## 2 Semi-Markov Process  
The semi-Markov process falls into the group of continuous-time finite-space Markov processes. Hitherto, several authors have proposed the semi-Markov process for the prediction of future condition based on condition ratings. [PERSON] and [PERSON] [34] outlined a procedure to model bridge deterioration based on the semi-Markov process, which was implemented on the National Bridge Inventory (NBI) database consisting of information on bridges in the United States. [PERSON] [35] presented the application of the semi-Markov process in optimizing decisions regarding the renewal of large infrastructure assets, such as water transmission pipes and trunk sewers. [PERSON] et al. [36] implemented the semi-Markov process with the Weibull distribution of sojourn times on the Serbian bridge database. [PERSON] et al. [37] presented the life-cycle optimization model using the semi-Markov process also based on the Weibull distribution. [PERSON] et al. [5] compared several stochastic models including the semi-Markov process, implementing them in bridge decks in Portugal. In the semi-Markov process, a structure is considered to be in a particular condition state (CS) for a random length of time distributed depending on the state. This length is called sojourn time (sometimes also holding or waiting time). The state of the process at any time \(t\) can be denoted by an n-dimensional vector \(p(t)\):  
\[p(t) = [p_{1}(t),p_{2}(t)\ldots p_{n}(t)], \tag{1}\] \[\sum_{i\ =\ 1}^{n}P_{i}(t)\ =\ 1 \tag{2}\]  
where:  
\(p_{i}(t)\) is the probability that the process is in the condition state \(i\) at the time \(t\), and  
\(n\) is the number of possible condition states. The probability that the process will in the subsequent period \(k\) assume one of the condition states \(i\) is expressed through the probability vector \(p(t+k)\):  
\[p(t+k)\ =\ p(t)\cdot P^{t,t+1}\cdot P^{t+1,t+2}\ldots P^{t+k-1,t+k}, \tag{3}\]  
where:  
\(P^{t,t+1}\) is the time-dependent probability matrix (i.e., transition matrix), and  
\(k\) is the subsequent period being observed. Each transition matrix \(P^{t,t+1}\) is populated with single (time) step transition probabilities \(p_{ij}^{t,t+1}\), which are defined as shown in Equation (4):  
\[p_{ij}^{t,t+1}\ =\ \Pr[X(t+1)\ =\ j|X(t)\ =\ i]. \tag{4}\]  
It is often assumed that an infrastructure asset can deteriorate only one state at a time, supposing no maintenance/repair/replacement occurred [35]. Under this assumption, the probability matrix \(P^{t,t+1}\) takes its simplified form:  
\[P^{t,t+1}\ =\ \left[ \begin{array}{cccc}p_{11}^{t,t+1}&p_{12}^{t,t+1}&0& \ldots&0\\ 0&p_{22}^{t,t+1}&p_{23}^{t,t+1}&\ldots&0\\ \ldots&\ldots&\ldots&\ldots&\ldots\\ 0&\ldots&\ldots&p_{n-1,n-1}^{t,t+1}&p_{n-1,n}^{t,t+1}\\ 0&0&\ldots&0&1\end{array} \right], \tag{5}\]  
where:  
\(p_{ii}^{t,t+1}\) is the probability that the process will stay in the same state, and  
\(p_{i,i+1}^{t,t+1}\) is the probability that the process will progress to next subsequent state. To determine the single step transition probabilities, sums of sojourn times need to be known. Random variables denoting the length of sojourn times in states 1, 2,..., \(n-1\) can be expressed as \(T_{1}\), \(T_{2}\),..., \(T_{n-1}\), respectively. Furthermore, a random variable denoting the sum of these times in states \(i\), \(i+1,\ldots,k-1\) can be expressed as \(T_{i\to k}\), and it denotes the time the process will need to go from state \(i\) to \(k\). If it is assumed that the process in state 1 represents a new structure (i.e., a structure without damages), the single step transition probabilities can be calculated as shown in Equation (6):  
\[p_{i,i+1}^{t,t+1}\ =\ Pr[X(t+1)\ =\ i+1\mid X(t)\ =\ i]\ =\ \frac{f_{1\to i}(t)}{S_{1\to i}(t)-S_{1\to(i-1)}(t)}, \tag{6}\]  
where:  
\(f_{1\to i}(t)\) is the probability density function (PDF) of the sum of sojourn times from state 1 to \(i\), and \(S_{1\to i}(t)\) is the survival function (SF) of the sum of sojourn times from state 1 to \(i\).