## The Value of Travel Time Variance

### Authors

M Fosgerau, DTU/CTS/ ENS, DK; L Engelson, KTH, DK

### Description

This paper considers the value of travel time variability.

### Abstract

This paper considers the value of travel time variability under scheduling preferences that are defined in terms of linearly time-varying utility rates associated with being at the origin and at the destination. The main result is a simple expression for the value of travel time variability that does not depend on the shape of the travel time distribution. The related measure of travel time variability is the variance of travel time. These conclusions apply equally to travellers who can freely choose departure time and to travellers who use a scheduled service with fixed headway. Depending on parameters, travellers may be risk averse or risk seeking and the value of travel time may increase or decrease in the mean travel time.

Recently, Fosgerau and Karlstrom (FK) (Transportation Research Part B 2010) presented a derivation of the value of travel time variability based on scheduling preferences adapted from Vickrey (1969) and Small (1982). They derived the time cost for a trip of random duration for a traveller who could freely choose his departure time, with these scheduling preferences and optimal choice of departure time. The time cost is the value of travel time multiplied by the mean travel time plus a constant, the value of travel time variability, times the standard deviation of travel time.

There are a number of advantages associated with the FK result. First, maximum expected utility is just a linear combination of the mean travel time ì and the travel time variability ó. Second, the result holds for essentially any distribution of travel times Ö. Third, the preferred arrival time does not appear in the expression. It is then not necessary to know the preferred arrival times of travellers in order to apply scheduling preferences. Previously, this was thought to be an obstacle as such information is hard to find. Fourth, the result provides a basis for including a measure of scale of the distribution of travel times directly in the specification of preferences. This has been done in a range of papers, but lacked the justification that is obtained from defining preferences in terms of travel times outcomes rather than the travel time distribution. Fifth, FK show that their expression remains a good approximation when ì and ó are allowed to depend (in a limited way) on the departure time.

There are however also disadvantages associated with á-â-ã preferences and the FK result. First, the value of travel time variability depends on the shape of the travel time distribution. Second, the GTC expression is not additive over parts of a trip. Additivity would have been a desirable property of a measure of the value of travel time variability, since then the time cost could have been computed separately for different parts and then added. This would have made easier the application of FK to links in a network. Third, and perhaps most importantly, it is not given that á-â-ã preferences is the best representation of the scheduling preferences of travellers. Finally, the traveller must be able to freely choose his departure time, which is not true for a scheduled service.

Just as many travellers may care about not being late for some activity, they might also care about not leaving some other activity too early. The á-â-ã preferences treat departure time differently from arrival time. There is a special time for arrivals but no special time for departures. A priori it is not clear why this should be so.

Consider travellers who differ in one respect only, the duration of the trip. The á-â-ã preferences imply that the traveller with the longer duration would depart earlier but arrive at the same time as the traveller with the shorter duration. This is an empirically testable proposition which may be used to refute (in an appropriately loose sense) á-â-ã preferences.

Vickrey (1973) considered another type of scheduling preferences. This type of scheduling preferences associates a time varying utility rate with time spent at the origin and a similar time varying utility rate with time spent at the destination. The scheduling utility associated with a trip departing at time a and arriving at time t is the utility gained from being at the origin until time a and at the destination after time t. This is appealing since it connects scheduling preferences with the activities before and after the trip in a symmetric way. These scheduling preferences form the basis of this paper.

#### Publisher

Association for European Transport