Modeling and Simulation of Train Networks using MaxPlus Algebra Hazem Al-Bermanei 1 Jari M. Böling 2 Göran Högnäs 3 1 Faculty of Business ICT and Life Sciences, Turku University of Applied Sciences, Turku, Finland, hazem.al-bermanei@turkuamk.fi . 2 Department of Chemical Engineering, Åbo Akademi University, Turku, Finland, jboling@abo.fi . 3 Department of Mathematics and Statistics, Åbo Akademi University, Turku, Finland Abstract Max-plus algebra provides mathematical methods for solving nonlinear problems that can be given the form of linear problems. Problems of this type, sometimes of an administrative nature, arise in areas such as manufacturing, transportation, allocation of resources, and information processing technology. Train networks can be modelled as a directed graph, in which nodes correspond to arrivals and departures at stations, and arcs to travelling times. A particular difficulty is represented by meeting conditions in a single-track railway system. Compared to earlier work which typically include numerical optimization, max-plus formalism is used throughout this paper. The stability and sensitivity of the timetable is analyzed, and different types of delays and delay behavior are discussed and simulated. Interpretation of the recovery matrix is also done. A simple train network with real world background is used for illustration. Keywords: train schedules, meeting conditions, max- plus algebra, discrete-event systems, delay sensitivity, recovery matrix 1. Introduction The increasingly saturated European railway infrastructure has, among other concerns, drawn attention to the stability of train schedules as they may cause of domino effect delays across the entire network. A train timetable must be insensitive with regard to small disturbances so that recovery from such disturbances can occur without external control. After a break of self-regulation, this behavior schedule requires the distribution of accurate recovery times and buffer times to reduce delays and prevent the propagation of delay, respectively. Schedule models for railways are usually based on deterministic process times (running times, and transfer times). Moreover, running times are rounded and train tracks are modified to fit the schedule or constraints. The validity of these decisions and streamline schedules must be evaluated to ensure the viability and stability and durability, with respect to network mutual relations and differences in process times. Train networks can be modeled using max-plus algebra (D’Ariano et al., 2007). Stability can be evaluated by calculating the eigenvalue of the matrix in max-plus algebra (Baccelli et al., 1992; van den Boom and De Schutter, 2004; van den Boom et al., 2012; Corman et al., 2012). This eigenvalue is the minimum cycle time required to satisfy all of the schedule and progress constraints, where the timetable operating with this eigenvalue time is given by the associated eigenvector (Baccelli et al., 1992; De Schutter and van den Boom, 2008). Thus, if the eigenvalue λ is more than the intended length of the schedule T, then the schedule is unstable. If λ<T the schedule will be stable, and critical if λ=T (van den Boom et al., 2012; Corman et al., 2012). If individual trains are delayed, the effect on the whole network is quite difficult to predict. Smaller delays can typically be absorbed by speeding up the trains, and this can be handled by using max-plus algebra. Larger delays are often handled by rescheduling, typically using optimization, see for example De Schutter et al., (2002); D’Ariano et al., (2007); Corman et al., (2012); and van den Boom and De Schutter, (2004). In this paper we study the impact of both permanent and dynamic delays in a train network, but restrict ourselves to using max-plus algebra, and thus we do not consider rescheduling. So in practice our study is limited to delays up to half of the cycle time. Meeting conditions, including those introduced by having single tracks, are also fully handled using max-plus state- space formalism, by extending the state with delayed states. When constructing a recovery matrix (van den Boom et al., 2012). Of this extended system, it naturally results in redundancy, as the same physical state appears many times. This redundant recovery information can however be incorrect, due to that no constraints are specified for the delayed states, which are only shifted copies of the most recent state. The parts of the recovery matrix corresponding to the most recent states are still valid. EUROSIM 2016 & SIMS 2016 612 DOI: 10.3384/ecp17142612 Proceedings of the 9th EUROSIM & the 57th SIMS September 12th-16th, 2016, Oulu, Finland