Probability and Statistics with Reliability, Queuing and Computer Science Applications: Chapter 4 on Expected Value and Higher Moments

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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_at_ee.duke.edu, kst_at_ee.duke.edu

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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

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Higher Moments

• RVs X and Y (F(X)). Then,
• F(X) Xk, k1,2,3,.., EXk kth moment
• k1? Mean k2 Variance (Measures degree of
  variability)
• Example Exp(?) ? EX 1/ ? s2 1/?2

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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 EX(t)

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Binomial Random Variable (cont.)

• Y(t) is binomial with parameters n,p

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Poisson Distribution

• Probability mass function (pmf) (or discrete
  density function)
• Mean EN(t)

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Exponential Distribution

• Distribution Function
• Density Function
• Reliability
• Failure Rate
• failure rate is age-independent (constant)
• MTTF

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Exponential Distribution

• Distribution Function
• Density Function
• Reliability
• Failure Rate (CFR)
• Failure rate is age-independent (constant)
• Mean Time to Failure

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Weibull Distribution (cont.)

• Failure Rate
• IFR for DFR for
• MTTF
• Shape parameter ? and scale parameter ?

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Using Equations of the underlying Semi-Markov
Process (Continued)

• Time to the next diagnostic is uniformly
  distributed over (0,T)

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Using Equations of the underlying Semi-Markov
Process (Continued)
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E of a function of mutliple RVs

• If ZXY, then
• EXY EXEY (X, Y need not be independent)
• If ZXY, then
• EXY EXEY (if X, Y are mutually
  independent)

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Variance function of Mutliple RVs

• VarXYVarXVarY (If X, Y independent)
• CovX,Y EX-EXY-EY
• CovX,Y 0 and (If X, Y independent)
• Cross Cov terms may appear if not independent.
• (Cross) Correlation Co-efficient

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Moment Generating Function (MGF)

• For dealing with complex function of rvs.
• 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,

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MGF (contd.)

• Complex no. domain characteristics fn. transform
  is
• If X is Gaussian N(µ, s), then,

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MGF Properties

• If YaXb (translation scaling), then,
• Uniqueness property

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MGF Properties

• For the LST
• For the z-transform case
• For the characteristic function,

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MFG of Common Distributions

• Read sec. 4.5.1 pp.217-227

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MTTF Computation

• R(t) P(X gt t), X Lifetime of a component
• Expected life time or MTTF is
• In general, kth moment is,
• Series of components, (each has lifetime Exp(?i)
• Overall lifetime distribution Exp( ),
  and MTTF

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Series system (Continued)

• Other versions of Equation (2)

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Series System MTTF (contd.)

• RV Xi ith comps life time (arbitrary
  distribution)
• Case of least common denominator. To prove above

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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

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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

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TMR (Continued)

• Assuming that the reliability of a single
  component is given by,
• we get

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TMR (Continued)

• In the following figure, we have plotted RTMR(t)
  vs t as well as R(t) vs t.

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Homework 5

• specialize the bridge reliability formula to the
  case
• where
• Ri(t)
• find Rbridge(t) and MTTF for the bridge

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MTTF Computation (contd.)

• Parallel system life time of ith component is rv
  Xi
• X max(X1, X2, ..,Xn)
• If all Xis are EXP(?), then,
• As n increases, MTTF also increases as does the
  Var.

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Standby Redundancy

• A system with 1 component and (n-1) cold spares.
• Life time,
• If all Xis 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

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Cold standby
X
Y

• Lifetime of
• Active
• EXP(?)

Lifetime of Spare EXP(?)
Total lifetime 2-Stage Erlang
Assumptions Detection Switching perfect spare
does not fail
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Warm standby

• With Warm spare, we have
• Active unit time-to-failure EXP(?)
• Spare unit time-to-failure EXP(?)
• 2-stage hypoexponential
  distribution

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Warm standby derivation

• First event to occur is that either the active or
  the spare will fail. Time to this event is
  minEXP(?),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 ? .

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Hot standby

• With hot spare, we have
• Active unit time-to-failure EXP(?)
• Spare unit time-to-failure EXP(?)
• 2-stage
  hypoexponential

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The WFS Example
File Server
Computer Network
Workstation 1
Workstation 2
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RBD for the WFS Example
Workstation 1
File Server
Workstation 2
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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

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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

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Comparison Between Exponential and Weibull
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Homework 2

• For a 2-component parallel redundant system
• with EXP( ) behavior, write down expressions
  for
• Rp(t)
• MTTFp

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Solution 2
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Homework 3
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Homework 3

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

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Solution 3
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Homework 4

• Specialize formula (3) to the case where
• Derive expressions for system reliability and
  system meantime to failure.

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Homework 4
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Control channels-Voice channels Example
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Homework 5
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Homework 5

• specialize the bridge reliability formula to the
  case
• where
• Ri(t)
• find Rbridge(t) and MTTF for the bridge

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Bridge conditioning
C1
C2
C3 fails
S
T
C1
C2
C5
C4
C3
S
T
C3 is working
C5
C4
C1
C2
S
T
Factor (condition) on C3
C4
C5
Non-series-parallel block diagram
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Bridge Rbridge(t)
When C3 is working
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Bridge Rbridge(t)
When C3 fails
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Bridge Rbridge(t)
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Bridge MTTF
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Homework 7
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Homework 7

• Derive compare reliability expressions for
  Cold, Warm and Hot standby cases.

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Cold spare
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Warm spare
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Hot spare
EXP(2?)
EXP(?)
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Comparison graph
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Homework 8
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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.

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Solution 8
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TMR and TMR/simplexas hypoexponentials
TMR/Simplex
TMR