Probability and Statistics with Reliability, Queuing and Computer Science Applications: Chapter 5 on Conditional Probability and Expectation

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Probability and Statistics with Reliability,
Queuing and Computer Science Applications
Chapter 5 on Conditional Probability and
Expectation

• Dept. of Electrical Computer engineering
• Duke University
• Email bbm_at_ee.duke.edu, kst_at_ee.duke.edu

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Conditional pmf

• Conditional probability
• Above works if x is a discrete rv.
• For discrete rvs X and Y, conditional pmf is,
• Above relationship also implies,
• Hence we have another version of the theorem of
  total probability

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Independence, Conditional Distribution

• Conditional distribution function
• Using conditional pmf,

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Example

• Two servers

• p prob. that the next job
• goes to server A

k jobs
A
p
Bernoulli trial
Poisson ( ?) Job stream
n jobs
1-p
B

• n total jobs, k are passed on to server A

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Conditional pdf

• For continuous rvs X and Y, conditional pdf is,
• Also,
• Independent X, Y ?
• Marginal pdf (cont. version of the TTP),
• Conditional distribution function

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Conditional Reliability

• Software system after having incurred (i-1)
  faults,
• Ri(t) P(Ti gt t) (Ti inter-failure times)
• Ti independent exponentially distributed
  Exp(?i).
• ?i Failure rate itself may be random, then
• Conditional reliability

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Mixture Distribution

• Conditional distribution continuous and discrete
  rvs combined.
• Examples (Response time that there k
  processors), (Reliability k components) etc. (Y
  continuous, Xdiscrete)
• Compute server with r classes of jobs
  (i1,2,..,r)
• Hence, Y follows an r-stage HyperExpo
  distribution.

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Mixture Distribution (contd.)

• What if fYX(yi) is not Exponential?
• The unconditional pdf and CDF are,
• Using LST,
• Moments can now be found as,

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Mixture Distribution (contd.)

• Such Mixture distrib. arise in reliability
  studies.
• Software system Modules (or objects) may have
  been written by different groups or companies,
  ith group contributes ai fraction of modules and
  has reliability characteristic given by Fi.
• Gp1 EXP( ?1) (a frac) Gp2 r-stage Erlang (1-
  a frac)

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Mixture Distribution (contd.)

• Ycontinuous X continuous or uncountable, e.g.,
  life time Y depends on the impurities X.
• Finally, Ydiscrete X continuous

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Mixture Distribution (contd.)

• X web server response time Y of requests
  arriving while a request being serviced.
• For a given value of Xx, Y is Poisson,
• The joint pdf is, f(x,y) pYX(yx)fX(x)
• Unconditional pmf pY(y) P(Yy)
• With (?µ)x w,

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Conditional Moments

• Conditional Expectation is EYXx or EYx
• EYx a.k.a regression function
• For the discrete case,
• In general, then,

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Conditional Moments (contd.)

• This can be specialized to
• kth moment of Y EYkXx
• Conditional MGF, MYX(? x) Ee?YXx
• Conditional LST, LYX(sx) Ee-sYXx
• Conditional PGF, GYX(zx) EzYXx
• Total Expectation
• Total moments

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Conditional Moments (contd.)

• Total transforms
• In the previous example,
• Total expectation
• Therefore, we can also talk of conditional MTTF
• MTTF may depend on impurities or operating temp.

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Conditional MTTF

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

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Imperfect Fault Coverage

• Hybrid k-out of-n system, with m cold standbys.
• Reliability depends on recovery from a failure.
  What if the failed module cannot be substituted
  by a standby? These are called not covered
  faults.
• Probability that a fault is covered is c
  (coverage factor)

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Fault Handling Phases

• Fault handling involves 3-distinct phases.
• Finite success probability for each phase ?
  finite coverage.
• c P(ok recoveryfault occurs)
• P(fault detected fault located
  fault corrected fault occurs)
• cd.cl.cr

Fault Processing
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Near Coincident Faults

• Coincident fault 2nd fault occurs while the 1st
  one has not been completely processed.
• Y Random time to process a fault.
• X Time at which coincident fault occurs
  (EXP(?)).
• Fault coverage prob. that Y lt X

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Near Coincidence Fault Coverage

• Fault handling has multiple phases. This gives
• XLife time of a system with one active one
  standby
• ? Active components failure rate
• Y 1 ? fault covered Y 0 ? fault not
  covered.
• c0 or c1?

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Life Time Distribution-Limited Coverage

• fXY(t0) life time of the active comp. EXP(?)
• fXY(t1) life time of activestandby 2-stage
  Erlang
• Joint density fn
• Marginal density fn
• Reliability