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Dependability Evaluation through Markovian model

Dependability Evaluation through Markovian model. Markovian model. The combinatorial methods are unable to: - take care easily of the coverage factor - model the maintenance The Markov model is an alternative to the combinatorial methods. Two main concepts: - state

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Dependability Evaluation through Markovian model

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  1. Dependability Evaluationthrough Markovian model

  2. Markovian model The combinatorial methods are unable to: - take care easily of the coverage factor - model the maintenance The Markov model is an alternative to the combinatorial methods. Two main concepts: - state - state transition

  3. State and state transitions State: the state of a system represents all that must be known to describe the system at any given instant of time For the reliability/availability models each state represents a distinct combination of faulty and fault-free components State transitions govern the changes if state that occur within a system For the reliability/availability models each transition take place when one or more components change state for the event of a fault or for a repair action

  4. State and state transitions (cnt.) • State transitions are characterized by probabilities, such probability of fault, fault coverage and the probability of repair • The probability of being in any given state, s, at some time,t+Dt depends both: • the probability that the system was in a state from which it could transit to state state s given that the transition occurs during Dt • the probability that the system was in state s at instant t and there was no event in the interval time Dt • The initial state should be any state, normally it is that representing all fault-free components • IMPORTANT: IN A MARKOV CHAIN THE PROBABILITY TRANSITION DEPEND ONLY BY THE ACTUAL STATE, i.e. THE STATE CAPTURES THE HISTORY OF THE OLD TRANSITIONS

  5. C1 I O r/n C2 C3 TMR reliability evaluation • There are 4 components (1 voter + computation module), therefore each state is represented by 4 bit: • if the component is fault-free then the bit value is 1 • otherwise the bit value is 0. • For example (1,1,1,1) represents the faut-free state • For example (0,0,0,0) represents all components faulty

  6. TMR reliability evaluation: states diagram

  7. Markov chain reliability evaluation methodology • State transition probability evaluation: • If the fault occurence of a component is exponentially distributed (e-t) with fault rate equal to (l), then the probability that the fault-free component at istant t in the interval Dt become faulty is equal to:  • 1 – e-t

  8. Probability property Prob{there is a fault between t e t+Dt} = = Prob{there is a fault before t+Dt/the component was fault-free at t} = = Prob{there is a faul before t+Dtand the component was fault-free at t} Prob{the component was fault-free at t} = Prob{there is a fault before t+Dt} Prob{there is a fault before t} = Prob{the component was fault-free at t} = (1 – e-(t+t))  (1 – e-t)1 – e-(t+t) 1 + e-t = e-t e-t

  9. Probability property = e-t– e-(t+t) = e-t = e-t _ e-(t+t) = 1  e-t e-t e-t If we expand the exponential part we have the following series: 1 – e-t = 1  1 + (t) + (t)2 + … 2! t (t)2… 2! For value of lDt << 1, we have the following good approximation: 1 – e-tt

  10. TMR reliability evaluation: reduced states diagram State (3,1)  (1,1,1,1) State (2,1)  (0,1,1,1) + (1,0,1,1) + (1,1,0,1) State (G)  all the other states Transition probability (in the interval between t and t+Dt): ·     from state (3,1) to state (2,1) -> 3eDt ; ·  from state (3,1) to state (G) -> vDt ; from state(2,1) to state (G) -> 2eDt+vDt .

  11. TMR reliability evaluation Transition probability (in the interval between t and t+Dt): ·     from state (3,1) to state (2,1) -> 3eDt ; ·  from state (3,1) to state (G) -> vDt ; from state(2,1) to state (G) -> 2eDt+vDt .

  12. TMR reliability evaluation Given the Markov process properties, i.e. the probability of being in any given state, s, at some time, t+Dt depends both: • the probability that the system was in a state from which it could transit to state state s given that the transition occurs during Dt • the probability that the system was in state s at instant t and there was no event in the interval time Dt  we have that P(3,1) (t+t) = (13e tvt) P(3,1) (t) P(2,1) (t+t) = 3e t P(3,1) (t) + (1 2e tvt) P(2,1) (t) P(G) (t+t) = vt P(3,1) (t) + (2e t+ vt) P(2,1) (t) + P(G) (t)

  13. TMR reliability evaluation With algebric operations: t  0 P(3,1) (t+t)  P(3,1) (t) =  (3e + v) P(3,1) (t) = d P(3,1) (t) t dt t  0 P(2,1) (t+t)  P(2,1) (t) = 3e P(3,1) (t)  (2e + v) P(2,1) (t) = d P(2,1) (t) t dt t  0 P(G) (t+t)  P(G) (t) = v P(3,1) (t) + (2e + v) P(2,1) (t) = d P(G) (t) t dt

  14. TMR reliability evaluation i.e: P'3,1 (t) =  (3e + v)P3,1 (t) P'2,1 (t) = 3e P3,1 (t)  (2e + v)P2,1 (t) P'G (t) = v P3,1 (t) + (2e + v)P2,1 (t) That in matrix notation can be expressed as: (t) = (t)Q(t) dt (P'3,1 P'2,1 P'G) = (P3,1 P2,1 PG) * Q 

  15. TMR reliability evaluation the reliability is the probability of being in any fault-free state, i.e, in this case of being in state (3,1) or (2,1). R(t) = P3,1 (t) + P2,1 (t) = 1  PG (t) with the initial conditionP3,1(0) = 1

  16. aaTMR reliability evaluation where: (3e+v) 3ev Q = 0 (2e+v) (2e+v) 0 00 P = Q + I  Q = P  I 1(3e+v) 3ev P = 0 1(2e+v)(2e+v) 001

  17. Properties of Laplace’s transformation

  18. Markov Processes for maintenable systems Two kind of events: - fault of a component (module or voter) - repair of the system (of a module or the voter or both) Hypothesis: the maintenance process is exponentially distributed with repair rate equal to m

  19. Availability evaluation of TMR system P3,1(t) +P2,1(t) + PG(t) = 1 P3,1(0) = 1 P’3,1(t) =  (3e + v ) P3,1(t) + P2,1(t) + PG(t) P’2,1(t) = 3e P3,1(t)  (2e + v + ) P2,1(t) P’G(t) = v P3,1(t) + (2e + v) P2,1(t) PG(t) d(t) = (t)Q(t) dt i.e. (P'3,1 P'2,1 P'G) = (P3,1 P2,1 PG) * Q

  20. (3e+v) 3e v Q =  (2e+v+) (2e+v)  0  1 (3e+v) 3e v P =  1  (2e+v+) (2e+v)  0 1  Availability evaluation of TMR system Q = P  I P = Q + I

  21. Istantaneous Availability evaluation of TMR system the Istantaneous Availability is the probability of being in any fault-free state, i.e, in this case of being in state (3,1) or (2,1). A(t) = P3,1 (t) + P2,1 (t) = 1  PG (t) with the initial conditionP3,1(0) = 1

  22. Limiting or steady state Availability evaluation of TMR system P3,1(t) +P2,1(t) + PG(t) = 1 P3,1(0) = 1 with t 00we have thatP’(t) = 0 P’3,1(t) = 0 =  (3e + v ) P3,1(t) + P2,1(t) + PG(t) P’2,1(t) = 0 = 3e P3,1(t)  (2e + v + ) P2,1(t) P’G(t) = 0 =v P3,1(t) + (2e + v) P2,1(t) PG(t)

  23. Limiting or steady state Availability evaluation of TMR system P3,1(t) +P2,1(t) + PG(t) = 1 P3,1(0) = 1 with t 00we have thatP’(t) = 0 and P(t) = P P’3,1(t) = 0 =  (3e + v ) P3,1 + P2,1 + PG P’2,1(t) = 0 = 3e P3,1 (2e + v + ) P2,1 P’G(t) = 0 =v P3,1 + (2e + v) P2,1(t) PG

  24. Limiting or steady state Availability evaluation of TMR system P3,1 +P2,1 + PG= 1 P3,1 = P2,1 = PG =

  25. Safety evaluation Four kind of events: - fault of a component (module or voter) correcttly diagnoticated - fault of a component not detected - correct repair of the system (of a module or the voter or both) - uncorrect repair of the system  fault rate  repair rate Cg fault detection coverage factor Cr correct repair coverage factor

  26. Single component Safety evaluation Hypothesis: • if a fault is not well diagnosticated then it will never be detected • If a reconfiguration is not wel done then it will be never detected • Therefore GI is an absorbing state 0 fault free state GS  safe fault state GI unsafe fault state

  27. Single component Safety evaluation Safety = probability to stay in state 0 or GS PO(t) +PGS(t) = 1  PGI(t) PO(0) = 1 P’O(t) =  ((1 Cg) + Cg)) PO(t) + Cr PGS(t) P’GS(t) = Cg PO(t)  ((1 Cr)+ Cr) PGS(t) P’GI(t) = (1 Cg) PO(t) + ((1 Cr)PGS(t)

  28.  Cg (1-Cg) Q = Cr   (1-Cr) 0 0 0 Single component Safety evaluation d(t) = (t)Q(t) dt i.e. (P'3,1 P'2,1 P'G) = (P3,1 P2,1 PG) * Q

  29. Performability Index taking into account even the performance of the system given its state (related to the number of fault-free components) We will discuss it when we will know how evaluate the performance of a system

  30. C2 C1 C21 C11 C112 C111 C22 C12 C23 Reliability/Availability/Safety evaluation of complex system I O R = R 1 . R 2 R 11 = R 111 . R 112 R 1 = 1 - (1 - R 11). (1 - R 12) R 2 = 1 - (1 - R 21). (1 - R 22) . (1 - R 23)

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