# Simulation-Based Evaluation of V2X System with Variable Computational Infrastructure
## 3 Materials and Methods
### Mathematical Model  
We will now briefly review the mathematical framework employed for implementing the V2X system model and its subsequent simulation. In the authors' earlier work [17], a rectangular metric was used for organizing the road network due to its widespread adoption and ease of description. In the present study, the road structure model is represented generally as a graph, where intersections are vertices and roads are edges. This approach is significantly more flexible and allows for an expanded class of optimization problems. Note that the choice of metric, or the method of distance calculation, becomes secondary and depends on the specific tasks and road maps used for the simulation. We outline the core principles concerning traffic organization. Vehicle generation, which is managed subsequently, is described by Equation (1), defining the speed distribution with average \(\mu\) and \(\sigma^{2}\) dispersion density for each OBU on a road section within a single RSU's coverage area. The speeds represent a stream of independent, identically distributed random variables equally distributed according to a truncated normal distribution with values ranging from a specified minimum to maximum speed, i.e., \(u_{min}\leq u\leq u_{max}\). \[ \begin{split}\varphi(u)=\frac{2}{\sigma\sqrt{2\pi}}e^{-\frac{(u- \mu)^{2}}{2\sigma^{2}}}\cdot&\frac{1}{erf\left(\frac{u_{max}-\mu }{er\sqrt{2}}\right)-erf\left(\frac{u_{min}-\mu}{er\sqrt{2}}\right)}\end{split}  \tag{1}\]  
where \(erf()\) is the normal distribution function or Laplace function. OBUs interact with RSUs by exchanging data blocks, such as information regarding traffic congestion on a route section. Let each OBU and each RSU generate information as a vector \(M\) of uncorrelated data elements \(M=\{1,\;2,\;...,\;m\}\). For numerical simulation, we assume that the random variable representing the probability of an element \(i\) in the tuple \(M\) being requested by an OBU follows a Zipf distribution [45], calculated using Equation (2):  
\[p(i)=\frac{i^{-A}}{\sum_{k=1}^{m}k^{-A}}, \tag{2}\]  
where \(A\) is the distribution asymmetry coefficient and corresponds to the loading degree of the vehicle system. If an OBU accelerates within a road section serviced by RSU, \(s=u_{i}\) m/s, and the coverage area of this section spans \(L_{s}\) meters, then the maximum number of requests that the RSU can process from a specific OBU is calculated using Equation (3):  
\[k=\left\lfloor\frac{L_{s}}{u\cdot\alpha_{s}}\right\rfloor, \tag{3}\]  
where \(\alpha_{s}\) refers to the average processing time per OBU. It is worth noting that in real-world conditions, signal attenuation occurs naturally as the vehicle moves away from the roadside unit within the RSU's coverage area. Our simulation scheme employs the well-known Hata (Okumura-Hata) model [46] to account for the building coefficient and its influence on attenuation. However, this model is designed for frequencies up to 2.5 GHz and is not quite applicable to the 5.9 GHz frequency used in V2X systems. Therefore, we compared and additionally conducted a simulation based on the 3 GPP Urban Canyon NLOS model [47]. The use of the Hata model was due to its similarity to the authors' modeling conditions in earlier work on the IEEE 802.11p interconnection protocol. 3 GPP Urban Canyon NLOS model is designed for higher frequencies and specifically accounts for signal propagation in urban environments with narrow streets and tall buildings. The comparison of the two models reveals significant differences in signal attenuation behavior, especially at larger distances. As shown in the graph (Figure 3), the Hata model predicts relatively lower path losses compared to the 3 GPP UMi Urban Canyon model, particularly in urban canyon conditions. This discrepancy arises from the Hata model's inability to account for high-frequency signal attenuation and specific propagation challenges in dense urban environments, such as reflection, diffraction, and shadowing caused by tall buildings and narrow streets.