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Stochastic modeling and analysis of road–tramway intersections


In the last decades, the socio-demographic evolution of the population has substantially changed mobility demand, posing new challenges in minimizing urban congestion and reducing environmental impact. In this scenario, understanding how different modes of transport can efficiently share (partially or totally) a common infrastructure is crucial for urban development. To this aim, we present a stochastic model-based analysis of critical intersections shared by tram traffic and private traffic, combining a microscopic model of the former with a macroscopic model of the latter. Advanced simulation tools are typically used for such kind of analyses, by playing various traffic scenarios. However, simulation is not an exhaustive approach, and some critical, possibly rare, event may be ignored. For this reason, our aim is instead to adopt suitable analytical solution techniques and tools that can support instead a complete, exhaustive analysis, so being able to take into account rare events as well. Transient analysis of the overall traffic model using the method of stochastic state classes is adopted to support the evaluation of relevant performance measures, namely the probability of traffic congestion over time and the average number of private vehicles in the queue over time. A sensitivity analysis is performed with respect to multiple parameters, notably including the arrival rate of private vehicles, the frequency of tram rides, and the time needed to recover from traffic congestion.

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  1. 1.

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    By traffic congestion we identify the situation when the number of vehicles arriving is higher than the number of vehicles that can flow across the intersection.


  1. 1.

    ACEA (2016) The 2030 urban mobility challenge. Technical report, European Automobile Manufacturers Association

  2. 2.

    Agarwal A, Lämmel G (2016) Modeling seepage behavior of smaller vehicles in mixed traffic conditions using an agent based simulation. Transp Dev Econ 2(2):12

    Article  Google Scholar 

  3. 3.

    Albrecht AR, Howlett PG, Pudney PJ, Vu X (2013) Energy-efficient train control: from local convexity to global optimization and uniqueness. Automatica 49(10):3072–3078

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    Bernardi S, Campos J, Merseguer J (2011) Timing-failure risk assessment of UML design using Time Petri Net bound techniques. IEEE Trans Ind Inform 7(1):90–104

    Article  Google Scholar 

  5. 5.

    Biagi M, Carnevali L, Paolieri M, Vicario E (2017) Performability evaluation of the ERTMS/ETCS-level 3. Transp Res Part C Emerg Technol 82:314–336

    MATH  Article  Google Scholar 

  6. 6.

    Carnevali L, Grassi L, Vicario E (2009) State-density functions over DBM domains in the analysis of non-Markovian models. IEEE Trans Softw Eng 35(2):178–194

    Article  Google Scholar 

  7. 7.

    Carnevali L, Flammini F, Paolieri M, Vicario E (2015) Non-Markovian performability evaluation of ERTMS/ETCS level 3. In: Lecture notes in computer science 9272, EPEW 2015. Springer, pp 47–62

  8. 8.

    Carnevali L, Fantechi A, Gori G, Vicario E (2018) Analysis of a road/tramway intersection by the ORIS tool. In: International conference on verification and evaluation of computer and communication systems. Springer, pp 185–199

  9. 9.

    Charypar D, Axhausen K, Nagel K (2007) Event-driven queue-based traffic flow microsimulation. Transport Res Res 2003:35–40

    Article  Google Scholar 

  10. 10.

    Choi H, Kulkarni VG, Trivedi KS (1994) Markov regenerative stochastic Petri nets. Perform Eval 20(1–3):333–357

    MathSciNet  Google Scholar 

  11. 11.

    Dobler C, Lämmel G (2013) Integration of a multi-modal simulation module into a framework for large-scale transport systems simulation. In: Pedestrian and evacuation dynamics 2012. Springer, pp 739–754

  12. 12.

    ERTRAC (2009) ERTRAC road transport scenario 2030+ “road to implementation”. Technical report, European Road Transport Research Advisory Council

  13. 13.

    Fujii H, Uchida H, Yoshimura S (2017) Agent-based simulation framework for mixed traffic of cars, pedestrians and trams. Transp Res C Emerg Technol 85:234–248

    Article  Google Scholar 

  14. 14.

    Galpin V, Zon N, Wilsdorf P, Gilmore S (2018) Mesoscopic modelling of pedestrian movement using C arma and its tools. ACM Trans Model Comput Simul 28:11

    Article  Google Scholar 

  15. 15.

    Gawron C (1998) An iterative algorithm to determine the dynamic user equilibrium in a traffic simulation model. Int J Mod Phys C 9(3):393–407

    Article  Google Scholar 

  16. 16.

    Ghazel M (2009) Using stochastic Petri nets for level-crossing collision risk assessment. IEEE Trans Intell Transp Syst 10(4):668–677

    Article  Google Scholar 

  17. 17.

    Ghazel M, El-Koursi E (2014) Two-half-barrier level crossings versus four-half-barrier level crossings: a comparative risk analysis study. IEEE Trans Intell Transp Syst 15(3):1123–1133

    Article  Google Scholar 

  18. 18.

    González-Gil A, Palacin R, Batty P, Powell J (2014) A systems approach to reduce urban rail energy consumption. Energy Convers Manag 80:509–524

    Article  Google Scholar 

  19. 19.

    Higgins A, Kozan E, Ferreira L (1996) Optimal scheduling of trains on a single line track. Transp Res B Methodol 30(2):147–161

    Article  Google Scholar 

  20. 20.

    Horváth A, Paolieri M, Ridi L, Vicario E (2012) Transient analysis of non-Markovian models using stochastic state classes. Perform Eval 69(7–8):315–335

    Article  Google Scholar 

  21. 21.

    Huang Y, Weng Y, Zhou M (2010) Critical scenarios and their identification in parallel railroad level crossing traffic control systems. IEEE Trans Intell Transp Syst 11(4):968–977

    Article  Google Scholar 

  22. 22.

    Kerner BS, Klenov SL, Wolf DE (2002) Cellular automata approach to three-phase traffic theory. J Phys A Math Gen 35(47):9971–10013

    MATH  MathSciNet  Article  Google Scholar 

  23. 23.

    Krajzewicz D, Hertkorn G, Rössel C, Wagner P (2002) SUMO (Simulation of Urban MObility)-an open-source traffic simulation. In: 4th middle east symposium on simulation and modelling, pp 183–187

  24. 24.

    Krajzewicz D, Erdmann J, Behrisch M, Bieker L (2012) Recent development and applications of SUMO-Simulation of Urban MObility. Int J Adv Syst Meas 5(3&4):128–138

    Google Scholar 

  25. 25.

    Krajzewicz D, Erdmann J, Härri J, Spyropoulos T (2014) Including pedestrian and bicycle traffic into the traffic simulation SUMO. In: ITS 2014, 10th ITS European congress, 16–19 June 2014, Helsinki, Finland

  26. 26.

    Li X, Yang X (2013) A stochastic timetable optimization model in subway systems. Int J Uncertain Fuzziness Knowl Based Syst 21(supp01):1–15

    MATH  MathSciNet  Article  Google Scholar 

  27. 27.

    Mubasher MM, ul Qounain JSW (2015) Systematic literature review of vehicular traffic flow simulators. In: 2015 international conference on open source software computing (OSSCOM), pp 1–6

  28. 28.

    Ondráček J, Schwarz J, Ždímal V, Andělová L, Vodička P, Bízek V, Tsai CJ, Chen SC, Smolík J (2011) Contribution of the road traffic to air pollution in the Prague city (busy speedway and suburban crossroads). Atmos Environ 45(29):5090–5100

    Article  Google Scholar 

  29. 29.

    Paolieri M, Biagi M, Carnevali L, Vicario E (2019) The ORIS tool: quantitative evaluation of non-Markovian systems. IEEE Trans Softw Eng.

  30. 30.

    Peng G, Cai X, Liu C, Cao B, Tuo M (2011) Optimal velocity difference model for a car-following theory. Phys Lett A 375(45):3973–3977

    MATH  Article  Google Scholar 

  31. 31.

    PTV GROUP (2011) Ptv vissim

  32. 32.

    Shi J, Sun Y, Schonfeld P, Qi J (2017) Joint optimization of tram timetables and signal timing adjustments at intersections. Transp Res C Emerg Technol 83:104–119

    Article  Google Scholar 

  33. 33.

    Tang T, Wang Y, Yang X, Wu Y (2012) A new car-following model accounting for varying road condition. Nonlinear Dyn 70(2):1397–1405

    MathSciNet  Article  Google Scholar 

  34. 34.

    Taplin M (2019) New tramways for 2019. Tramw Urban Transit 975:89–94

    Google Scholar 

  35. 35.

    Tonguz OK, Viriyasitavat W, Bai F (2009) Modeling urban traffic: a cellular automata approach. IEEE Commun Mag 47(5):142–150

    Article  Google Scholar 

  36. 36.

    Transportation Research Board of the National Academies (2015) Traffic and transportation simulation. Looking back and looking ahead: celebrating 50 years of traffic flow theory. A workshop. Technical report E-C195, Washington DC

  37. 37.

    Ullrich O, Franz S, Speckenmeyer E, Lückerath D (2012) Simulation and optimization of Cologne’s tram schedule. Simul Notes Europe (SNE) 22:69–76

    Google Scholar 

  38. 38.

    Ullrich O, Lückerath D, Speckenmeyer E (2015) A robust schedule for Montpellier’s Tramway network. Simul Notes Europe (SNE) 25:1–8

    Article  Google Scholar 

  39. 39.

    Vicario E, Sassoli L, Carnevali L (2009) Using stochastic state classes in quantitative evaluation of dense-time reactive systems. IEEE Trans Softw Eng 35:703–719

    Article  Google Scholar 

  40. 40.

    Yang J, Deng W, Wang J, Li Q, Wang Z (2006) Modeling pedestrians’ road crossing behavior in traffic system micro-simulation in China. Transp Res A Policy 40(3):280–290

    Article  Google Scholar 

  41. 41.

    Yoshimura S (2006) MATES: multi-agent based traffic and environmental simulator-theory, implementation and practical application. Comput Model Eng Sci 11(1):17–25

    Google Scholar 

  42. 42.

    Zeng W, Chen P, Nakamura H, Iryo-Asano M (2014) Application of social force model to pedestrian behavior analysis at signalized crosswalk. Transp Res C Emerg Technol 40:143–159

    Article  Google Scholar 

  43. 43.

    Zheng LJ, Tian C, Sun DH, Liu WN (2012) A new car-following model with consideration of anticipation driving behavior. Nonlinear Dyn 70(2):1205–1211

    MathSciNet  Article  Google Scholar 

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This work has been partially funded by the Fondazione Cassa di Risparmio di Firenze (Grant No. 2014.0771). We thank GEST for disclosing data about actual tram operation in Florence.

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Correspondence to Alessandro Fantechi.

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Carnevali, L., Fantechi, A., Gori, G. et al. Stochastic modeling and analysis of road–tramway intersections. Innovations Syst Softw Eng 16, 215–230 (2020).

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  • Stochastic modeling of traffic flows
  • Markov regenerative processes
  • Regenerative transient analysis
  • Stochastic Time Petri Nets
  • ORIS tool