# Enhanced Scalability and Security in Blockchain-Based Transportation Systems for Mass Gatherings
## 3 Methodology
### Mathematical Model for Scalability in Blockchain-Based Transportation System  
In this study, we present a mathematical model that serves as a key analytical framework for optimizing blockchain-based transportation solutions through load forecasting. The model aims to predict transaction volumes by utilizing advanced statistical and machine learning techniques, enabling efficient resource allocation while maintaining scalability, security, and reliability. By manipulating key variables such as resource utilization, latency, and security overlays, the system parameters in the model change dynamically to maintain high efficiency in unfavorable situations. This ability to predict enables proactive scaling, mitigates bottlenecks, and ensures smooth operations. Second, the model provides security by allowing the anticipation of transaction surges and the reduction of potential vulnerabilities before they can become attack sources. This integration into our research underscores the need for a data-driven approach to modern transportation networks featuring adaptive and resilient blockchain implementations. The model accurately predicts future transaction loads, enabling more intelligent allocation of financial resources, reduced energy waste, and an enhanced user experience. Ultimately, this mathematics aligns with the study's objectives: to strengthen transportation infrastructure by developing an adaptable and secure digital ecosystem. Practically, our research ensures the incorporation of blockchain-based transportation solutions that are  
Figure 6: Blockchain-based transportation system architecture with scalability features. responsive, cost-effective, and robust in response to evolving demands. This equation is shown in Equation (1) [23]. \[T(t+1)=\propto.T_{h}.\beta.T_{r}.\gamma.\]  
Anomalies  
In Equation (1), historical loads \((T_{h})\), real-time loads \((T_{r})\), and anomaly-induced loads forecast future transaction load at time \(T\left(t+1\right)\). Further, \(T_{h}\) is used in the weight \(\alpha\), which reflects the influence of past trends derived from statistical or machine learning models. A high \(\alpha\) implies that historical trends are significant in predictions, while a low \(\alpha\) suggests that real-time factors are more important. The weight \(\beta\) contributes to Tr obtained from real-time analytics. A firm reliance on immediate transaction data is crucial in volatile environments, as a high \(\beta\) represents. Estimates of disruption, such as delays or security threats, are fed into an anomaly factor, \(\gamma\), using machine learning techniques, including clustering and outlier detection. This implies that anomalies highly affect transaction loads, so a resilient forecasting model is needed. Furthermore, State Channels and Rollups reduce on-chain activity, bringing scalability. In that case, multiple off-chain transactions are allowed, but only the final state is settled on the chain. Data submitted via Rollups is compressed into batches before being rolled up. Throughput, reduced fees, and speed--all these methods are increasing throughput, lowering costs, and improving efficiency while maintaining security and decentralization. It is this layered approach that makes the system scalable for peak transaction periods. Equation (2) [24]:  
\[\lambda_{total}=\lambda_{on-chain}+\lambda_{off-chain}\]  
Above, Equation (2) provides a quantitative model of the total transaction throughput (\(\lambda_{total}\)) as the off-chain \((\lambda_{on-chain})\) and off-chain \(\left(\lambda_{off-chain}\right)\) transactions showing the scalability impact of off-chain solutions. The decrease in network congestion that results from shifting transactions off the chain increases efficiency. The dynamic adjustment of the off-chain throughput enables a precise estimation of the scalability benefits, as determined by Equation (2). This offers adaptability in responding to traffic spikes with optimal performance.