197 Appendix A Appendix: Introduction A.1 Example of a Multi-State Deterioration model with two states Here we demonstrate that in a two-state Markov chain model with a single transition (from nominal to non-nominal behaviour) the state probability of being in the nominal state SN(t) is equivalent to the reliability function R(t). For this, Figure A.1 presents a two-states Markov chain, where N represents nominal state and ¬N is the non-nominal state. By using the master equation of Markov chains (Eq. 4.3), we define the corresponding system of di!erential equations in Eq. A.1. N ¬N ω(t) Figure A.1: Markov chain with 2 states. N represents nominal state and ¬N is the non-nominal state. ω(t) is the hazard rate function. φSN(t) dt =→ϖ(t)SN(t) (A.1a) φS¬N(t) dt =ϖ(t)SN(t) (A.1b) InEq.A.1, SN(t) andS ¬N(t) represent respectively the state probability of being in the nominal and non-nominal states; and ϖ(t) is the hazard rate. In Appendix C.1, more fundamental relations commonly used in reliability are presented. There, the following relations are derived:
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