A max-plus model deriving the effect of a skip-stop policy on the frequency of a mass transit line

Skip-stop policies are known to bring benefits to both transport line operators and passengers under certain circumstances. To evaluate the gains in frequency, comparing skip-stop to all-stop scenarios is necessary. We propose a Max-plus algebra model for the train dynamics on a metro line with a skip-stop policy. Our model allows the derivation and interpretation of the physics of traffic and the effect of applying a skip-stop policy on the line frequency. The model allows us to analytically derive a fundamental traffic diagram for the line (i.e. a diagram giving the average train-time headway or frequency as functions of the number of running trains), with three distinct phases: free-flow, train capacity, and train congestion phases. In addition, we provide and interpret analytical formulas that reveal the impact of the parity of the number of running trains on the train time-headway and frequency. Furthermore, we evaluate the gains induced by a skip-stop scenario compared to an all-stop scenario. The model is applied on a unidirectional loop line, and we take metro line 1 of the Paris network as the application case. The results indicate that the line capacity, i.e. the maximum train-frequency, can indeed be improved and that the gain function is piecewise linear and depends on the number of running trains. Besides, we show the existence of an optimal number of running trains for which the gain is maximum. The proposed model can be generalised for other skip-stop policies and network configurations.