Optimality conditions for public transport schedules with timepoint holding

Holding buses to scheduled departure time at timepoints involves a tradeoff between reliability and speed, with impacts on user and operating cost. Two new measures of user cost, excess waiting time and potential travel time, are proposed. They relate to the early extreme of a bus’s departure time distribution from a passenger’s origin stop, and the late extreme of a bus’s arrival time distribution at the destination stop. A route with long headway service is modeled assuming that segment running times are independently distributed. Operating impacts of unreliability are captured by requiring enough recovery time that delay does not systematically grow with each cycle. Based on an objective of minimizing a sum of operating cost and user costs, optimality conditions are derived for the strictness of a timepoint and for dispatching reliability at the terminal, which are related to the amount of slack within the running time schedule and within the scheduled layover. It is shown that a timepoint’s optimal strictness (probability of holding) increases with the demand for boardings at the timepoint, with the effect diminishing as stops become farther from the start of the route; however, welfare benefits compared to using a uniform percentage of slack across the route may be small. It is also shown that there is no universally optimal dispatch reliability; the more slack is built into the running time schedule, the less reliable should be the dispatch from the terminal. Up to a point, as scheduled running time increases, the optimal recovery time decreases, and slack time spent holding en route substitutes one-to-one for slack time spent holding at the terminal, so that holding at timepoints does not necessarily increase operating cost.