PETER, a performance evaluator for railway timetables

Transcription

1 PETER, a performance evaluator for railway timetables G. Soto y Koelemeijer*, A.R. lounoussov*, R.M.P. Goverde^ & R.J. van Egmond* ^Faculty ofinformation Technology and Systems, Delft University of Technology, the Netherlands ^Faculty of Civil Engineering and Geosciences, Delft University of Technology, the Netherlands Abstract An important step in the design of a timetable for a railway system is the evaluation of a candidate timetable. This evaluation mainly checks the robustness of the timetable with respect to delays, usualy by means of simulation. Recently the Dutch railways has developped a timetable generator DONS and a simulator SI- MONE for this purpose. However, such simulation studies are time consuming and defining a representative simulation study is not straightforward. An analytical tool for performance evaluation is provided by event graph theory. Event graph theory provides the minimal cycletime, which together with the timetable cycletime gives a rough measure on the robustness. Furthermore, critical circuits which can be seen as the bottleneck of the timetable, can be used as a starting point for a simulation study for a more detailed analysis. This analysis of event graph theory is the basis of PETER: Performance Evaluation Toolbox for Event graphs in Railway systems. 1 Introduction The design of a timetable for a railway system is a highly complex problem because of the large number of constraints. An important step in this direction is the evaluation of a candidate timetable. This evaluation mainly checks the robustness of the timetable with respect to delays, usualy by means of simulation. Once a train is delayed this may cause severe delay propagation. By incorporating recovery times or buffer times one can manage the delays. A robust timetable should

2 406, C.A. Brebbia J.Allan, R.J. Hill, G. Sciutto & S. Sone (Editors) performance measures performance measures: cycle time critical circuit Figure 1: Design of a timetable be able to deal with a certain amount of delay without operational intervention. Recently, the Dutch railways developed a timetable generator, called DONS, and a simulator, called SIMONE, which tests the candidate timetables on robustness. One of the disadvantages of such simulation studies is that is time consuming. An analytical tool for performance evaluation is provided by event graph theory. One of the advantages of this theory that it provides the minimal cycletime. Comparing this with the cycletime of the timetable gives a rough measure on the robustness of the timetable. Furthermore, with this theory the critical circuit, which can be seen as the bottleneck of the system, can easily be determined. This critical circuit can be used as a starting point for a simulation study for a more detailed analysis. This analysis of event graph theory forms the basis forpeter: Performance Evaluation Toolbox for Event graphs in Railway systems. In Figure 1 the relation between DONS, SIMONE and PETER is shown. These kind of analysis is implemented in PETER by using the Howard algorithm (cf. [2]). This algorithm can be used to obtain the minimal cycletime as well as the critical circuit. PETER transforms an input timetable (in DONS format) into an event graph. The paper is organized as follows. In Section 2 we will give some definitions, which will be needed to construct the model. In Section 3 we introduce a timed marked graph. In Section 4 we give a short introduction to the max-plus algebra. In Section 5 we give a theorem concerning the eigenvalue of the matrix. In Section 6 we do some analysis, concerning stability margin and delay propagation. Finally in Section 7 we give conclusions.

3 Computers in Railways Editorial ' rnoname.net Shl_lHl, dep: 4l Rtd_lHl, dep: < Atw_lHl, dep: Rtd_lTl, dep: 5 Shl_lTl, dep: 1 Asdz_lH2, dep; Shl_lH2, dep: 1 Rtd_lH2, dep: : Atw_lH2, dep: Rtd_lT2, dep: 2 Shl_lT2, dep: 4 Gvc_2Hl, dep: 2 Rtd_2Hl, dep: : Bd_2Hl, dep: 5: Tb_2Hl, dep: 3 Ehy 2Hl.dep:^; Figure 2: Marked graph»jci eai*mib«,iiuii «Jia.V VI OldllUI imuuuii IIIIIICIUL ## 111 HI 4 ShL Arrival H1 5 Shl_ Departur H1 9 Rtd_ Arrival H1 10 Rtd_ Departur HI 17 Atw_ Arrival H2 1 Asdz_ Departur H2 4 Shl_ Arrival H2 5 Shl_ Departur H2 9 Rtd_ Arrival H2 10 Rtd_ Departur H2 17 Atw_ Arrival T1 1 Atw_ Departur 2 1 T1 8 Rtd_ i Arrival T1 9 Rtd_ i Departur T1 13 Shl_ Arrival T1 14 Shl_ Departur Alm_ Alm_ Amf_ Amf_ Amf_ Amf_ Amf_ 9H2 19T1 19T2 21 H1 121T1 121T2 122T1 224H1 224H2 16H1 37T1 7H1 7T1 7T1 8H1 8T1 19H J227.I 9T Ahb_ I 9T Ahp_ ! 164H1 500 Ahpr_ T Ahwa_ l 224T Ahz_ : 21T Akl_ HI 8.00 Akm_ H Alm_ H Almb_ 136 H T Almboj 139.: H Almm_130.:i80.! 14T AImp_ 132.H78.; 30T Almpt_129.,* H Arna_ 150X T Amf_ ! _ CQorddJnates File: Figure 3: Dons input file

4 2 Building the model In order to build the model we need a DONS inputfile.an example can be found in Figure 3. The line data file which can be found in Figure 3 contains seven columns. Thefirstcolumn is the line number. A line number can be characterized by a route between two terminal stations and all intermediate served stations. The second column contains information about the direction and about the frequency. To be more precise, an H stands for the direction to (dutch: heen) and T stands for the direction back (dutch: terug). The number in the second column determines the number of the train on that particular line. In Figure 3, we have HI and H2 on line 1, which means that per hour two trains depart from station Asdz to Atw. Column 3 contains the stage or segment which is a part of the line between two stations. The fourth column contains the name of the station. The fifth column states whether the constraint is an arrival or a departure constraint. The sixth column contains the departure times according to the timetable and the last column represents the buffer time, planned on that particular segment. From the timetable and the buffer times, the travel times can be computed. For example, in Figure 3, information is given about line 1. We have already mentioned that the frequency is 2 times an hour. Thefirsttrain (HI) departures at.38 from Asdz to Atw. The second train (H2) departures at.08 from Asdz to Atw. When thefirsttrain arrives at Atw at.38 it turns around and goes back to Asdz at.20. Note that these trains (HI and H2) use the same stages. Each row of the synchronization data file contains a synchronization constraint. It consists of four columns. The first column contains the station at which the synchronization is realized. The second column contains information of the first (feeder) train. The third column contains information of the second (connecting) train. Finally, the fourth column contains the minimal transfer time between the arrival of the feeder train and the departure train of the connecting train. Also a coordinates datafilecan be found in Figure 3. This needed to visualize the marked graph (which will be treated in the next section) in a realistic way. 3 Timed Marked Graph We already mentioned that a railway system is a typical example of a discrete event system. One way to analyze a discrete event system is with the help of Petri net theory. In this section we shortly describe a timed marked graph which is a special kind of Petri net. In [4] an algorithm is given to translate the line data file and the synchronization data file into a timed marked graph. This timed marked graph can be written as a max-plus model. A Petri net is a bipartite directed graph where the nodes are partitioned into two disjoint subsets P and Q, where P is the set of places and Q is the set of transitions. Each place contains a nonnegative number of tokens, the flow of which reflects the dynamic behaviour of the system (for example train movements over a railroad network). A timed Petri net also contains a holding time at each place,

5 Computers in Railways 171 which represents the processing time of an activity associated with the place, for example a train run. Definition 3.1 (Deterministic timed Petri net) A deterministic timed Petri net is a 5-tuple G = {P, Q, E, p,, r}, where L P is afiniteset o/places, \P\ ~ m; 2. Q = {qi,..., qn] is afiniteset of transitions, P fl Q = 0; 3. E is a set of directed arcs connecting the places and transitions; 4. fj, IN is the initial marking of all places; 5. r G R+ is the vector q/" holding times of all places. A marked graph is a Petri net in which each place has exactly one input and one output transition. In Figure 2 a timed marked graph is visualized. The boxes represent transitions, the circles represent places. The different colors of the places in Figure 2 indicate that the places have different numbers of tokens. A circuit in a timed marked graph is a directed sequence of places and transitions over the arcs, where thefirstand last place coincides. A Petri net is strongly connected if any pair of places is contained in a circuit. For further reading the reader is referred to [6]. 4 The Max-Plus Algebra The max-plus algebra can be used to analyze discrete event systems. In this section we give a simple motivation for the use of the max-plus algebra concerning transportation networks where synchronization plays an important role. We also give a short introduction to the max-plus algebra. For further reading the reader is referred to [1]. A motivation of the max-plus algebra can be explained by means of an example. Consider the network in Figure 4. In this figure the boxes represent stations, which are indicated by 1,2 3 and 4. An arc between two stations denotes a segment. Suppose we have three trains, one on the segment 1-4-3, one on the segment 2 > 3 and one on the segment 3 -> 4. The running times plus dwell time of each segment is also given. For example the running time plus dwell time of the segment 1 -> 3 is given by 5. Assume that the train that departs from station 3 can not leave before the trains departing from station 1 and station 2 have arrived at station 3. Let now Xi denote the departure time, departing from station i. Given x\ and X2 we are interested in the value xg. By simple reasoning, if the train which departs from station 1 is the second train that arrives at station 3, then xg > x\ -f- 5. Likewise, if the train that departs from station 2 is the last one to arrive at station 3 we have that x% > x% + 7. This implies that xg = max(xi + 5,X2 + 7). (1)

6 station 1 _ 12 station 3 station 4 station 2 Figure 4: Motivation max-plus algebra In this equation two operators play a role, namely the operator max and the operator +. For simplicity, we write 0 for max and <g> for +, then equation 1 translates into %3 = (%i(g)5)0(%2(2)7). (2) We now give some examples to get familiar with the max-plus algebra notations. Example )3 = = = max(6, 7) = (88 = 40 (7 # 8) = max(4,15) = 15. Note that, like in the conventional algebra, <8> has a higher priority than 0. This was used in Example The neutral element 0 in the conventional algebra translates into oo in the max-plus algebra, and the one in the conventional algebra translates into 0 in the max-plus algebra. Let e denote the max-algebraic -oo and let e denote the max-algebraic 0. Note that if there does not exist an arc between two station we write e for the running time. Example e = 3-i-6 = 3 + -oo = -oo = e 2. 40s = max(4,s) = max(4, oo) = (g)e = 34-e = = e = max(4, e) = max(4,0) = 4. Translating a timed marked graph into a max-plus model yields that we construct matrices, which contain the running times, dwell times and synchronization constraints. The max-plus algebra can also be applied to matrices. Let A, B R"*". Then addition in the max-plus algebra is defined by jj, J%j), Vz, V?.

7 A i i If A 6 E *^ andb W* by then multiplication in the max-plus algebra is defined fc=l for i = 1,...,raand j 1,..., n. fc-l,..-,r A^ # B^) (3) Example 4.3 Let 1 2 \ j 4 / and then (1 8) 8 (2 (8> 5) \ _ / 987 (308)8(405) / \ 1189 ^ " ^11 Let Xj (k) denote the fc-th departure from station j. If for example the k-th train from station j has to wait for the k-th train from station i and the (k - l)-st train from station / we obtain the following equation A more general form is given by - 1). (4) - ^) VA; > 0. (5) In this model the term t denote the number of tokens of a place in the timed marked graph that corresponds to the segment i -» j, which is indicated in the matrix as A^. the expression t G E+ simply means that the number of tokens in the places of the marked graph are nonnegative as we mentioned before. Note that eqn ( 5) can be translated into afirstorder model which can be written as %(&) = A0z(&- 1). (6) 5 The eigenvalue and the eigenvector The eigenvalue of the matrix determines the minimal cycle time and the eigenvector determines the timetable. Let us define A = 0^^+(At)07^. The generalized spectral problem can be written as where v G M^x is the eigenvector (which is not the trivial solution, i.e. the vector containing e only) of the matrix A and A G E is the eigenvalue of the matrix A. If the marked graph is strongly connected, the corresponding polynomial matrix A is said to be irreducible.

8 Theorem 5.1 An irreducible polynonial matrix A for which holds that the graph of the matrix AQ contains no circuits admits a unique eigenvalue. The proof can be found in [1]. It also can be proven that the eigenvalue is equal to the maximum average weight of all circuits in the marked graph. To compute the eigenvector and eigenvalue of the irreducible matrix A we can apply for example the power algorithm [5] or Howard's algorithm [2]. PETER uses Howard's algorithm for computing the eigenvalue and the eigenvector, because of the following advantages: If the network is not irreducible, Howard's algorithm also can be applied. This in contrast to the power algorithm, where in some special cases it gives an answer, but in general it does not work. Howard's algorithm can directly be applied to systems of the form (5). If we want to use the power algorithm we first have to translate it into afirstorder model of the form (6). The amount of space required for Howard's algorithm is less than the amount of space required for the power algorithm. The advantage of the power algorithm is that it is straightforward and easy to implement. The Howard algorithm is an algorithm based on policy iteration. A policy can be interpreted in terms of a marked graph as a subgraph of Q(A) in which every transition has exactly one incoming arc. A timetable and a minimal cycle time for this subsytem is computed for which this subsystem can operate. Policy iteration is applied to compute a better policy. 6 Analysis In this section we shortly describe how we can analyze the model with the help of event graph theory. 6,1 Critical circuit In the previous section we have mentioned that the Howard algorithm is an algorithm that computes the eigenvalue A, which is the minimal cycletime, and eigenvector v which represents a timetable. The eigenvalue A is directly related to the critical circuit, the bottleneck of the system. The critical circuit can be found by investigating the optimal policy of the Howard algorithm. If the eigenvalue A is smaller than the period of the timetable, we call the timetable stable. 6.2 Stability margin If we have for example an hourly pattern, hence the period T of the timetable equals 60, and A = 50 we have a stable timetable, i.e. any delay will disappear after afinitenumber of steps.

9 This brings us to the following definition. Definition 6.1 Stability margin The stability margin 6 of a system is the minimum amount of time that can simultaneously be added to all holding times in the system for which no longer a stable timetable exists with period length T. 6.3 Propagation of delays An important aspect in timetable performance analysis is to investigate how initial delays propagate over the network. Timetable may contain buffertimes between arrival and departure of trains between stations. These buffer times can be used to compensate or even eliminate delays. The following definition is due to van Egmond [3]. Definition 6.2 Consider the max-plus system. The element (R}ij of the recovery matrix R is the maximal delay of Xj(k) for which for all m > k Xi(m) is not delayed. PETER determines the recovery matix by using the following formula. where M(T-i)) = ^"' and A+ = 7 Conclusion In this paper we presented a software tool PETER: Performance Evaluation Toolbox for Event graphs in Railway system. It is based on event graph theory, which can be described by the so-called max-plus algebra. Once the max-plus algebra model has been build several computation can be done, concerning the minimal cycletime, the stability margin as well as the propagation of delays. References [1] Baccelli, F.L., Cohen, G., Olsder, G.J., Quadrat, J.P., Synchronization and Linearity: An Algebra for Discrete Event Systems, Wiley, Chichester, [2] J. Cochet-Terrasson, G. Cohen, S. Gaubert, M. M. Gettrick, J.-P. Quadrat, Numerical computation of spectral elements in max-plus algebra, in : Proc of the IFAC Conference on System Structure and Control, Nantes, July [3] Egmond van, R.J., Propagation of delays in public transport, Reports of the Faculty of Technical Mathematics and Informatics, No , Delft University of Technology, Delft, 1998.

10 A] A [4] G. Soto y Koelemeijer, R.M.R Goverde, Algorithms for performance evaluation of periodic railway timetables, in progress, [5] Subiono, Woude, J.W. van der, Power Algorithms for (max,+) and Bipartite Systems, accepted for publication in Discrete Event Dynamic Systems. [6] Wang, J., Timed Petri Nets: Theory and Applications, Kluwer, Boston, 1998.