94 Part II: Multi-state deterioration modelling II.2 Nomenclature Markov chains for Multi-State Deterioration Modelling: T Parameter space. t Time (e.g., component age), with t ↓T Xt Stochastic process over t. S State space of Xt. k Severity index k ↓S. p (0) k Initial state distribution over S. p (t) k State probability distribution over S at t. M Markov chain hypothesis. i Sojourn state, where i ↓S. j Arrival state, where j ↓S. Pij(·) Transition probability matrix (function). Qij(·) Transition rate matrix (function). Survival Analysis: f(·) Probability density function. S(·) Survival function. ϖ(·) Hazard rate function. II.3 Related work Sewer networks, crucial for social and economic welfare, present management challenges due to limited budgets, environmental changes, and complex, hard-tomodel deterioration processes. As these systems approach the end of their design life, predictive tools for deterioration become vital for e"cient maintenance and logistics (Marc Ribalta and Rubión, 2023). Robust models for sewer main deterioration help in identifying high-risk pipes, thus facilitating proactive maintenance, decisionmaking, and strategic planning (Scheidegger, Hug, Rieckermann, et al., 2011; Egger, Scheidegger, Reichert, et al., 2013; Caradot, Sonnenberg, Kropp, et al., 2017). Deterioration models for sewer mains are typically developed using inspection data adhering to standards such as the EN 13508:1 and EN 13508:2. These standards guide the classification of damages observed via Closed Circuit Television (CCTV) inspections into severity levels. Comprehensive reviews by Ana and Bauwens, 2010; Malek Mohammadi, Najafi, Kaushal, et al., 2019; Hawari, Alkadour, Elmasry, et al., 2020; Saddiqi, Zhao, Cotterill, et al., 2023; Zeng, Z. Wang, H. Wang, et al., 2023 categorise sewer main deterioration models into three main types: physics-based, Machine Learning (ML)-based, and probabilistic models. Physics-based models utilise mathematical relations grounded in physical principles. ML-based models are increasingly recognised for their ability to identify complex patterns in large datasets and use these insights for predictive and decision-making applications. Comparisons of di!erent
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