MATHEMATICAL ANALYSIS OF STOCHASTIC TRANSITION MATRICES IN MARKOV MODELS USING MONTE CARLO AND MCMC SIMULATION

Road network control remains a mathematically and practically complex challenge in optimizing urban traffic systems. Traffic congestion, particularly at unsignalized intersections, continues to be a critical issue in densely populated cities. These intersections often allow vehicles to navigate without external control, leading to randomized vehicle behavior and increased risks of conflicts or accidents. Traditional methods have addressed intersection congestion through various control strategies, yet few have formally modeled these dynamics using rigorous mathematical structures. In this paper, we construct a finite-state Markov chain model to describe the stochastic evolution of traffic states at an unsignalized intersection. The system is represented by a row-stochastic transition matrix, where each state corresponds to a discrete traffic condition (e.g., low flow, moderate flow, queued, and congested). The stationary distribution vector is derived under the condition that its multiplication with the transition matrix leaves it unchanged, capturing the stable, long-term distribution of traffic across states. A spectral analysis of the transition matrix is conducted, with emphasis on the subdominant eigenvalue and the spectral radius, to evaluate the rate of convergence toward equilibrium and assess intersection stability. The results demonstrate that the spectral properties of the transition matrix serve as robust indicators of system performance and congestion potential. Intersections with slowly decaying eigenvalues exhibit persistent traffic buildup, whereas rapidly converging systems suggest smoother flow. These findings establish a formal mathematical foundation for diagnosing and potentially optimizing traffic behavior at critical road network points.