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SPEAKER: |
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TITLE: |
"Semi-Markov Processes and Reliability" |
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DAY: |
Wednesday, April 1, 1998 |
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TIME: |
3:30 p.m. [Coffee in BSB-202 at 3:00 p.m.] |
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PLACE: |
BSB-108 |
Semi-Markov process is a natural generalization of Markov and Renewal processes. Consider the two dimensional ExR_{+}-valued Markov chain (J_{n},X_{n})_{n>=0}, with transition function Q(x,A,t)=P(J_{n+1} in A, X_{n+1}<=t | J_{n}=x), x in E, A in B(E), t in R_{+}. The J_{n} is the state occupied at the n-th jump and S_{n}=X_{1}+...+X_{n} is the time of the n-th jump. This process (J_{n}, S_{n})_{n>=0} is called a Markov Renewal Process.
Define now, for each t, the r.v. N(t) which gives the number of jumps before t and Z_{t}=J_{N(t)}, i.e. N(t)=sup{n : S_n<=t}. The process (Z_{t})_{t>=0} with state space E is the semi-Markov process associated to MRP (J_{n}, S_{n})_{n>=0}. We will present the basic results concerning the semi-Markov process Z when E is a finite set (i.e. processes associated to Markov renewal kernel Q, Markov renewal theory and limit theorems) and formulate Reliability related indicators (Reliability, Availability, Mean time to failure,...) for systems of semi-Markov random behavior. After that, we will present a method to approximate a semi-Markov process by a Markov one via Ph-distributions. This is an important technique that allows us to use Markov results in the semi-Markov analysis. Finally, we will present recent results on nonparametric estimation of semi-Markov kernel, of Markov renewal function and of Reliability and Availability functions. Illustration of results by numerical examples will be given.
Dr. Nikolaos Limnios received his Ph.D. from France working on Semi-Markov models. He is a Professor in the Department of Applied Mathematics at the Université de Technologie de Compiègne, France. His main interests are on applied probability and reliability applications. He has written many papers on the subject of Markov and Semi-Markov processes and their applications to reliability problems. He is a co-organizer of an International Conference on this topic that is to take place at Compiègne, France, in December of this year.