Delay effect on schedule in shuttle bus transportation controlled by capacity

We study a bus schedule in a shuttle bus transportation system controlled by capacity. The motion of shuttle buses depends on the inflow rate of passengers, the number of buses, and the delayed increase of buses. The bus schedule is closely related to the dynamic motion of buses. The delayed increase of buses has an important effect on the arrival of buses. We present the delayed piecewise map model for the dynamics of shuttle buses with the delayed increase. The motion of buses changes from a stable state to an unstable state and vice versa with increasing the inflow rate. The arrival of shuttle buses oscillates with various periods in the unstable state. The dynamic transitions change highly with the delayed increase of buses. We clarify the effect of the delayed increase on the bus schedule.