Exactly solvable dual bus-route model

In 1998, O'Loan et al. introduced a simplified bus-route model to illustrate bus dynamics. However, fluctuations in passenger numbers make it challenging to achieve an exact solution for the model's stationary state, as these fluctuations can impact bus behavior. In this study, we present an exactly solvable model for a dual bus-route system that builds upon O'Loan et al.'s model. This dual model allows us to comprehensively analyze bus-route dynamics. Our model introduces additional parameters not previously considered by O'Loan et al. to account for neighboring effects and align with an observation in the original model, which can influence the average stationary current and velocity of buses. When the neighboring effect is weak, our model behaves similarly to O'Loan et al.'s bus-route model. However, with a strong neighboring effect, our model exhibits intriguing characteristics. Additionally, in some limiting cases, our model recovers well-known models such as the simple exclusion process, its generalization, the cooperative exclusion process, and the RNA polymerase model during the elongation stage.