218 Appendix C. Appendix: Multi-State Deterioration Finally, we get: P[t ⇓T <t +∆t|T ⇒t] = P[t ⇓T <t +∆t] P[T ⇒t] (C.5) By replacing Eq. C.5 in Eq. C.4 we get: ϖ(t)= lim ∆t⇐0 P[t ⇓T <t +∆t] ∆tP[T ⇒t] (C.6) For ∆t ↑0, from Eq. C.1 we get that: P[t ⇓T ⇓t +∆t] = t+∆t t fT(t)dt ≈f(t) · ∆t (C.7) By plugging-in Eq. C.7 and Eq. C.3 into Eq. C.6, we get: ϖ(t)= f(t) S(t) (C.8) From Eq. C.3 we get that: R↑(t)=→f(t) (C.9) Then by replacing Eq. C.9 in Eq. C.8 we get: R↑(t) R(t) =→ϖ(t) (C.10a) dR(t) R(t) =→ϖ(t)dt (C.10b) t 0 dR(t) R(t) =→ t 0 ϖ(s)ds (C.10c) ln|R(s)| t 0 =→ t 0 ϖ(s)ds (C.10d) ln|R(t)| →ln|R(0)| =→ t 0 ϖ(s)ds (C.10e) R(t)=exp→ t 0 ϖ(s)ds (C.10f)
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