Modelling the Tendency of Buses to Form Pairs

Abstract
The paper describes a simple mathematical model of the behaviour of buses, on a relatively frequent service, after one has arrived at a stop at a time different from that scheduled. The model is similar to that used in some earlier papers on the pairing of buses, but derives a different objective function. Results are presented in terms of parameters relating to the scheduled headway (h) and the size of the initial deviation from it (hd), the usual time spent by a bus at a stop, and the ratio (k) of the passenger arrival rate to the boarding rate. A graphical method was employed to find P, the number of the stop at which the first two buses form a pair. For the typical values of parameters used the relationship P = 1/6dk was deduced which, although no generality is claimed, can be used to set operational limits so that the bunching of buses does not occur. Limitations of the model are discussed.

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