## The Value of Reliability and the Distribution of Random Durations

### Authors

M Fosgerau, Danish Technical University, DK

### Description

This paper considers the problem of when to initiate an activity of random duration when there is a fixed opportunity cost per time unit of starting early and a greater fixed cost per time unit of finishing late relative to some fixed deadline.

### Abstract

This paper considers the problem of when to initiate an activity of random duration when there is a fixed opportunity cost per time unit of starting early and a greater fixed cost per time unit of finishing late relative to some fixed deadline. The structure of this scheduling problem occurs in many contexts with timing of travel demand under congested conditions being a prominent case. It turns out that the optimal starting time, the head start, as well as the minimal expected cost is linear in the mean and standard deviation of duration, regardless of the form of the standardised distribution of durations. There is thus a link between the individual costs of lateness and earliness and aggregate characteristics of the duration distribution. The marginal expected cost of standard deviation, the value of reliability, depends on the duration distribution in a specific way that may be employed to transfer the value of reliability from one setting to another. These results generalise approximately to the case where the mean and standard deviation of duration depend on head start as is the case during peak-load demand. The results are illustrated empirically using the travel time distribution from a typical congested urban road.

#### Publisher

Association for European Transport