# Simulation-Based Evaluation of V2X System with Variable Computational Infrastructure
## 3 Materials and Methods
### Mathematical Model
Several other studies [48; 49] have also explored the use of various path loss models, including the Hata and 3 GPP models. Data loss occurs during OBU-RSU interaction, as described by Equation (4). Let us enter the non-negative loss function when processing RSU-OBU requests. Let the OBU at number \(i\) form a request \(D(i)=\sum_{k=1}^{m}C\cdot p(i)\), then the interaction loss on the length portion of the \(L_{s}\) will be as follows:  
\[V(i)= \begin{cases}D(i)-C\cdot k,&C\cdot k<D(i)\\ 0,&C\cdot k\geq D(i)\end{cases} . \tag{4}\]  
If some of the requested information needs to be updated, then the total interaction time will be as follows (5):  
\[\sum_{i=1}^{k}\Bigl{(}t_{i}^{(0)}+\gamma\cdot t_{i}^{(1)}\Bigr{)}\leq\frac{L_ {s}}{u}, \tag{5}\]  
where the time to process a request from a particular OBU for data block \(i\) is \(t_{i}^{(0)}\) if the data at that memory location are up-to-date. \(t_{i}^{(1)}\) is the time required to update outdated data if the data require updating; \(\gamma=1\) if the data are outdated, and \(\gamma=0\) if the data do not require updating for each OBU-RSU pair. Figure 3: The comparison of the Hata model and 3 GPP Urban Canyon model. Describing the mathematical framework for mRSU operation requires a more detailed explanation of pathfinding algorithms in graphs since we assume that in all scenarios, mRSUs move along dedicated lanes without encountering traffic jams. It is convenient to utilize Estimated Time of Arrival (\(ETA\)) [50] as a cost function: the resulting route will be the shortest among all possible routes. In the simplest case, travel time comprises two factors: time spent traversing graph edges and additional time for maneuvering when transitioning between graph vertices, formalized by Equation (6):  
\[ETA=\sum_{i=1}^{n}t_{i}+\sum_{i=1}^{m}c_{i}, \tag{6}\]  
where the first summation represents the travel time along graph edges, and the second represents penalties for maneuvers. If the mRSU moves along a dedicated lane on the previously described route, the average travel time remains constant. If mRSU movement to a specific map point is considered, then at each stage, the element of our dynamic system closest in terms of arrival time must be selected. The problem with the chosen cost function, \(ETA\), lies in the need to consider route parameters beyond just travel time. The identified route may be highly inconvenient for several reasons: poor road quality (e.g., poor surface, numerous potholes, lack of markings) or complex turns or intersections that should be avoided. Iterating through all possible route options would consume significant computational resources, impacting the overall system. Therefore, our study employs statistical analysis to determine preferred edges for traversal, formalized by Equation (7):  
\[F_{cost}(route)=\sum_{i=1}^{n}\frac{t(edge_{i})}{f\Big{(}p_{edge_{i}}\Big{)}}+ \sum_{i=1}^{m}\ c^{*}(turn_{i})+\sum_{i=1}^{k}r_{i}^{*} \tag{7}\]  
where \(f(p)\) is a normalized weight function approximating the attractiveness of each edge on the route, and \(r_{i}^{*}\) is the portion of penalties accumulated at traversed intersections. Experimental results indicate that a linear function \(f(p)=p\) yields the best results. Systemic errors in speed determination can occur on certain roads (e.g., due to insufficient data), resulting in a suboptimal calculated route. However, the proposed approach significantly reduces the average error magnitude.