The Marginal Social Cost of Headway for a Scheduled Service
M Fosgerau, DTU Transport, DK
The paper derives a simple formula for the contribution of headway to the generalised travel cost for users of a scheduled service such as bus or rail. Model account both for planning and unplanning travellers.
This brief paper derives the marginal social cost of headway for a scheduled service, i.e. the cost for users of marginal increases to the time interval between departures. In brief we may call it the value of headway in analogy with the value of travel time and the value of reliability. Users have waiting time costs as well as schedule delay costs measured relative to their desired time of arrival at the destination. They may either arrive at the station to choose just the next departure or they may plan for a specific departure in which case they incur also a planning cost. Then planning for a specific departure is costly but becomes more attractive at longer headways. Simple expressions for the user cost result. In particular, the marginal cost of headway is large at short headways and smaller at long headways. The difference in marginal costs is the value of time multiplied by half the headway.
For a frequent service we may imagine users do not plan to use a specific departure, they just arrive at their chosen time in order to catch the next departure and they might arrive at any time between two departures. Their waiting time at the station can be zero if they are lucky enough to arrive just in time for a departure. At worst, they arrive immediately after a departure and have to wait for a length of time equal to the headway. In general, we expect the waiting time at the station to follow a uniform distribution ranging from zero to the headway. This observation has led some to include just the average waiting time, half the headway, into the user cost. In the present case that accounts for scheduling considerations, they are seen to incur costs of waiting as well as costs due to the uncertainty about the time at which they will arrive at their destination.
For a less frequent service we may imagine that users plan which departure they want to use. In this case, the choice of service is based on scheduling considerations. The choice between planning and not planning we can imagine is governed by a cost of planning, which may include the effort involved in consulting the time table, the timing of the trip to the station as well as a planned wait at the station. We may suppose that the planning cost for a specific trip depends on the frequency of use of the service. Frequent users are able to distribute their planning cost over several trips. They therefore have a small planning cost for a specific trip and will tend more often to plan for a specific departure. Conversely, occasional travellers have a comparatively large planning cost and will tend less often to plan for a specific departure. In general we suppose the planning cost to have a distribution in the population of users such that no user will plan for a specific departure at very short headways, while all users will plan for a specific departure at very long headways. The distribution of planning costs in the population of users implies a smooth transition between the two cases as headway increases.
Association for European Transport