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# «Spatial, Temporal and Size Distribution of Freight Train Delays: Evidence from Sweden Niclas A. Krüger* a,b,c Inge Vierth a,b Farzad Fakhraei ...»

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s = 2.3263 ⋅ 1 ⋅ 200 / 12 = 39 (6) If the standard deviation is 1 hour, in 99 % of all cases the deliveries will be within 8.23 hours, that is, at most 2.32 hours delayed. The buffer stock is thus what is needed in production during the 2.32 hours of maximum delay, roughly 39 parts (since we need 200/12 parts per hour as input in production). Thus, in total we have to order 139 parts each time, 100 parts for production during the expected transport time and 39 as a buffer for the stochastic part of transport time.

The finding in this paper is however that rail transports are not normally distributed. This affects the calculation of buffer stocks twofold: once, the standard deviation itself might be biased downwards and second, the standard deviation is not a good risk measure for variability.

The empirical delay distribution shows that the 99th percentile between Malmö and Hallsberg is more than 2.32 times the standard deviation of 1 hour (that is 2 hours and 20 minutes); it is instead almost 4 hours in total. Hence, the necessary buffer stock becomes 67 parts (demand in production per hour times 4 hours of maximum delay) instead of 39 parts, an increase by 70 percent compared to the calculation based on standard deviation and the assumption of normality.

Thus, there are huge amounts to be saved by reducing the extreme delays. However, because of the possible complex causal-structure behind large delays we do not know if it will be effective to target them. Still, the buffer stock example here illustrates the importance of the tail for freight transports. It can be argued that we will seldom see rail-based just-in-time production and that most transport chains include collection and distribution transports by road. However, this is an adaption to the present state of the rail network by companies. Keeping buffer stocks, localisation of supply industry close to factory (agglomeration buyer-supplier) and transporting freight by other modes are possible strategies to cope with rail transport time variability.

However, all these measures tentatively increase costs for companies. Using standard deviation and normality of delays would be misleading for calculation of this cost.

5.2 General transport policy implications We believe that this paper has important, more general policy implications. The results suggest that the general association between rail breakdowns and weather conditions might be just a revelation of symptoms and not the sole cause of such breakdowns. For example, at many times and places bad weather occur without train operations failing. Hence, we think that even if exogenous factors matter for the probability and severance of breakdowns, the causes are to some extent endogenous to the rail transport system and most probably there is some interdependence and combinations between various endogenous and exogenous causes.

Endogenous in this context means that the rail transport system has a complex structure that is prone to be robust with respect to disturbances at most times and places but that at some times and places the network is fragile to disturbances, propagating initial small and geographically limited disturbances through connecting links, nodes and the rolling stock itself to larger and more widespread breakdowns in the transport network. This is the reason why we consider them as endogenous: the network itself causes the breakdowns because of its own structure. One example is the track capacity utilization that is assumed to have a pronounced effect on freight train delays. But one can argue that large delays occur where capacity is used the most, because there are many links, nodes and train that can be affected and once affected influence other links, nodes and trains. In other words, the extremely unevenly capacity utilization that is characteristic to rail networks (and other complex networks) is in itself a cause for large delays. Once we account for the number of trains the average delay per train (and per train kilometre) is not significantly affected. Hence, large delays are just a sign that we increased transport demand and there will be no easy way to prevent delays. But there are nevertheless some possible policy measures that can be used to target the problem. The reason is the revealed extremely uneven distribution of transport demand and delays over time and space. These distributions suggest that investment in capacity, reinvestments and ordinary maintenance measures should be spread not uniformly across the rail network but be focused on problematic hotspots. Even now investments and maintenance are not carried out evenly but capacity investments and reinvestments should probably be even more extremely distributed.

Often, but not always, we can identify a certain cause for how a major break-down started.1 We have to acknowledge that many delays are unpredictable. Our data analysis reveals that the delays that matter most have extremely low probability. Even if we know based on our analysis that during the next year a major breakdown of a certain size will occur at least one time, we still do not know where and when. If we could predict them they would not occur. However, based on our analysis of distributions delay distributions at stations we can say that measures undertaken to reduce the importance of central nodes (regardless whether it is a steering computer or a marshalling yard) would milder the consequences of disturbances. For example, backup capacity at steering and information centrals that can overtake responsibility temporarily.

Similarly, measures undertaken for improving reliability outside the central nodes are with a high An example: On 23rd July 2011 the city of Norrköping south of Stockholm received 43 mm of rain within just two hours. Due to this heavy rainfall the computer central of the Swedish Transport Administration, responsible for steering and providing information for rail traffic in large parts of Southern Sweden located in Norrköping, were flooded. As a result, train operations in major parts of Sweden between Malmö and Stockholm had to be cancelled and redirected (with large delays) for several days.

probability without any major effect on overall performance of the rail network since they are of minor importance of the functioning of the whole network. For this reason, our results provide some (even if limited) insights on the impact of measures on performance outcomes and are therefore of some value for cost-benefit analysis of measures targeted at improving reliability and increasing capacity.

6 Conclusions This paper analyzes the distribution of freight train arrival delays on the spatial, temporal and frequency-size scale. Since the spatial and frequency-size and temporal distributions describe the vulnerability of a rail transport system it has potentially important policy implications.

Considering the tail of arrival delays at the final destinations within the rail system we find that it is exponentially distributed. This implies that the tail makes up the biggest part of total delay time. The 20 % largest delays contribute to about 74% of total delay minutes. For the spatial scale, we find that more than 50% of the total arrival delay per year occurs in just 7% of stations, all of those being marshaling yards. The contributions of the paper to the literature are manifold, mainly because this paper: i) analyzes all freight train delays in a national rail network covering 2 full years, ii) analyzes the spatial and temporal concentration of delays, iii) uses the exogenous chock caused by the financial crisis to investigate the relationship between capacity utilization and average delay and iv) analyzes the implications of observed delay distribution on the inventory management of firms which has, as we argue, implications for the valuation of transportation time variability as well.

With the help of regression analysis we analyze how delays are propagated in the network.

We find that delays at the origin increase arrival delay but that some part of the initial delay is gained at arrival, probably due to large slack in the timetables. Finally, in the temporal scale we analyze arrival delays in different time scales such as monthly, weekly and daily delays. We expected that the significant reduction of ton-kilometers transported due to the economic recession would reduce not only total but also the average arrival delay since there would be more free capacity. The results show, however, that the average delay did not decrease as the number of freight trains decreased due to the economic contraction in 2009.

To the best of our knowledge this is the first study in its kind related to rail freight. Based on our findings we would advocate further research in this area with focus on passenger trains, the interaction between passenger and freight trains and further analysis of Swedish rail network structure. The analysis of delay distribution facilitates the selection of appropriate methods for valuation of rail reliability and thus for cost-benefit analysis of rail investments and other measurements. More specific, our analysis suggests that standard deviation is not an appropriate measure of transport time variability of rail freights. Moreover, it seems that very few extremely large delays matter most for the total amount of delay and not the many small delays. However, only a sound valuation method can answer the question whether society would benefit from reducing the number of delays in general or from preventing the extremely large delays.

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