## 2 Materials and Methods
Let us also assume that the values \(X_{1},X_{2},\;\ldots\) correspond to the duration of life times (a position error below the allowable value according to the IHO orders), and \(Y_{1},Y_{2},\;\ldots\) correspond to their failure times (a position error above the allowable value according to the IHO orders). In this way, a consequence of the change in the position error is a change in the operational status of a system represented by variable \(\alpha(t)\)(Figure 3). Let us also introduce additional designations so that the moments of  
Figure 2: The position error as a function of time (the upper diagram) and three diagrams corresponding to the operational status for the IHO orders: special (green colour), 1a/1b (blue colour), and 2 (red colour). time \(Z^{\prime}_{n}\ =\ X_{1}+Y_{1}+\ X_{2}+Y_{2}+\ \ldots\ +\ Y_{n-1}+\ X_{n}\) become moments of failure, while moments \(Z^{\"}_{n}\ =\ Z^{\prime}_{n}+Y_{n}\) are moments of renewal. Let us additionally assume that the random variables \(X_{i}\), \(Y_{i}\) for \(i\ =\ 1,\ 2,\ \ldots\) are independent, and that the life and failure times have identical distributions. For the model, it is necessary to introduce several additional designations and assumptions resulting from its mathematical properties [34]. Let us assume that the distribution functions of life times \(F(x)\) and failure times \(G(y)\) are right-continuous. Then:  
\[P(X_{i}\leq x)=F(x) \tag{2}\]  
\[P(Y_{i}\leq y)=G(y)\ \ \mbox{for}\ i=1,2,\ldots \tag{3}\]  
Let us also introduce designations of the expected value and variance, which can be expressed as:  
\[E(X_{i})=E(x) \tag{4}\]  
\[E(Y_{i})=E(y) \tag{5}\]  
\[V(X_{i})=\sigma_{1}^{2} \tag{6}\]  
\[V(Y_{i})=\sigma_{2}^{2}\ \ \mbox{for}\ i=1,2,\ldots \tag{7}\]  
where:  
\(E(X_{i})\)--life time expected value,  
\(E(Y_{i})\)--failure time expected value,  
\(V(X_{i})\)--life time variance, and  
\(V(Y_{i})\)--failure time variance. Moreover, let us also assume that:  
\[\sigma_{1}^{2}+\sigma_{2}^{2}>0 \tag{8}\]  
Based on the adopted assumptions, let us establish the reliability process in which its status (of either life or failure) is determined by the relationship between a single measurement error \(\delta_{n}\) and the parameter \(U\) assigned to a particular IHO order. Let \(\alpha(t)\) be a binary interpretation of the reliability status of this process in the following form:  
\[U=\left\{ \begin{array}{ll}1\ \mbox{for}\ Z^{\"}_{n}\leq t<Z^{\prime}_{n+1}\\ 0\ \mbox{for}\ Z^{\prime}_{n+1}\leq t<Z^{\prime}_{n+1}\end{array} \right.\ \ \mbox{for}\ n=0,1,\ldots \tag{9}\]  
The status \(\alpha(t)\ =\ 1\) denotes that at the moment \(t\), the single measurement error was smaller than or equal to the value of the allowable position error \(U\) determined according to Equation (1). Otherwise, for \(\delta_{n}\ >\ U\), let us assume that the system has a failure status. To be able to determine a positioning system's availability corresponding to the defined position error value of the analyzed system, let us define this value as the probability that at any given moment  
Figure 3: The operational statuses: life (a system’s ability to satisfy the positioning requirements for IHO orders) and failure (an opposite event). Own study based on: [33]. of time \(t\), the position's error \(\delta_{n}\) will be smaller than or equal to the arbitrarily adopted value \(U\). Let us designate it with variable \(A(t)\) while assigning to it a mathematical formula in the following form:  
\[A(t)=P[\delta[t]\leq U] \tag{10}\]  
\[A(t)=1-F(t)+\int\limits_{0}^{t}{[1-F[t-x]]dH_{\Phi}(x)} \tag{11}\]  
where:  
\[H_{\Phi}(x)=\sum\limits_{n=1}^{\infty}{\Phi_{n}(x)} \tag{12}\]  
is a function of the renewal stream made up of the renewal moments of the navigation system complying with a specific IHO order, while \(\Phi_{n}(t)\) is a distribution function of the random variable \(Z_{n}\).