The optimal step toil mechanism is a representative flexible toll scheme for the bottleneck model in the field of transport economics. However, no information about the individual behavior of departure time movement from the no-toll to the tolled case is available from the previous literature for practical application. To solve this problem, we derived a series of mathematical formulas for each motorist's departure time, queue length, and entry time before and after implementing the optimal n-step toll scheme (n = 1, 2, 3, ...) by the conservation principle of equilibrium cost. All the derived formulas show significant regularities as the number of charging steps (n) increased one by one. By comparing the formulas before and after implementing such a toll scheme, we found that all the tolled motorists benefited not only from reduced queuing times but also from extended departure times after the change from the no-toll to the tolled case. In addition, regular departure time movements from the no-toll to the optimal n-step toiled cases for every single motorist were obtained. The regularities in all the results obtained from our mathematical formulas could be helpful for authorities making management decisions regarding the implementation of an electronic optimal n-step toll collection system.
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