In this paper, we introduce a mathematical approach for modeling the bus priority system. The main objective is to highlight the problem of the bus priority dilemma. The mathematical approach is based on Petri Nets with inhibitor arc. Based on dioid algebra, we add a new constraint that models the resource sharing problem with a priority user. The system behavior is described by equations -linear equations. As a result, the proposed approach provides us with interesting performance evaluation, such as real-time counts of cars and buses which correspond to best priority-configurations. An example is worked out to highlight the bus priority dilemma. This example allows obtaining of interesting observations about the relevance of bus priority in a congested traffic network.
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