Continuous Time Markov Chains

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Chapter 8

• Continuous Time Markov Chains

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Markov Availability Model
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2-State Markov Availability Model

• 1) Steady-state balance equations for each state
• Rate of flow IN rate of flow OUT
• State1
• State0
• 2 unknowns, 2 equations, but there is only
  one independent equation.

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2-State Markov Availability Model(Continued)

• 1) Need an additional equation

Downtime in minutes per year
876060
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2-State Markov Availability Model(Continued)

• 2) Transient Availability
• for each state
• Rate of buildup rate of flow IN - rate of flow
  OUT
• This equation can be solved to obtain assuming
  P1(0)1

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2-State Markov Availability Model(Continued)

• 3)
• 4) Steady State Availability

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Markov availability model

• Assume we have a two-component parallel redundant
  system with repair rate ?.
• Assume that the failure rate of both the
  components is ?.
• When both the components have failed, the system
  is considered to have failed.

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Markov availability model (Continued)

• Let the number of properly functioning components
  be the state of the system. The state space is
  0,1,2 where 0 is the system down state.
• We wish to examine effects of shared vs.
  non-shared repair.

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Markov availability model (Continued)
2
1
0
Non-shared (independent) repair
2
1
0
Shared repair
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Markov availability model (Continued)

• Note Non-shared case can be modeled solved
  using a RBD or a FTREE but shared case needs the
  use of Markov chains.

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Steady-state balance equations

• For any state
• Rate of flow in Rate of flow out
• Consider the shared case
• ?i steady state probability that system is in
  state i

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Steady-state balance equations (Continued)

• Hence
• Since
• We have
• or

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Steady-state balance equations (Continued)

• Steady-state unavailability ?0 1 - Ashared
• Similarly for non-shared case,
• steady-state unavailability 1 - Anon-shared
• Downtime in minutes per year (1 - A) 876060

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Steady-state balance equations
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Homework 5

• Return to the 2 control and 3 voice channels
  example and assume that the control channel
  failure rate is ?c, voice channel failure rate is
  ?v.
• Repair rates are ?c and ?v, respectively.
  Assuming a single shared repair facility and
  control channel having preemptive repair priority
  over voice channels, draw the state diagram of a
  Markov availability model. Using SHARPE GUI,
  solve the Markov chain for steady-state and
  instantaneous availability.

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Markov Reliability Model
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Markov reliability model with repair

• Consider the 2-component parallel system but
  disallow repair from system down state
• Note that state 0 is now an absorbing state. The
  state diagram is given in the following figure.
• This reliability model with repair cannot be
  modeled using a reliability block diagram or a
  fault tree. We need to resort to Markov chains.
• (This is a form of dependency since in order
  to repair a component you need to know the status
  of the other component).

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Markov reliability model with repair (Continued)
Absorbing state

• Markov chain has an absorbing state. In the
  steady-state, system will be in state 0 with
  probability 1. Hence transient analysis is of
  interest. States 1 and 2 are transient states.

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Markov reliability model with repair (Continued)

• Assume that the initial state of the Markov chain
• is 2, that is, P2(0) 1, Pk (0) 0 for k 0,
  1.
• Then the system of differential Equations is
  written
• based on
• rate of buildup rate of flow in - rate of flow
  out
• for each state

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Markov reliability model with repair
(Continued)
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Markov reliability model with repair
(Continued)

• After solving these equations, we get
• R(t) P2(t) P1(t)
• Recalling that
  , we get

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Markov reliability model with repair
(Continued)

• Note that the MTTF of the two component
  parallel redundant system, in the absence
• of a repair facility (i.e., ? 0), would
  have
• been equal to the first term,
• 3 / ( 2? ), in the above expression.
• Therefore, the effect of a repair facility is
  to
• increase the mean life by ? / (2?2), or by a
  
• factor

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Markov Reliability Model with Imperfect Coverage
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Markov model with imperfect coverage

• Next consider a modification of the above
• example proposed by Arnold as a model of
• duplex processors of an electronic
• switching system. We assume that not all
• faults are recoverable and that c is the
• coverage factor which denotes the
• conditional probability that the system
• recovers given that a fault has occurred.
• The state diagram is now given by the
• following picture

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Now allow for Imperfect coverage
c
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Markov modelwith imperfect coverage (Continued)

• Assume that the initial state is 2 so that
• Then the system of differential equations are

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Markov model with imperfect coverage (Continued)

• After solving the differential equations we
  obtain
• R(t)P2(t) P1(t)
• From R(t), we can system MTTF
• It should be clear that the system MTTF and
  system reliability are
• critically dependent on the coverage factor.

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SOURCES OF COVERAGE DATA

• Measurement Data from an Operational system
  Large amount of data needed
• Improved Instrumentation Needed
• Fault/Error Injection Experiments
• Costly yet badly needed tools from
• CMU, Illinois, Toulouse

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SOURCES OF COVERAGE DATA (Continued)

• A Fault/Error Handling Submodel
• Phases of FEHM
• Detection, Location, Retry, Reconfig, Reboot
• Estimate Duration Prob. of success of each
  phase
• IBM(EDFI), HARP(FEHM), Draper(FDIR)

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Homework 6

• Modify the Markov model with imperfect coverage
  to allow for finite time to detect as well as
  imperfect detection. You will need to add an
  extra state, say D. The rate at which detection
  occurs is ? . Draw the state diagram and using
  SHARPE GUI investigate the effects of detection
  delay on system reliability and mean time to
  failure.

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