Markov Models

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Markov Models

• Difficulties with the combinatorial models
• Complex systems cannot be modeled easily
  difficult to construct reliability block diagrams
  and the expressions can be very complex
• Fault coverage is sometimes difficult to
  incorporate in general
• The process of repair is extremely difficult to
  incorporate

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• Two main concepts in the Markov model
• System state
• State transition
• For reliability models, each state of the Markov
  model represents a distinct combination of faulty
  and fault-free modules.
• The state transitions govern the changes of state
  that occur within a system

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• Several assumptions made to construct transitions
  for this example
• System does not contain repair permanent faults
• Only one fault can occur at a time
• The system starts in the perfect state

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Example

• Markov Model
• Reliability of TMR system
• A B C 1 Good 0 Faulty
• 1 1 1 (3 Good)
• 1 1 0 (2 Good / 1 Faulty)
• 1 0 1 (2 Good / 1 Faulty)
• 1 0 0 (1 Good / 2 Faulty)
• 0 1 1 (2 Good / 1 Faulty)
• 0 1 0 (1 Good / 2 Faulty)
• 0 0 1 (1 Good / 2 Faulty)
• 0 0 0 (3 Faulty)

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Markov Model of the TMR system showing possible
states, State transition, and state transition
probabilities
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• P3(t?t) (1-3??t)P3(t)
• P2(t?t) (3??t)P3(t) (1-2??t)P2(t)
• PF(t?t) (2??t)P2(t) PF(t)

Reduced Markov Model of the TMR system with a
minimal Number of states
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• P3(t?t) 1- 3??t 0 0
  P3(t)
• P2(t?t) 3??t (1-2??t) 0
  P2(t)
• PF(t?t) 0
  2??t 1 PF(t)
• P(t?t) AP(t)
• P3(t?t)
• P(t?t) P2(t?t)
• PF(t?t)
• (1-3 ??t) 0 0
• A 3 ??t (1-2 ??t) 0
• 0 2 ??t 1

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• P(?t) AP(0)
• P(?t ?t) AP(?t) AAP(0)
• P(n?t) AA..AP(0) AnP(0)
• P(n?t) AnP(0)
• P3(t) P2(t) PF(t) 1
• RTMR(t) 1 PF(t) P3(t) P2(t)
• P3(0) 1
• P2(0) 0
• PF(0) 0

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• Continuous-Time
• P3(t ?t) - P3(t)
• -------------------- -3 ? P3(t)
• ?t
• P2(t ?t) - P2(t)
• -------------------- 3 ? P3(t) -2 ? P2(t)
• ?t
• PF(t ?t) - PF(t)
• -------------------- 2 ? P2(t)
• ?t
• dp3(t)
• -------- -3 ? P3(t)
• dt

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• sP3(s) p3(0) -3?P3(s)
• sP2(s) p2(0) 3?P3(s) - 2?P2(s)
• sPF(s) pF(0) 2?P2(s)
• P3(s) 1 / ( s3?)
• P2(s) 3? / ((s2?)(s3?))
• PF(s) 6 ?2 / (s(s2 ?)(s3 ?))

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• P3(s) 1 / ( s3?)
• P2(s) (3/(s2?)) (-3/(s3?))
• PF(s) 1/s (-3/(s2 ?)) (2/(s3?))
• P3(t) e-3?t
• P2(t) 3e-2?t - 3e-3?t
• PF(t) 1 3e-2?t 2e-3?t
• RTMR(t) p3(t) p2(t) e-3?t 3e-2?t -
  3e-3?t 3e-2?t - 2e-3?t
• R(t) 3e-2?t - 2e-3?t

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Comparison of results from computer solution of
the discrete-time Markov and the combinatorial
model for the TMR system. The failure Rate, ?, is
0.1 failure per hour, and the time step, ?t is
0.1 seconds.
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Hybrid Redundancy
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Hybrid Redundancy

• Disagreement with voter output ? switch out the
  faulty unit
• After the first fault the system is reconfigured
  as a duplex system
• Switch continues to compare and perform
  diagnosis
• After the second fault
• Switch will isolate the second faulty unit.
• Switch can now be disconnected
• Finally the last good unit operates until
  failure

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Hybrid Redundancy
Switch coverage C For C 0 The
system TMR Rsys 3R2-2R3 For
C 1 The system parallel Rsys
1-(1-R) 3
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Discrete-time Markov Model of the Hybrid
Redundancy
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Reliability as a function of fault coverage for
the system using Markov model. The failure rate,
?, is 0.1 failures per hour, and The time step,
, is 0.1 seconds
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Markov Model TMR with Hot Spare

• The system fails when the spare switched in is
  faulty along with a fault in one of the main
  units
• The system fails when there is a failure in the
  switch as well as two modules failing

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Markov Model TMR with Hot Spare
1-3??t
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Safety Modeling
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With non-zero coverage, the safety is enhanced
1-(1-R)2
Reliability
R
0.25
0.5
0.75
1.0
Fault Coverage (C)
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Dual Redundant Architecture for the Electronics
Controller
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Equivalent Dual Redundant System
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Four state Markov Model of Reconfigurable
Duplication
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Five state Markov Model of the Standby Spare
System
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Reliability of Standby Sparing and Reconfigurable
Duplication
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Reliability of Standby Sparing and Reconfigurable
Duplication
1.00
0.56
Standby sparing
0.71
0.57
Reliability
? 3.5 10-4 failures per hour Cd 0.95 t 3000
hours
0.43
Reconfigurable Duplication
0.29
0.14
0.00
0.00
0.25
0.50
1.00
0.75
Self-Diagnostics Coverage (Cd)
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Comparison of standby sparing and reconfigurable
duplication safety versus fault coverage