126 Chapter 6. Comparing Homogeneous and Inhomogeneous Time Markov Chains for Modelling Deterioration in Sewer Pipe Networks 1 2 3 4 5 F ω12(t, ϑ) ω1F(t, ϑ) ω23(t, ϑ) ω2F(t, ϑ) ω34(t, ϑ) ω3F(t, ϑ) ω45(t, ϑ) ω4F(t, ϑ) ω5F(t, ϑ) Figure 6.1: Markov chain structure modelling the deterioration of a sewer main considering five deterioration states and a functional failure state. For formal definitions of IHTMCs, HTMCs, CTMCs, DTMCs, see Section II.4.1 on page 95. 6.2.1 Multi-state deterioration modelling for sewer networks using parametrised Markov chains We first outline the structure of our Markov chain model (Figure 6.1) and then describe its parameterisation. The pipe element is defined with K=6 sequential states p= [p1, p2, . . . , pK], where p1 represents the pristine state and pK themost deteriorated state. The six states account for severity levels 1 to 5 and functional failure, as reported in sewer network inspections. Further details on the type of Markov chain in Figure 6.1 are provided in Section II.4.4 (p. 104), applicable to both IHTMCs and HTMCs. We parametrise our Markov chains using probability density functions to model hazard rates. Specifically, we employ Exponential, Gompertz, Weibull, Log-Logistic, and Log-Normal density functions. For the Log-Normal function, lacking a closed-form hazard rate, we compute it as the ratio f(t; ς)/S(t; ς). Hazard rates and hyper-parameter ranges for the other functions are specified in Eq. 6.1. Exponential: ϖ(t; ↼)=↼ (6.1a) Rate: ↼>0 Gompertz: ϖ(t; ↼, ↽)=↼↽eϱt (6.1b) Shape: ↼>0, Scale: ↽ >0 Weibull: ϖ(t; ↼, ↽)= ↽ ↼ t ↼ ϱ↗1 (6.1c) Scale: ↼>0, Shape: ↽ >0 Log-Logistic: ϖ(t; ↼, ↽)= (↽/↼)(t/↼)ϱ↗1 1+(t/↼)ϱ (6.1d) Scale: ↼>0, Shape: ↽ >0 From Eq. 4.3 (page 98), we derive the system of di!erential equations related to the Markov chain in Figure 6.1 and present them in Eq. 6.2. To solve numerically
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