Statistical Analysis of Reliability Data for Markov Multi-state System via Output Performance Observation

1. Statistical Analysis of Reliability Data for MarkovMulti-state System via Output Performance Observation Anatoly Lisnianski The Israel Electric Corporation

2. ... ... ... ... ... 1 N N-1 2 ... ... ... ... Markov Multi-state System State-space representation

3. GA(t) T t MSS Performance G(t) as a Stochastic Process

4. Dataderived from performance stochastic process observation during timeT • Sample of system sojourn times in state i during observation time T . • Number of system transitions from state i to any possible state j during observation time T. • Number of system residences in state i (or number of system exits from state i to any other possible state) during observation time T.

5. Estimate transition intensities based onsingle realization of discrete-state continuous-time stochastic process observed during time T. The problem

6. conditional sojourn timein statei unconditional sojourn timein statei ... ... ... ... one-step transition probability 1 N i j ... Probabilitythat the process will transit from stateito statej up to timet, if at Initial time instant t=0, the process was In state i. ...

8. Distribution of unconditional sojourn time Ti in any state i Ti is exponentially distributed random variable with mean

9. Estimation of mean unconditional sojourn time can be obtained by using the sample

10. One-step transition probabilities for embedded Markov chain Therefore

11. can be easily estimated as a ratio of corresponding transitions frequencies Transition Intensities Estimates

12. Example

13. Transition intensities should be estimated by using the following expression

14. Diagonal transition intensities Resulting matrix of point estimations of transition intensities