114 Chapter 5. Deterioration Modelling of Sewer Pipes via Discrete-Time Markov Chains: A Large-Scale Case Study in the Netherlands Chain ‘Multi’: 1 2 3 4 5 0.9571 0.0347 0.0074 0.0006 0.0002 0.9995 0.0005 0 0 0.9999 0.0001 0 0.9999 0.0001 1.00 (a) Chain ‘Single’: 1 2 3 4 5 0.955 0.045 0.972 0.028 0.979 0.021 0.988 0.012 1.00 (b) Figure 5.1: DTMCs modelling the deterioration of sewer mains considering five deterioration states and no repairs. (a) Chain “Multi”; (b) Chain “Single”. where )k →S p (0) k =1 at any step n. To calculate the state probabilities associated with the n-step p(n), we apply the Chapman-Kolmogorov equation in Eq. 4.5 (page 99). From here, the n-step transition probability matrix P(n) ij is obtained by multiplying the matrix Pij by itself n times. If n is a positive real number instead of a natural number, one can compute the fractional power of the matrix using, for example, the scipy.linalg function in Python (Virtanen, Gommers, Oliphant, et al., 2020). We compute the expected severity E(n), as shown in Baik, Jeong, and Abraham, 2006, by multiplying the state probability distribution p(n) at step nby the severity class vector ϑ↓S. In this context, 1 indicates pristine condition, and 5 represents the maximal severity that can be assigned to a type of damage. Then E(n) is computed as follows: E(n) =p(n)ϑ (5.2) We adopt the structures of Markov chains types “Multi” and “Single” as discussed in Section II.4.4, page 104. These are typical Markov chain structures used to model the deterioration of sewer mains via DTMCs (Ana and Bauwens, 2010). Figure 5.1 shows an example of these structures used in this chapter. The number on the arrows indicate the probability of moving from one state to another.
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