200 Appendix A. Appendix: Introduction A.2 Example of a Multi-State Deterioration model with three states Figure A.2 presents a Markov chain with three states. The corresponding system of di!erential equations formulated from the master equation of the Markov chains (Eq. 4.3) is presented in Eq. A.11. A B C ωA,B(t) ωB,C(t) Figure A.2: Markov chain with 3 states. φSA(t) dt =→ϖA,B(t)SA(t) (A.11a) φSB(t) dt =ϖA,B(t)SA(t) →ϖB,C(t)SB(t) (A.11b) φSC(t) dt =ϖB,C(t)SB(t) (A.11c) To solve the system of di!erential equations in Eq. A.11, we follow a similar approach to the two-state Markov chain described in Section A.1. Let’s start first with Eq. A.11a. φSA(t) dt =→ϖA,B(t)SA(t) SA(t)=SA(0)RA,B(t) (A.12a) To get the solution of Eq. A.11b, we first make use of the following expression: u(t)=exp t 0 ϖ(ϱ)dϱ (A.13a) u↑(t)=ϖ(t)u(t) (A.13b) By multiplying u(t) to Eq. A.11b, and making use of the relation u↑B,C(t) = ϖB,C(t)uB,C(t), we get: dSB(t) dt uB,C(t)+SB(t) duB,C(t) dt =uB,C(t)ωA,B(t)SA(t) d dtSB(t)uB,C(t) =uB,C(t)ωA,B(t)SA(t) SB(t) = 1 uB,C(t) t 0 uB,C(ϱ)ωA,B(ϱ)SA(ϱ)dϱ +C→
RkJQdWJsaXNoZXIy MjY0ODMw