  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.
