Modeling of Railway Networks Using Timed Automata

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1 Applied Mathematical Sciences, Vol. 10, 2016, no. 49, HIKARI Ltd, Modeling of Railway Networks Using Timed Automata Dieky Adzkiya Department of Mathematics Institut Teknologi Sepuluh Nopember, Surabaya 60111, Indonesia Alessandro Abate Department of Computer Science University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK Copyright c 2015 Dieky Adzkiya and Alessandro Abate. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We model a railway network with synchronization feature using timed automata (TA). The procedure consists of two steps: first, each train in the railway network is modeled as a TA; then, the final model is obtained by taking the parallel composition of all TA obtained in the first step. Mathematics Subject Classification: 68Q60, 03B70 Keywords: railway networks, timed automata 1 Introduction Timed automata (TA) [3] are a modeling framework for a wide range of realtime systems, such as in web services [15], audio/video protocols [10], bounded retransmission protocols [7], collision avoidance protocols [1, 13] and commercial field bus protocols [8]. Originally, TA has been defined as a Büchi automaton containing finitely many states extended with real-valued variables modeling clocks. In order to restrict the behavior of the automaton, constraints

2 2430 Dieky Adzkiya and Alessandro Abate on the clock variables are used. Furthermore Büchi accepting conditions are used to enforce progress properties. A simplified version called timed safety automata has been introduced in [12] to enforce progress properties using local invariant conditions. In this work, we use timed safety automata and refer them as TA for simplicity (cf. Definition 2.1). Several modeling frameworks have been used to model railway networks, e.g. Petri nets [9], cellular automata [14], TA [6] and max-plus-linear (MPL) systems [4, 11]. The TA model developed in this work is different with [6]. The authors of [6] discuss a TA model of a railway control system, which controls access to a bridge for several trains. In this work, we develop a TA model of a railway network with synchronization between several trains. This opens up the application of techniques in TA for analyzing railway networks. Let us remark that this work is related with [2], which proposes a formal approach to analyze a railway network modeled as MPL systems. As mentioned in the previous paragraph, we propose to model a railway network with synchronization features using TA. The procedure to build the TA consists of two steps. In the first step, each train in the railway network is modeled as a TA. Then, the TA model of the entire railway network is obtained by taking the parallel composition of all TA obtained in the first step. 2 Preliminaries 2.1 Timed Automata As discussed in the Introduction, a TA can be used to model a wide range of real-time systems. A TA is a directed graph extended with real-valued variables (called clocks) that model the logical clocks. Clock constraints (i.e. guards on edges and location invariants) are used to restrict the behavior of the automaton. A clock constraint is a conjunctive formula of atomic constraints of the form x n or x y n for x, y C, {, <, =, >, } and n N. We use B(C) to denote the set of clock constraints. Moreover we denote the power set of a set C by 2 C. Definition 2.1 A timed automaton TA is a sextuple (L, l 0, Act, C, E, Inv) where L is a set of finitely many locations (or vertices); l 0 L is the initial location; Act is the set of actions; C is a set of finitely many real-valued clocks; E L B(C) Act 2 C L is the set of edges; Inv : L B(C) assigns invariants to locations. The location invariants are restricted to constraints of the form: c n or c < n where c is a clock and n is a natural number. The semantics of a TA are defined as a transition system where a state consists of the current location and the current value of clocks. There are two types of transitions between states: delay for some time (delayed transition) or

3 Modeling of railway networks using timed automata 2431 take an enabled edge (discrete transition). Each edge is labeled with a guard, an action and a reset. The guard is described as a clock constraint ( B(C)). An edge can be taken when the value of clocks satisfies the guard associated with the edge. A subset of all the clocks ( 2 C ) may be reset to zero when a discrete transition is taken. Finally each edge is associated with an action ( Act). The set of actions is used for synchronous communication between a pair of TA. It is done by hand-shake synchronization using input and output actions [5]. The input and output actions are denoted by d i? and d i! for i N, respectively. An action that can be taken independently of other actions is called internal action, denoted by *. When there is a single TA, all actions are internal because synchronous communication is not possible. 2.2 Railway Networks In this paper, we adopt the railway network and its synchronization mechanism discussed in [11, Section 0.1]. The railway network consists of a number of stations connected by several circuits. A circuit can be divided into one or more tracks. Each track contains zero or more trains. Two circuits can be synchronized in a station. In this case, we define the same departure time from the station for all trains in those circuits Figure 1: A simple railway network [11, Figure 0.1]. In order to illustrate the railway networks considered in this paper, we use the example in Figure 1. This railway network has two stations connected by three circuits (left, middle and right). The left and right circuits consist of one track, whereas the middle one consists of two tracks. There is one train placed in each track. The number near each track represents the travel time of the track. In this network, the left and middle circuits are synchronized in station 1, whereas the right and middle circuits are synchronized in station 2.

4 2432 Dieky Adzkiya and Alessandro Abate 3 Construction of the TA Model for the Railway Network In this section, we discuss a procedure to construct a TA model of a given railway network. The procedure consists of two steps. In the first step, each train in the railway network is modeled as a TA. Then, the TA model of the railway network is obtained by taking the parallel composition of all TA (obtained in the first step). Let us discuss the procedure to construct a TA from a train. We denote m as the number of trains. Furthermore in the railway network, there are n stations that are denoted by s 1,..., s n. The time needed using a train to travel from s i to s j is denoted by t i,j for i, j {1,..., n}. Notice that each train is assigned to a circuit. Each circuit is indexed with a unique natural number for identification purposes. Furthermore every circuit is also denoted by a sequence of stations connected by arrows, i.e. s c(1) s c(2) s c(n ) s c(1) where c(1),..., c(n ) {1,..., n}. The synchronization SyncC is formally defined as an n-tuple (SyncC 1,..., SyncC n ) where SyncC i is a set containing two circuit indexes for i {1,..., n}. As an example, the synchronization for the railway network in Figure 1 is SyncC = ({1, 2}, {2, 3}). Definition 3.1 The TA generated by a train assigned to j-th circuit s c(1) s c(2) s c(n ) s c(1) where the synchronization happens in stations SyncS {1,..., n } is given by (L, l 0, Act, C, E, Inv): L = n i=1 {s c(i)} n 1 i=1 {g c(i),c(i+1)} {g c(n ),c(1)}; l 0 = s c(1) ; { {dc(i)!}, if j = max SyncC c(i) Act = { } i SyncS {d c(i)?}, if j = min SyncC c(i) C = {x}; E = { n (gc(i),c(i+1), x = t 1 c(i),c(i+1),, {x}, s c(i+1) ), if i / SyncS i=1 (g c(i),c(i+1), x = t c(i),c(i+1),,, s c(i+1) ), if i SyncS { (gc(n ),c(1), x = t c(n ),c(1),, {x}, s c(1) ), if i / SyncS (g c(n ),c(1), x = t c(n ),c(1),,, s c(1) ), if i SyncS (s c(i), x = 0,,, g c(i),c(i+1) ), if i / SyncS n 1 i=1 (s c(i), x 0, d c(i)!, {x}, g c(i),c(i+1) ), if i SyncS and j = max SyncC c(i) (s c(i), x 0, d c(i)?, {x}, g c(i),c(i+1) ), if i SyncS and j = min SyncC c(i)

5 Modeling of railway networks using timed automata 2433 (s c(n ), x = 0,,, g c(n ),c(1)), if n / SyncS (s c(n ), x 0, d c(n )!, {x}, g c(n ),c(1)), if n SyncS and j = max SyncC c(n ) (s c(n ), x 0, d c(n )?, {x}, g c(n ),c(1)), if n SyncS and j = min SyncC c(n ) Inv(s c(i) ) = {x = 0}, if i / SyncS Let us describe the TA generated by Definition 3.1 in more detail. Location s c(i) is active when the train stops at c(i)-th station. Location g c(i),c(i+1) is active when the train is on the way from c(i)-th station to c(i + 1)-th station. Initially, the train stops in c(1)-th station. The action d i corresponds to the departure synchronization in the i-th station, for all i. All actions are defined as urgent, i.e. it will be taken as soon as it is enabled. The clock variable x represents the time elapsed since the last departure. The core of the definition is the construction of the set of edges E. The construction of E can be divided into two parts: with and without synchronization. First we focus on the edges with synchronization. In this case, the clock variable is reset to zero when the train departs from a station. The output actions are associated with the train assigned to higher circuit index, whereas the input actions are associated with the train assigned to lower circuit index. Notice that both trains depart as soon as those two trains have arrived in the station. Finally when the value of the clock variable equals the travel time of the corresponding track, the train enters the destination. Next we focus on the edges without synchronization. In this case, the clock variable is reset to zero when the train enters a station. After that, the train departs immediately. Let us construct the TA associated with the railway network in Figure 1. First we associate a unique number to each circuit: the left, middle and right circuits correspond to 1, 2 and 3, respectively. Then we associate a unique number to each train: the left, bottom, top and right trains correspond to 1, 2, 3 and 4, respectively. The TA generated by each train is depicted in Figure 2. The locations and edges are represented by circles and arrows, respectively. The initial location is represented by double circles. The name and invariant of each location is located above and below the circle, respectively. The guard, action and reset are located near the corresponding arrow. Notice that initially, the first and third trains depart from station 1, whereas the second and fourth trains depart from station 2. 4 Conclusions and Future Work In this paper, we have modeled a railway network as a TA. The TA model can be used to verify some properties of the railway network. For example, we

6 2434 Dieky Adzkiya and Alessandro Abate g 1,1 x == 2 s 1 g 2,2 x == 3 s 2 x 2 d 1? x 3 d 2! s 2 s 2 x == 3 d 2? x = 0 x == 3 d 2? x = 0 g 1,2 g 2,1 g 1,2 g 2,1 x 3 x 5 x 3 x 5 d 1! x = 0 s 1 x == 5 d 1! x = 0 s 1 x == 5 Figure 2: The TA in the top-left and top-right parts is associated with the first and fourth trains, respectively. The TA in the bottom-left and bottom-right parts is associated with the second and third trains, respectively. can check whether the delay between two consecutive departures of a train is upper bounded by a given constant. We leave it as a future work. References [1] L. Aceto, A. Burgueño and K.G. Larsen, Model checking via reachability testing for timed automata, Chapter in Tools and Algorithms for the Construction and Analysis of Systems (TACAS 98), B. Steffen, editor, volume 1384 of Lecture Notes in Computer Science, Springer, Heidelberg, 1998, [2] D. Adzkiya, B. De Schutter and A. Abate, Finite abstractions of maxplus-linear systems, IEEE Transactions on Automatic Control, 58 (2013), no. 12, [3] R. Alur and D.L. Dill, A theory of timed automata, Theoretical Computer Science, 126 (1994), no. 2,

7 Modeling of railway networks using timed automata 2435 [4] F. Baccelli, G. Cohen, G.J. Olsder and J.-P. Quadrat, Synchronization and Linearity: An Algebra for Discrete Event Systems, John Wiley and Sons, [5] C. Baier and J.-P. Katoen, Principles of Model Checking, The MIT Press, [6] G. Behrmann, A. David and K.G. Larsen, A tutorial on uppaal, Chapter in Formal Methods for the Design of Real-Time Systems (SFM-RT 04), M. Bernardo and F. Corradini, editors, volume 3185 of Lecture Notes in Computer Science, Springer, Heidelberg, 2004, [7] P.R. D Argenio, J.-P. Katoen, T.C. Ruys and J. Tretmans, The bounded retransmission protocol must be on time!, Chapter in Tools and Algorithms for the Construction and Analysis of Systems (TACAS 97), E. Brinksma, editor, volume 1217 of Lecture Notes in Computer Science, Springer, Heidelberg, 1997, [8] A. David and W. Yi, Modelling and analysis of a commercial field bus protocol, In 12th Euromicro Conference on Real-Time Systems, (2000), [9] A. Giua and C. Seatzu, Modeling and supervisory control of railway networks using Petri nets, IEEE Transactions on Automation Science and Engineering, 5 (2008), no. 3, [10] K. Havelund, A. Skou, K.G. Larsen and K. Lund, Formal modeling and analysis of an audio/video protocol: an industrial case study using UPPAAL, Proceedings of the 18th IEEE Real-Time Systems Symposium (RTSS 97), (1997), [11] B. Heidergott, G.J. Olsder and J.W. van der Woude, Max Plus at Work Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications, Princeton University Press, [12] T.A. Henzinger, X. Nicollin, J. Sifakis and S. Yovine, Symbolic model checking for real-time systems, Information and Computation, 111 (1994), no. 2, [13] H.E. Jensen, K.G. Larsen and A. Skou, Modelling and analysis of a collision avoidance protocol using SPIN and UPPAAL, BRICS Report Series, 3 (1996), no

8 2436 Dieky Adzkiya and Alessandro Abate [14] K.-P. Li, Z.-Y. Gao and B. Ning, Modeling the railway traffic using cellular automata model, International Journal of Modern Physics C, 16 (2005), no. 06, [15] A.P. Ravn, J. Srba, and S. Vighio, Modelling and verification of web services business activity protocol, Chapter in Tools and Algorithms for the Construction and Analysis of Systems (TACAS 11), P.A. Abdulla and K.R.M. Leino, editors, volume 6605 of Lecture Notes in Computer Science, Springer, Heidelberg, 2011, Received: November 15, 2015; Published: July 25, 2016

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