Delay Indices for Train Punctuality

Indicators of expected quality of service in public contracts are often based on some kind of “punctuality”, usually defined in terms of the percentage of trains arriving at the final destination (and/or at intermediate significant stops) within a given delay. Passengers, however, tend to use the word “punctuality” with a more general meaning, mostly as a synonym for expected delay at their own destination, and especially in case of commuters are much less tolerant of even smaller delays than train operators would normally allow. In particular, measuring the delay only at the final destination is perceived as largely inadequate, leading to underestimation of the actual percentage of late trains, and in turn undermining passengers’ trust in official performance statistics. In this paper, we take the passengers' perspective, introducing a family of delay indices called D-indices aimed at capturing the overall performance of a train “as a whole”, taking into account both the delays at the sampling points and the mutual location and order of such sampling points. In this paper, all indicators have the physical dimension of time in order to be easily replaceable to other delay measures. We first present typical approaches and definitions of punctuality in the literature, then introduce D-indices while exploring their features, pros and cons, and relevant properties. We validate and discuss our approach by comparing this model with existing approaches both theoretically and by comparison with selected datasets consisting of about one hundred trains transcribed over the last three years.