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Martingales and reliability theory: marked point signatures processes

Grant number: 15/02249-1
Support type:Regular Research Grants
Duration: June 01, 2015 - May 31, 2017
Field of knowledge:Physical Sciences and Mathematics - Probability and Statistics
Principal Investigator:Vanderlei da Costa Bueno
Grantee:Vanderlei da Costa Bueno
Home Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

At the 26-th European Conference on Operational Research, Rome, Italy, 2013, I presented results in marked point signature processes. Summarizing, a system failure point process decomposition (related to the sigma algebra generated by its components failures) as a combination of the component failures point processes with ordered lifetimes. The combination coefficients is called marked point signature processes. The expected value of this process results in the classic decomposition for the system lifetime distribution indicating that the new concept is an extension of the classical signature definition, due to Samaniego (1985), which assumes continuous component lifetimes independent and identically distributed."The marked point signature processes allows us to work with stochastic dependence and time dynamics."In a recent work I actualize the compensator of the marked point signature process as a combination of the compensators of the component failures point processes with ordered lifetimes. In this context, in a proposal first part, I intend to study conditional distribution classes related to the observed passed (The sigma algebra generated by the component failures) as in Arjas (1981). Such a conditional distributions with increasing (decreasing) failure rate, new is better (worse) than used and others, relatives to the sigma algebra generated by the component failures, are very useful in Reliability Theory. A result should be about the distribution class preservation under the coherent system formation. Other can be the production of systems reliability bounds, etc.. The decomposition representation of the system failure marked point signature processes, related to the sigma algebra generated by the component failures, through the combination of the component failures point processes with ordered lifetimes become very convenient to work with conditioned distributions classes related to the same family of sigma algebras.In a second part of the proposal I intend to analyse systems maintenance policies in the failure marked point signature processes as in Jensen (1989). Maintenance policies can be conveniently connected with stopping time problems. To minimize the rate of the cycle cost process expected value and the cost process of a cycle expected value can be set as minimize the rate of two semimartingales expected values. The semimartingale of the marked point signature processes system representation is determined by its compensator. We must define some conditionsin the compensator process to allow the use of stopping rule ILA (infinitesimal -look-ahead) as in Jensen (1989).Vanderlei da Costa Bueno (AU)