called the "regression function", the "regression equation", the "smoothing function", or the "smooth". [

52. Conditional MTTF
Y: time-to-failure may depend on the temperature, and the conditional MTTF may be:
Let Temp be normal,
Unconditional MTTF is: