First-passage time analysis Successive events and Energy Networks for Markovian deteriorating model Markovian deteriorating model Gejza Dohnal Introduction Model Gejza Dohnal One-way Model Repairable model Example Center of Quality and Reliability of Production, CTU in Prague Lifetime of the Czech Republic system First-affect time Renewal period Conclusions Modeling Smart Grids - A new Challenge for Stochastics and Optimization September 10-11, Prague
Introduction First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model One-way Model Repairable model Example Lifetime of the system First-affect time Renewal period Conclusions
Introduction First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model One-way Model Repairable model Example Lifetime of the system First-affect time Renewal period Conclusions
Introduction First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model One-way Model Repairable model Example Lifetime of the system First-affect time Renewal period Conclusions
Introduction First-passage time analysis × Failure Disaster for Markovian deteriorating model Gejza Dohnal Introduction Model One-way Model Repairable model Example Lifetime of the system First-affect time Renewal period Conclusions
Introduction First-passage time analysis × Failure Disaster for Markovian deteriorating random occurrence in time random occurrence in time model independent recurrence growing spread Gejza Dohnal renewal process domino effect Introduction quick repair or renewal repairs slower than spreading Model One-way Model Repairable model Example Lifetime of the system First-affect time Renewal period Conclusions
Introduction First-passage time analysis × Failure Disaster for Markovian deteriorating random occurrence in time random occurrence in time model independent recurrence growing spread Gejza Dohnal renewal process domino effect Introduction quick repair or renewal repairs slower than spreading Model One-way Model equipment failure explosion, fire Repairable model Example accidents, health damage disease, epidemic Lifetime of the information system failure system breakdown, piracy attack system motor vehicle accident traffic collapse First-affect time natural accident natural disaster Renewal ........ ....... period Conclusions
Introduction First-passage time analysis × Failure Disaster for Markovian deteriorating random occurrence in time random occurrence in time model independent recurrence growing spread Gejza Dohnal renewal process domino effect Introduction quick repair or renewal repairs slower than spreading Model One-way Model equipment failure explosion, fire Repairable model Example accidents, health damage disease, epidemic Lifetime of the information system failure system breakdown, piracy attack system motor vehicle accident traffic collapse First-affect time natural accident natural disaster Renewal ........ ....... period Conclusions Model → Prediction → Prevention
Introduction First-passage time analysis × Failure Disaster for Markovian deteriorating random occurrence in time random occurrence in time model independent recurrence growing spread Gejza Dohnal renewal process domino effect Introduction quick repair or renewal repairs slower than spreading Model One-way Model equipment failure explosion, fire Repairable model Example accidents, health damage disease, epidemic Lifetime of the information system failure system breakdown, piracy attack system motor vehicle accident traffic collapse First-affect time natural accident natural disaster Renewal ........ ....... period Conclusions Model → Prediction → Prevention Preventive Maintenance Policy Disaster Recovery Plan
Model First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model One-way Model Repairable model Example Lifetime of the system First-affect time Renewal period Conclusions
Model First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model One-way Model Repairable model Example Lifetime of the system First-affect time Renewal period Conclusions
Model First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model One-way Model Repairable model ω ( t ) = ( ω 1 ( t ) , ω 2 ( t ) , . . . , ω n ( t )) , t ≥ 0 , State of the system: Example Lifetime of the system First-affect time Renewal period Conclusions
Model First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model One-way Model Repairable model ω ( t ) = ( ω 1 ( t ) , ω 2 ( t ) , . . . , ω n ( t )) , t ≥ 0 , State of the system: Example Lifetime of the (i) at the beginning (at the time 0), the system is in the state system ω 0 = ( 0 , 0 , . . . , 0 ) , First-affect (ii) the process starts by deterioration of an object i with time probability π i , i = 1 , 2 , . . . , n , Renewal period (iii) when at the time t an object i was affected, there was a random time period τ after which the event moved onto some Conclusions of the unaffected objects, (iv) states of the system in time t create a stochastic process { X ( t ) , t ≥ 0 } in continuous time. Values of this process lie within the set Ω = { 0 , 1 } n
Model First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model One-way Model Repairable model ω ( t ) = ( ω 1 ( t ) , ω 2 ( t ) , . . . , ω n ( t )) , t ≥ 0 , State of the system: Example Lifetime of the (i) at the beginning (at the time 0), the system is in the state system ω 0 = ( 0 , 0 , . . . , 0 ) , First-affect (ii) the process starts by deterioration of an object i with time probability π i , i = 1 , 2 , . . . , n , Renewal period (iii) when at the time t an object i was affected, there was a random time period τ after which the event moved onto some Conclusions of the unaffected objects, (iv) states of the system in time t create a stochastic process { X ( t ) , t ≥ 0 } in continuous time. Values of this process lie within the set Ω = { 0 , 1 } n
Model First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model One-way Model Repairable model ω ( t ) = ( ω 1 ( t ) , ω 2 ( t ) , . . . , ω n ( t )) , t ≥ 0 , State of the system: Example Lifetime of the (i) at the beginning (at the time 0), the system is in the state system ω 0 = ( 0 , 0 , . . . , 0 ) , First-affect (ii) the process starts by deterioration of an object i with time probability π i , i = 1 , 2 , . . . , n , Renewal period (iii) when at the time t an object i was affected, there was a random time period τ after which the event moved onto some Conclusions of the unaffected objects, (iv) states of the system in time t create a stochastic process { X ( t ) , t ≥ 0 } in continuous time. Values of this process lie within the set Ω = { 0 , 1 } n
Model First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model One-way Model Repairable model ω ( t ) = ( ω 1 ( t ) , ω 2 ( t ) , . . . , ω n ( t )) , t ≥ 0 , State of the system: Example Lifetime of the (i) at the beginning (at the time 0), the system is in the state system ω 0 = ( 0 , 0 , . . . , 0 ) , First-affect (ii) the process starts by deterioration of an object i with time probability π i , i = 1 , 2 , . . . , n , Renewal period (iii) when at the time t an object i was affected, there was a random time period τ after which the event moved onto some Conclusions of the unaffected objects, (iv) states of the system in time t create a stochastic process { X ( t ) , t ≥ 0 } in continuous time. Values of this process lie within the set Ω = { 0 , 1 } n
Model First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model One-way Model Repairable model Example Example ( n = 7 π = ( 1 , 0 , 0 , 0 , 0 , 0 , 0 ) ) Lifetime of the From all 2 7 possible states only the following 14 are admissible: system First-affect (1,0,0,0,0,0,0), (1,1,0,0,0,0,0), (1,0,1,0,0,0,0), (1,1,1,0,0,0,0), time (1,0,1,1,0,0,0), (1,1,1,1,0,0,0), (1,0,1,1,1,0,0), (1,1,1,1,1,0,0), Renewal (1,0, 1,1,1,1,0), (1,0,1,1,1,0,1),(1,1,1,1,1,1,0), (1,1,1,1,1,0,1), period (1,0,1,1,1,1,1), (1,1,1,1,1,1,1). Conclusions
Model 1 First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model One-way Model Repairable model Example Model 1 – One-way Model Lifetime of the system Let us consider the following supplementary assumptions: First-affect time Renewal period Conclusions
Model 1 First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model One-way Model Repairable model Example Model 1 – One-way Model Lifetime of the system Let us consider the following supplementary assumptions: First-affect (v) an event can only affect one object in one moment, time (vi) an event can only occur once on a particular object, Renewal (vii) the process moves to the next object with a probability which period depends only on the recent state, not on the path leading to Conclusions the recent state (the time sequence of events).
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