Assessing Practical Railway Network Capacity using Petri Nets
C. Szymulaa,∗ , N. Besinovicb and K. Nachtigallaa Technische Universität Dresden, Dresden, Germanychristopher.szymula@tu-dresden.deb Delft University of Technology, Delft, The Netherlands∗ Corresponding author

Extended abstract submitted for presentation at the 11th Triennial Symposium on
Transportation Analysis conference (TRISTAN XI)
June 19-25, 2022, Mauritius Island

March 31, 2022
Keywords: Petri Nets, Railway Capacity, Railway Networks, Macroscopic Fundamental Diagram

1INTRODUCTION

Railway capacity represents one of the main characteristics of railway networks. It is importantfor understanding system limitations, current pricing and future investment decisions of theinfrastructure manager, and consequently is an important factor for the current and future
performance of this transport system. Railway capacity is typically measured by the absolute
number of trains over a given time interval or the corresponding occupation times. It is impacted
by factors such as the number of trains, the average train speed, the heterogeneity of train traffic
and the timetable-induced time margins (s. UIC (2004)). Additionally, in complex networks
we observe additional spatio-temporal dependencies due to heterogeneous rolling stock fleets,train services (routes, paths and frequencies) and service-related constraints (transfers and trainorders).
Railway network capacity (RNC) can be defined as the maximum number of trains regarding
the given service characteristics (i.e. timetable) and infrastructure. It is measured by the numberof overall departing trains per time interval, regarding the resulting level-of-service (LOS) (i.e.
average scheduled waiting time (SWT) per train). This is also known as practical RNC (s. Abrilet al. (2008)). RNC is a recent research topic (s. Landex (2008), Jensen (2015), Rotoli et al.
(2016), N. Besinovic (2017), D’Acierno et al. (2019), Kianinejadoshah & Ricci (2020) or Weiket al. (2020) as examples). However, the practical capacity i.e. determining maximal number oftrains including the resulting LOS, has hardly been addressed in the network context.
This work aims for filling this gap by providing a new petri net (PN) approach for assessing theRNC, building upon the work of Burkolter (2005) and Farhi et al. (2017). Our main contributions
are extending the existing PN approach from single railway lines to networks, adding an extension
for assessing heterogeneous service frequencies, the identification of relevant capacity limiting (i.e.
critical) processes and by providing a framework for RNC assessment.

2METHODOLOGY

For assessing the practical RNC, a PN approach is developed. PNs are networks of places andtransitions, which are a frequently used tool for modelling and visualization of complex processes(s. Petri (1962)). Being already well researched, PNs provide plenty of theoretical features, whichhave already been used for capacity assessment in the railway context (i.e. Olsder & Subiono
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Original abstract submittal

C. Szymula, N. Besinovic and K. Nachtigall2(1998) or Burkolter (2005)). Among the first, Farhi et al. (2017) applied the PN paradigm formodelling the traffic dynamics on a single metro line. The extensions to this approach, allowing
for network wide capacity assessment, are shown in the following subsection.

2.1

Railway Network Occupation Model

The model takes as input and fixed: the infrastructure network, the service/line network, thelocal routing in stations and the line order/hierarchy. Furthermore, homogeneous rolling stock
characteristics and driving dynamics are also assumed.
The modelling process starts by individually modelling the lines of the given service network
as p-timed PNs, where the traversed sequence of infrastructure elements is captured. Assuming
fixed-block operations, the lines are modelled with the granularity of individual blocks, each
represented by two corresponding PN places, a running and a headway one. The associatedtimings represent the minimal running- and headway times. Introducing pairwise connection arcs,
the networks representation is then obtained by connecting the line representations at sectionsof shared infrastructure (s. fig. 1b). Trains are modelled by putting token on the running-timeplace of a block. Heterogeneous numbers of trains per line are incorporated by using multiple
token per train. The number of token per train for each line are thus set according to the least
common multiple (lcm) over all numbers of trains per line in the network. The obtained networkmodel is referred to as the railway network occupation model (RNOM). The relevant criticalcircuits of the RNOM are then identified by assessing the actual occupation (i.e. token) of thenetwork. The headway (i.e. cycle mean) of the most limiting critical circuit (i.e. the one withthe lowest headway) provides the capacity-limiting bound of network performance.

L1L2(a) Railway Network Topology(b) PN Graph Representation of theTopology (p-timings omitted for clarity)

Figure 1 – Example network of two symmetric lines L1 and L2
Figure 1 provides an example of a railway network. Two lines, L1 and L2, consisting of 4
blocks each, are connected by two connection arcs (dashed arrows in fig. 1b) at one common
block section (blue section in fig. 1a). In the PN, a train is represented by an occupied placeon the inner loops. Empty blocks are indicated by the corresponding occupation of places on
the outer loops. The differing numbers of trains per line and the resulting heterogeneous service
frequencies are represented by multiple token. An exemplary critical circuit is indicated in yellow.

2.2

Railway Network Capacity Framework

The RNC framework is proposed for determining the capacity of the network. It applies theRNOM for assessing different occupation scenarios. A scenario consists of a certain number oftrains per line, including their distribution in the network. Since capacity equals the maximum
possible occupation, the scenario providing the highest headway on its relevant critical circuit iswanted. It is identified by enumerating over all possible scenarios between 1 train per line andTRISTAN XI Symposium

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C. Szymula, N. Besinovic and K. Nachtigall

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the maximum number of trains in the network (at most 1 train per block). The framework ispresented in fig. 2.

Figure 2 – Railway network capacity framework
In step 1, the RNOM is initialized as described in section 2.1. In step 2, the occupation
scenarios to be assessed are generated. Ensuring to find the scenario representing the capacity (i.e.
maximum occupation) of the network, one scenario is made for each combination of numbers oftrains per line. The RNOM is then applied by setting the tokens according to the current scenario(step 3), identifying the relevant critical circuits (step 4) and calculating the corresponding
occupation, relying on the headway of the identified circuit (step 5). Enumerating over all
scenarios, the capacity is found by identifying the scenario with the maximum occupation (step6). The frameworks provides the obtained maximum capacity and the corresponding occupationand LOS, visualized in a combined macroscopic fundamental diagram (MFD) (s. fig. 4).

3

RESULTS AND DISCUSSION

The presented RNC framework is used for assessing the capacity of an example network presented
in fig. 3. The network consists of fixed-block infrastructure with homogeneous running andheadway times (s. fig. 3a). A service network of three lines is operated, overall sharing 4 blocks
in different parts of the network (s. fig. 3b).

(a) Infrastructure of the Example Network(incl. block-markings and directions)(b) Lineplan of the Example Network

Figure 3 – Example network with 3 lines (red, green and yellow) and 4 shared blocks
The results of the capacity assessment are shown in an MFD (s.fig. 4), presenting the absolute number of trains (representing the actual occupation) and the resulting maximum train
flows of all evaluated scenarios. Additionally, the avg. SWTs per train and scenario are given,
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representing the corresponding LOS. The RNC framework determines a network capacity of 10
trains/h for a range between 6 and 15 trains in total, proving the maximum flow is obtainable
for multiple occupation scenarios. However, the most efficient maximum flow (i.e. with the least
number of trains and the lowest avg. SWTs) can be reached for 6 trains only. As the maximum
flow represents the aggregated line flows, it can be observed on the shared infrastructure section
only.

Figure 4 – Combined MFD for the example network with the 3 traffic phases
The MFD shows the existence of three distinct traffic phases: the phases of (1) free-flow, (2)maximum flow and (3) congested flow. The first phase is characterized by a steadily increasing
flow with an increasing number of trains, since the network is empty and no train interactions
cause waiting. Thus, the avg. SWTs are close to zero. Phase (2) is characterized by an almost
constant1 train flow for increasing numbers of trains. Caused by starting train interactions,increasing numbers of trains do not lead to increasing flow values but linearly increasing avg.
SWTs. Thus, the flow-increasing potential of additional trains is getting compensated by theresulting increasing train interactions and the resulting hindrances. Phase (3) shows decreasingflows with increasing train numbers, caused by the saturation of the network. Thus, the occurring
train interactions cause hyperbolic increasing SWTs and thus lead to a strong flow decrease.
The experimental results show that the traffic phases of the well known fundamental diagram (FD) of traffic flow also emerge in complex railway networks. Furthermore, the results
highlight the relations between traffic flow and avg. SWTs, where each phase corresponds to acharacteristic behaviour of the SWT. Also, the shape of the SWT is in line with findings of othercapacitated systems i.e. queuing systems. Finally, the results show that the maximum flow canbe reached with different occupation scenarios, whereas the most efficient maximum flow can be
only reached for specific ones.
The obtained results provide an upper bound for the RNC. Testing the framework on different
networks and real-life instances is expected for future research. Also, for practical application,
future work should incorporate the effects of variable train orders and heterogeneous rolling stock
characteristics. Capturing additional characteristic railway features, the proposed framework
1The saw-tooth-like pattern emerges from the discrete nature of trains and resulting combinations of numbers oftrains (or frequencies) per line respectively. Since the waiting times become minimal for homogeneous frequencies
only, the corresponding distinct maximum points occur.

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could then be used for supporting strategic, capacity efficient planning and thus might support
the recent ambitions for efficient and eco-friendly transportation.

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