## 2 Materials and Methods
The concept of the method can also be presented in the graphical form (Figure 4) where the figure shows the distribution of subsequent positions of the analyzed system in relation to the actual values (the center of the diagram). In the diagram, a circle with a radius of 20 m is marked. This radius corresponds to the maximum allowable position error value for IHO order 2, which means that the positions within the circle correspond to the condition \(\delta_{n}\ \leq U\), i.e., the system has a life status, while the positions located outside the circle correspond to the system's failure statuses. In navigation applications, it is most frequently assumed that the distributions of life and failure times are exponential, hence their probability density functions can be expressed using commonly known formulas [34]:  
\[f(t)=\left\{ \begin{array}{c}\lambda\cdot e^{-\lambda t}\text{ for }t>0\\ 0\text{ for }t\leq 0\end{array} \right. \tag{13}\]  
\[g(t)=\left\{ \begin{array}{c}\mu\cdot e^{-\mu t}\text{ for }t>0\\ 0\text{ for }t\leq 0\end{array} \right. \tag{14}\]  
with the following respective distribution functions:  
\[F(t)=\left\{ \begin{array}{c}1-e^{-\lambda t}\text{ for }t>0\\ 0\text{ for }t\leq 0\end{array} \right. \tag{15}\]  
Figure 4: An example of geometric interpretation of the availability required for IHO order 2. \[G(t)=\left\{ \begin{array}{c}1-e^{-\mu t\cdot t}\text{ for }t>0\\ 0\text{ for }t\leq 0\end{array} \right. \tag{16}\]  
where:  
\(f(t)\)--life time probability density function,  
\(g(t)\)--failure time probability density function,  
\(\lambda\)--failure rate,  
\(\mu\)--renewal rate. Having adopted the above assumptions, the final form of the availability function can be expressed as:  
\[A_{\text{exp}}(t)=\frac{\mu}{\lambda+\mu}+\frac{\lambda}{\lambda+\mu}\cdot e^{ -(\lambda+\mu)\cdot t} \tag{17}\]  
where \(A_{\text{exp}}(t)\) is an availability function determined for a specific position error value according to the IHO orders, with the assumed exponential distribution of the positioning system's life and failure times. In practical applications, it is advisable to provide the availability limit value \(\left(A_{\text{exp}}\right)\)[9], referred to as the availability factor [34]:  
\[\lim_{t\rightarrow\infty}\bigl{[}A_{\text{exp}}[t]\bigr{]}=A_{\text{exp}}= \frac{\frac{1}{\lambda}}{\frac{1}{\lambda}+\frac{1}{\mu}}=\frac{\mu}{\mu+\lambda} \tag{18}\]  
A Measurement Assessment of the Compliance of DGPS, EGNOS, and Multi-GNSS with the Positioning Requirements for the IHO orders  
The assessment of particular positioning systems (DGPS, EGNOS, and multi-GNSS) in terms of compliance with the requirements set out for the IHO orders should be carried out based on measurement sessions that are as long as possible. In addition, it should be noted that the results of an accuracy assessment for a certain system in both stationary and dynamic sessions may be significantly statistically different, which is proven by the research carried out on the Gdansk Bay in the years 2014-2017 (Figure 5) [35; 36; 37; 38]. For this reason, analyses and assessments of the model should be carried out in relation to stationary and dynamic measurements in parallel. To carry out the analyses and calculations in accordance with the proposed model, the data were gathered from:  
Figure 5: The place of dynamic (D) and stationary (S) measurements of the DGPS and EGNOS systems, as well as multi-GNSS receivers. * Stationary measurements of the DGPS and EGNOS systems carried out on the radio beacon in the port of Gdynia in April and May 2014. As part of the study, two receivers were used: Leica MX-Marine (DGPS) and Topcon Legacy-E (EGNOS), which simultaneously recorded their position coordinates with a frequency of 1 Hz. During the several-day measurement campaign, the accuracy statistics of these systems were determined based on nearly 1 million positions (Table 1) [35]. * Dynamic measurements of the DGPS and EGNOS systems carried out during vessel manoeuvring on the Gdarsk Bay on 20 June 2017.