We develop a Markovian traffic equilibrium model for ride-hailing in which vehicles, whether empty or hired, make sequential order-acceptance and link-choice decisions over a traffic network to maximize total discounted return in an infinite-horizon semi-Markov decision process. The model endogenizes both competition among empty vehicles for passenger demand and traffic congestion arising from road usage at the link level. We characterize equilibrium as the solution to a fixed-point system, establish its existence, and develop relaxed fixed-point iteration algorithms for equilibrium computation, with convergence results for specialized network structures. Computational experiments on realistic networks demonstrate the model's practical value for transportation planning. Ablation analyses reveal that ignoring either traffic congestion or drivers' forward-looking behavior can lead to potentially substantial biases in policy evaluation.

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