Estimating the Reliability of Travel Time on Railway Networks for Freight Transportation

Railway freight transportation is an important transport system that its reliability causes economic issues. Freight carriers require predictable travel times to schedule their programs in competitive environment, so the estimation of reliability of travel time is very important. The present study proposes a travel time index that estimates the reliability of railway freight transportation and evaluates performance as well. Travel time reliability is estimated based on the shortest path between O-D pairs. Statistical measures of travel time, defining as the ratio of the 95th percentile travel time and the shortest path mean travel time as an ideal travel time, for each obtained route are calculated according to their selected links. Experimental data on Iranian rail network has been used as case study and results revealed that the routes less than 400 kilometers should be improved in terms of their reliabilities, because they are less reliable than long distance routes.

The theory of reliability has been associated with the design and management of communication and performance of mechanical equipment. It is defined as "the probability that an entity will perform its intended function(s) satisfactory or without failure for a specified length of time under the stated operation conditions at a given level of confidence" (Kececioglu, 1991).
Travel time reliability is one of the important system performance measures for transport operations. It is an indicator for facility operational consistency over an extended period (Emam et al., 2006). Travel time reliability is defined as the probability that a trip between a given origin-destination pairs can be successfully made within a specific period of time (Recker et al., 2005). There are many different definitions for travel time reliability and subsequently different measures for estimating that in transportation networks. All these measures relate to properties of the (day-to-day) travel time distribution and the distribution curve shape (Van Lint et al., 2008). In fact, the reliability of travel time may be more important than the travel time itself for travelers, shippers, and transport managers for planning and programming (Lyman et al., 2008). A reliable travel time is more important than a delay-free travel time for some segments of nations' economy. It is also important because urban residents react to unexpected travel time rather than mean/average travel time. Any interruptions can significantly decrease shippers and traveler satisfaction and increase frustration (Lomax et al., 2003).
Today, railway freight transportation is one of the most important transport systems that its disruption may cause remarkable economic issues including delayed delivery penalties and customer losses.
Shippers and freight carriers require predictable travel times to schedule their transport programs in the competitive environment, so, they seek the systems which can deliver the raw materials with the minimum or at least predictable delays. The estimation of the reliability of travel time is an effective method to find such systems which improve regional transportation planning systems and their operations.
The aim of the present research work is to estimate the reliability of railway network based on the reliability of routes' links. This article is organized as follows. The following section briefly recalls a number of common used travel time reliability measures. In the next section, another measure of travel time (un)reliability is derived. Then, the travel time reliability of a selected railway network is estimated using empirical data set followed by comparing the new and old measures based on experimental data. Discussion on obtained results at the final section offers some conclusions and recommendations for researchers and practitioners.

Common Used Travel Time Reliability Measures
The travel time reliability measures are more generally defined and directly applied to the data and derived from continuous probability distributions. There are two categories of travel time reliability measures. The first category is including measures based on the moment (e.g., mean, standard deviation, Skewness, Kurtosis, and coefficient of variance) and measures based on percentile are the second category, e.g., the 90 th or 95 th percentile, buffer index, planning time index (Yang et al., 2016).
The following provides a subset of travel time reliability measures that are used by different sources over time and collected in periodic special studies, estimates from continuous point-based detector data, or estimation created through simulation (FHWA, 2007).

Standard Deviation
Standard deviation is a classic stat which is suitable for travel time reliability in studies with classic mathematical or statistical models (Dong et al., 2009). In standard deviation, all late and early arrivals have equal weights; thus, U.S. DOT guide (Texas Transportation Institute and Cambridge Systems, 2009) and NCHRP Report (Cambridge Systematic, 2008) discouraged its use as a reliability performance measure. Segments with narrow curves of average travel time have insignificant travel time variation from day to day; hence, they are more reliable than others. This measure is incomplete because it shows travelers' acceptability of travel time less than variation in travel time (Emam et al., 2006).

95 th Percentile Travel Time
The 90 th or 95 th percentile travel time is the "simplest method to measure travel time reliability". It estimates the amount of delay that will be on the heaviest travel days (FHWA, 2007).

Planning Time Index
The total travel time including buffer time is the planning time that is calculated as the 90 th or 95 th

Buffer Index
Passengers usually add some buffer time and departure earlier to avoid travel delay and deal with travel time variability (Li et al., 2013). Buffer index is an extra time that travelers require to add to their average travel time to arrive on time. Buffer index, obtained by equation (2), is defined as the ratio of the difference of the 95 th percentile travel time and average travel time over the average travel time (Van Lint et al., 2008).

Buffer Index= (95 percentile travel time-average travel time) average travel time
Since, buffer index is defined based on the average travel time, it is preferred for commuters that are familiar with everyday congestion and the planning time index that is based on free-flow travel time may be preferred for those who aren't familiar with that (Pu, 2011). In comparison the differences among different paths, the path with bigger buffer index is less reliable than others (Li et al., 2013).

Travel Time Index
The ratio of actual average travel time over free-flow travel time is defined as travel time index which is shown in equation (3). "Strictly speaking, the travel time index is a congestion intensity measure rather than a reliability measure" (Pu, 2011).

Assuming Lognormal Distributed Travel Times
There is a significant variation in travel time distributions. In literatures, using real-life travel time without fitting any parametric or nonparametric statistical distributions is a frequent approach of developing customized travel time reliability measures to obtain real-life reliability measures. Because of the characteristics of the underlying statistical distribution, some measures developed by this approach could be misleading; moreover, it is hard to depict the relationships between measures which vary on a case-by-case basis by using analytic methods. Hence, developing reliability measures by assuming statistical distribution is a meaningful approach to develop travel time reliability measures.
This approach reveals the analytic relationships between the measures in cases which assumed and empirical distribution matches well (Pu, 2011). Acknowledging that reliability measures can derive from multimodal distributions, the first scope is confined to uni-modal travel time distributions to find an appropriate and best fitting simple traditional statistical distribution. The closest traditional statistical distribution is the lognormal distribution that describes the distribution of travel times (Guo et al., 2010). For simplifying, in this study lognormal distribution has chosen as the traditional statistical distribution for travel time and the following steps are all based on this assumption (NIST, 2006). In other words, this paper's results utilize only for lognormal distributed travel times.
The random variable X is distributed lognormal whose logarithm is distributed normally. If Y has a normal distribution with mean µ and variance σ 2 then Y=log(x-θ). The general formula for the lognormal distribution probability density function is defined by equation (4), where σ is a shape parameter (and is the standard deviation of the log of the distribution), θ is a location parameter and m is a scale parameter (Balakrishnan, 1999).

Derivation of New Measure for Travel Time Reliability
The ideal travel time is the shortest travel time, which is an ideal passenger travel time without waiting and jam (FHWA, 2007). Minimum travel time is used by finding the shortest path between origin-destination pairs using Dijkstra algorithm. This algorithm is effective to find the shortest path between nodes in graph. In the used transportation network, nodes represent stations and links' travel times are positive edge weights between pairs of stations connected by a direct link. Dijkstra algorithm will find shortest route between one station and all other stations.

Calculating Reliability Stats
Because of accessing to more information in detail from detectors, reliability measures are more widely discussed nowadays. The study of variation in travel time conditions gets easier utilizing travel information which is obtained directly or estimated from detection systems. Using the traffic management centers' information, the calculation and estimation of all measurement concepts is possible (Lomax et al., 2003). In this paper, the reliability stats are calculated using data collected by monitoring and information systems in freight transportation in Iran. For individual links, the mean and variance of travel time are calculated by equation (6) and (7). If X is a random variable of sum of independent random variables x 1 , x 2 …, x k , the mean and variance of X are calculated as µ X =µ x 1 +x 2+ …+x k =µ x 1 + µ x 2 + …+µ x k and σ X 2 =σ x 1 +x 2+ …+x k 2 =σ x 1 2 +σ x 2 2 +…+σ x k 2 in sequence and based on these statistics we can calculate the location and scale parameter of route travel time and its' probability density function (Walpole & Mayer, 2012).

Route Travel Time Reliability
Using the lognormal distribution presented in the previous section, travel time percentiles and thus the reliability can be easily expressed as a function of mean and variance. Here, we concentrate and reintroduce on how the ideal travel time, mean and variation of travel time can be estimated such that studied at (Wu et al., 2014). If the travel times are known, the mean and variances of individual links can be calculated and then the mean and variance of the total route can be achieved by summation.
Calculation of required percentile travel time of each route t T,95,route is required to its travel time distribution function and in sequence the lognormal probability density function which is defined by the mean T T ,route and variance σ T, route 2 . We defined how to calculate these two parameters and will get the value for 95 percentile travel time by using inverse cumulative density function.
The shortest path problem is one of the network flow problems. Here we present the linear programming formulation of the shortest path problem (Taha, 2008). Let G(V,A) a directed graph, c ij link costs or lengths for all (i,j)∈ A and path start node s∈ V, end node t∈ V, s≠t. x ij is a variable and  There are many algorithms for finding the shortest path between nodes in a graph. Dijkstra algorithm is the fastest known single source shortest path algorithm for arbitrary directed graphs with unbounded non-negative weights and can be used to find the shortest route between one station and all other stations. However, finding a shortest path to any given goal or set of goals is possible with a small modification (Ahuja et al., 1993). As mentioned before, the ideal travel time T ideal is needed to calculate travel time reliability and the ideal travel time is the shortest travel time, which is an ideal passenger travel time without waiting and jam (FHWA, 2007). Dijkstra algorithm is now performed to find the shortest path as is desirable. Finally, the route travel time reliability R route defines as the ratio of ideal travel time and 95 th percentile travel time and can be used by equation (12)

Proposed Procedure for Estimating Route Travel Time Reliability
The following procedure is utilized to estimate route travel time reliability as also illustrated in Figure 1.
It has been proposed for determining the travel time reliability in a route consisting of more than one

Data Description
Travel times during days of three months (2015) are considered on the whole studied rail freight network between all main stations and stations which are located in routes' intersections. Figure 2 depicts a railway system map of Islamic republic of Iran, containing main routes and stations' names.
Sixty-one stations have been selected according to their importance and locations and relevant data gathered for all OD pairs. Two types of stations are selected because they are more critical in finding the shortest path between OD pairs. Observed travel times were available from train traffic control centers for every departure during a day between OD pairs and data collected from those centers provide train movement details. Using observed data implies an estimate of the mean travel time for freight trains departing in a certain departure time period. The train traffic control data can obtain the time trains entering and exiting the stations. Sixty-one stations of Iran freight rail transit were selected and numbered as well as relevant data has been received for OD pairs. Since, a large amount of traffic data used to measure travel time reliability by the time of day and day of week (8), we selected data from 90 days in 2015 at the different times of days that fell within this definition.
Step 1: Determining the travel time of link l at time i Step 2: Calculating the mean of travel time Step 3: Calculating the variance of travel time Step 4: Determining the ideal travel time for OD pairs Step 5: Calculating the variance of the shortest path Step 6: Determining the percentile of the travel time Step 7

Travel Time Reliability Calculation
In order to calculate the defined travel time reliability index of each OD pair, the travel time probability density function is fitted to lognormal distribution. The calculation results are tabulated in Table 1 as a part of analysis results. Following that, route travel time reliability between station 1 to other stations can be compared. Having the 1-4 OD for example, T ideal , t T ,95, route and the travel time reliability of this route are 180 minutes, 199 minutes and 0.9 or 90%, respectively. As an example, let to consider a route between station 32 and 50 in studied area. The proposed procedure is applied for calculating the reliability of travel time. To calculate reliability, data has been received from RAI (The Railways of the Islamic Republic of Iran). The shortest path between two stations utilizing Dijkstra algorithm is (32-31-28-29-38-39-40-48-49-50) shown in Figure 3. The input parameters and results for illustrative example are given in Table 2

Route Prioritization Using Reliability Measure
The amount of travel time reliability is used here as the prioritizing measure. Routes which have two major features should be select to improve: a) less than 400 km in length, b) less than 10 links in segments. Routes have been categorized by length and put them into 200 km groups to find out which groups reliability are more critical. The results revealed that routes shorter than 400 kilometers are less reliable than the others as shown in Table 3 and Figure 4 (a). These routes include about 14 percent of all routes in the studied railway network. Figure 4 (b) depicts relationship between route length and travel time reliability. As mentioned, the main notable feature is that apparently the routes reliability increases significantly with route length and between routes which are shorter than 400 km, most of them consist of less than 10 links ( Figure 5). So, they should be prioritized under study to improve their reliabilities and get the more reliable network. In other words, some changes are required to tight their travel time variation and gain desired reliability as a critical reliability.

Discussion
Previous studies showed the importance of travel time reliability. Measuring accurate travel time reliability is the first step to improve it and ensure travelers' on-time arrivals. In this paper, with the use

Based on Number of Links
There is an outstanding limitation to conduct this research work is to assume travel time distribution log normality while in real cases it may be non-lognormal distributed for example bimodal, Weibull or something else. A distribution that can fit standardize travel time is a suitable distribution that can be use in further studies. Nonetheless, in future studies this paper methodology to calculate travel time reliability could be applied to non-lognormal distributed travel time. In this paper Statistical independence between the links travel time is assumed for simplification. In further researches the assumption of correlation of two adjacent links can give more accurate results and analysis.