II.4.4 Deterioration modelling in sewer mains using Markov Chains 105 0 [1993-1994) [1994-1995) [1995-1996) [1996-1997) [1997-1998) [1998-1999) [1999-2000) [2000-2001) [2001-2002) [2002-2003) [2003-2004) [2004-2005) [2005-2006) [2006-2007) [2007-2008) [2009-2010) [2010-2011) [2011-2012) [2012-2013) [2013-2014) [2014-2015) [2015-2016) [2008-2009) 5k Count 1 2 3 4 5 # Inspections % network inspected 85.50% 12.48% 1.66% 0.29% 0.06% 1993 1998 2004 2010 2016 Year (a) Frequency of inspections per year (c) Percentage of the sewer network inspected annually (b) Frequency inspections per pipe 0 2000 20% 15% 10% 5% 4000 6000 8000 Frequency 10k 15k 20k 25k Figure II.6: Histograms on inspection dataset (1,045 km of sewer network length). (a) Type "Single" p12 p23 p34 p45 1 2 3 4 5 p12 p1F p23 p23 p34 p4F p3F p45 p5F 2 3 4 5 F (b) Type "Multi" (c) Type "Single+Failure" p12 p25 p13 p14 p15 p35 p23 p24 p34 p45 1 2 3 4 5 1 Figure II.7: Types of Markov chain structures are discussed in this dissertation. For Discrete-Time Markov Chains, self-loops must be added to the states. All structures shown in Figure II.7 correspond to acyclic Markov chains, meaning transitions can only occur from the current state to worse-case states (pij =0 for i > j with i, j ↓ S), without the possibility of improving its condition without intervention (e.g., via repairs). Only the final state is absorbing, meaning p |S||S| =1. The Chain “Single” (Figure II.7(a)) permits transitions only between consecutive states from better to worse-case (i.e., 0 ⇓pij ⇓1 for all i and j =i +1, with pij = 0 for j > i +1). The Chain “Multi” (Figure II.7(b)) allows transitions between both consecutive and non-consecutive states from better to worse-case (i.e., 0 ⇓pij ⇓1 for all i ⇓j). As severity level k = 5 is not a functional failure, the Chain “Single+Failure” (Figure II.7(c)) extends the Chain “Single” by adding an additional state, resulting
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