SafetyandReliabilityAnalysisLecture5Yiliu LiuDepartment of Production and Quality EngineeringNorwegian Universityof ScienceandTechnologyyiliu.liu@ntnu.noNTNU- TrondheimNorwegian University ofScience and Technologywww.ntnu.edu
1 Safety and Reliability Analysis Lecture 5 Yiliu Liu Department of Production and Quality Engineering Norwegian University of Science and Technology yiliu.liu@ntnu.no
MARKOVPROCESSNTNU-TrondheimNorwegian University ofScience and Technologywww.ntnu.edu
2 MARKOV PROCESS
3AnintroductoryexampleConsidera parallel structureoftwo components.Each component is assumedtohavetwostates, a functioning state and a failed state.The structure has therefore 22=4 possiblestates,andthestatespaceisX=(0,1,2,3)StateComponent1Component23FunctioningFunctioning2FailedFunctioningFailed1Functioning0FailedFailedNTNU -TrondheimNorwegianUniversityofScienceandTechnologywww.ntnu.edu
3 An introductory example Consider a parallel structure of two components. Each component is assumed to have two states, a functioning state and a failed state. The structure has therefore 2ଶ ൌ 4 possible states, and the state space is X = {0,1,2,3} State Component 1 Component 2 3 Functioning Functioning 2 Functioning Failed 1 Failed Functioning 0 Failed Failed
AnintroductoryexampleTheMarkovprocessisdefinedbythestatetransitiondiagram22μ2212μμi222NTNU-TrondheimNorwegian University ofScience and Technologywww.ntnu.edu
4 An introductory example The Markov process is defined by the state transition diagram
5StatespaceLetX(t)denotethestateof systemattimet.Thestatespaceisthesetofallthepossiblesystemstates.IfwenumberthestatesbyintegersfromOtor,thestatespacecanbedescribedasx= [0,1,2,.,r)LetP;(t)=Pr(X(t)=i)betheprobabilitythatthesystemisinstateiattimet.ThestateprobabilitydistributionisdenotedP(t) = (Po(t),Pi(t),..,P(t))NTNU -Trondheim莎NorwegianLniversityofScienceandTechnologywww.ntnu.edu
5 State space Let ܺ ݐ denote the state of system at time ݐ. The state space is the set of all the possible system states. If we number the states by integers from 0 to ݎ ,the state space can be described as ߯ ൌ 0,1,2, . , ݎ Let ܲ ݐ ൌ Pr ܺ ݐ ൌ݅ be the probability that the system is in state ݅ at time ݐ. The state probability distribution is denoted ܲ ݐ ൌ ሺܲ ݐܲ, ଵ ݐܲ,., ሺݐሻሻ