Acyclic Phase-Type Distributions in Fault Trees Pepijn Crouzen Reza - PowerPoint PPT Presentation

Theory Practice Conclusion Acyclic Phase-Type Distributions in Fault Trees Pepijn Crouzen Reza Pulungan Dept. of Computer Science Jurusan Ilmu Komputer Saarland University Universitas Gadjah Mada Germany Indonesia The 9th International Workshop on Performability Modeling of Computer and Communication Systems Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Practice Conclusion Reliability analysis What is the likelihood of system failure ? given the likelihood of component failure ? Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Practice Conclusion Reliability analysis What is the likelihood of system failure ? given the likelihood of component failure ? Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Practice Conclusion Outline Theory 1 Fault Trees Phase-Type distributions Practice 2 Dynamic Fault Trees Case study Conclusion 3 Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Fault Trees Practice Phase-Type distributions Conclusion Fault Trees - from component failure to system failure T 2 / 3 B 1 B 2 B 3 Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Fault Trees Practice Phase-Type distributions Conclusion Fault Trees - from component failure to system failure T 2 / 3 B 1 B 2 B 3 Basic events ‘Component Failure’ Failure probability is given Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Fault Trees Practice Phase-Type distributions Conclusion Fault Trees - from component failure to system failure T Gate ‘ T occurs after 2 of B 1 , B 2 , B 3 occur’ 2 / 3 ‘ T fails after 2 of its 3 subcomponents fail’ B 1 B 2 B 3 Basic events ‘Component Failure’ Failure probability is given Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Fault Trees Practice Phase-Type distributions Conclusion Fault Trees - from component failure to system failure Top Event ‘System Failure’ T Gate ‘ T occurs after 2 of B 1 , B 2 , B 3 occur’ 2 / 3 ‘ T fails after 2 of its 3 subcomponents fail’ B 1 B 2 B 3 Basic events ‘Component Failure’ Failure probability is given Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Fault Trees Practice Phase-Type distributions Conclusion Fault Trees - from component failure to system failure Top Event ‘System Failure’ T Gate ‘ T occurs after 2 of B 1 , B 2 , B 3 occur’ 2 / 3 ‘ T fails after 2 of its 3 subcomponents fail’ B 1 B 2 B 3 Basic events ‘Component Failure’ Failure probability is given But actually... Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Fault Trees Practice Phase-Type distributions Conclusion A Fault Tree is a Boolean Function Some terminology Truth table of f The state of the basic events is a random boolean vector � B = ( B 1 , . . . , B n ) , B 1 B 2 B 3 T The fault tree is a function f from { 0 , 1 } n to 0 0 0 0 { 0 , 1 } . And now, 0 0 1 0 0 1 0 0 P ( T = 1 ) = P ( f ( � B ) = 1 ) . 1 0 0 0 0 1 1 1 1 0 1 1 This problem can be solved efficiently with 1 1 0 1 binary decision diagrams. 1 1 1 1 We only consider coherent fault trees where events are irrevocable. Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Fault Trees Practice Phase-Type distributions Conclusion A Fault Tree is a Boolean Function Some terminology Truth table of f The state of the basic events is a random boolean vector � B = ( B 1 , . . . , B n ) , B 1 B 2 B 3 T The fault tree is a function f from { 0 , 1 } n to 0 0 0 0 { 0 , 1 } . And now, 0 0 1 0 0 1 0 0 P ( T = 1 ) = P ( f ( � B ) = 1 ) . 1 0 0 0 0 1 1 1 1 0 1 1 This problem can be solved efficiently with 1 1 0 1 binary decision diagrams. 1 1 1 1 We only consider coherent fault trees where events are irrevocable. Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Fault Trees Practice Phase-Type distributions Conclusion Fault Trees with Time Adding Time... State of BEs at time t is a stochastic B ( t ) = ( B ( t ) process � 1 , . . . , B ( t ) n ) , P ( B ( t ) = 1 ) is the probability that event 1 B 1 has occurred on or before time-point t , and Again we have P ( T ( t ) = 1 ) = P ( f ( � B ( t ) ) = 1 ) . But now: How do we represent the distribution of basic events and the top event? Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Fault Trees Practice Phase-Type distributions Conclusion Fault Trees with Time Adding Time... 1 State of BEs at time t is a stochastic P(B (t) 1 =1) B ( t ) = ( B ( t ) 0.5 process � 1 , . . . , B ( t ) n ) , P ( B ( t ) = 1 ) is the probability that event 0 0 1 2 3 1 B 1 has occurred on or before time-point t t , and Again we have P ( T ( t ) = 1 ) = P ( f ( � B ( t ) ) = 1 ) . But now: How do we represent the distribution of basic events and the top event? Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Fault Trees Practice Phase-Type distributions Conclusion Fault Trees with Time Adding Time... 1 State of BEs at time t is a stochastic P(B (t) 1 =1) B ( t ) = ( B ( t ) 0.5 process � 1 , . . . , B ( t ) n ) , P ( B ( t ) = 1 ) is the probability that event 0 0 1 2 3 1 B 1 has occurred on or before time-point t 1 t , and P(T (t) =1) Again we have 0.5 P ( T ( t ) = 1 ) = P ( f ( � B ( t ) ) = 1 ) . 0 0 1 2 3 t But now: How do we represent the distribution of basic events and the top event? Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Fault Trees Practice Phase-Type distributions Conclusion Fault Trees with Time Adding Time... 1 State of BEs at time t is a stochastic P(B (t) 1 =1) B ( t ) = ( B ( t ) 0.5 process � 1 , . . . , B ( t ) n ) , P ( B ( t ) = 1 ) is the probability that event 0 0 1 2 3 1 B 1 has occurred on or before time-point t 1 t , and P(T (t) =1) Again we have 0.5 P ( T ( t ) = 1 ) = P ( f ( � B ( t ) ) = 1 ) . 0 0 1 2 3 t But now: How do we represent the distribution of basic events and the top event? Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Fault Trees Practice Phase-Type distributions Conclusion PH distribution overview Properties X A PH-distribution is represented by a CTMC with a single absorbing state, 1 3 3 Matrix characterization, 1 2 3 For a random variable Z PH-distributed with representation X we have, P ( Z ≤ t ) = P ( X ( t ) = 3 ) = � α e Qt � ω. Infinitely many different representations, Acyclic PH-distributions (APH), For FTs: = 1 ) = P ( X ( t ) = 3 ) . P ( B ( t ) 1 Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Fault Trees Practice Phase-Type distributions Conclusion PH distribution overview Properties X A PH-distribution is represented by a CTMC with a single absorbing state, 1 3 3 Matrix characterization, 1 2 3 For a random variable Z PH-distributed with representation X we have, 0 1 − 3 3 0 α = ( 1 , 0 , 0 ) , Q = P ( Z ≤ t ) = P ( X ( t ) = 3 ) = � � 0 − 3 3 α e Qt � @ A ω. 0 0 0 Infinitely many different representations, Acyclic PH-distributions (APH), For FTs: = 1 ) = P ( X ( t ) = 3 ) . P ( B ( t ) 1 Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Fault Trees Practice Phase-Type distributions Conclusion PH distribution overview Properties X A PH-distribution is represented by a CTMC with a single absorbing state, 1 3 3 Matrix characterization, 1 2 3 For a random variable Z PH-distributed with representation X we have, 0 1 − 3 3 0 α = ( 1 , 0 , 0 ) , Q = P ( Z ≤ t ) = P ( X ( t ) = 3 ) = � � 0 − 3 3 α e Qt � @ A ω. 0 0 0 Infinitely many different representations, Acyclic PH-distributions (APH), For FTs: = 1 ) = P ( X ( t ) = 3 ) . P ( B ( t ) 1 Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Fault Trees Practice Phase-Type distributions Conclusion PH distribution overview Properties X A PH-distribution is represented by a CTMC with a single absorbing state, 1 3 3 Matrix characterization, 1 2 3 For a random variable Z PH-distributed with representation X we have, 0 1 − 3 3 0 α = ( 1 , 0 , 0 ) , Q = P ( Z ≤ t ) = P ( X ( t ) = 3 ) = � � 0 − 3 3 α e Qt � @ A ω. 0 0 0 Infinitely many different representations, 1 Acyclic PH-distributions (APH), P(B (t) 1 =1) 0.5 For FTs: = 1 ) = P ( X ( t ) = 3 ) . P ( B ( t ) 1 0 0 1 2 3 t Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees

Theory Fault Trees Practice Phase-Type distributions Conclusion FT and PH ? Theorem The top event of a coherent fault tree with T PH-distributed basic events is itself PH-distributed. 2 / 3 Corollary The top event of a coherent fault tree with B 1 B 2 B 3 APH-distributed basic events is itself APH-distributed. PH PH PH Pepijn Crouzen, Reza Pulungan Acyclic Phase-Type Distributions in Fault Trees