# A Traffic Breakdown Model Based on Queueing Theory

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## Abstract

In this paper, we propose a queueing model to describe traffic breakdown phenomena caused by perturbations of on-ramp merging vehicles. In congested mainline traffic flow, we assume that a merging vehicle will trigger a jam queue formulation. If this jam queue cannot dissipate before the next vehicle merges into the main road, it could grow into a wide jam and eventually result in traffic breakdown. Different from many existing models that focused on the propagation of jam waves, the proposed model emphasizes the size evolution of a jam queue (local congested vehicle cluster) instead of its spatial evolutions. This new approach reduces analysis difficulties and allows us to directly link microscopic driving behaviors with macroscopic breakdown phenomena. Test results show that the simulated breakdown probability curve (parameter tuning using Next Generation Simulation vehicular trajectories) fits with the empirical breakdown probability curve that is estimated by Performance Measurement System (PeMS) data. This indicates that this new model can account for the probability of breakdown phenomena and help us set an appropriate inflow rate so as to void breakdown and meanwhile maintain a high road capacity.

## Keywords

Traffic breakdown Queueing theory Newell’s simplified model Headway## Notes

### Acknowledgments

This work was supported in part by National Basic Research Program of China (973 Project) 2012CB725405, The National Science and Technology Support Program 2013BAG18B00, National Natural Science Foundation of China 51278280, 51138003.

## References

- Ahn S, Cassidy MJ, Laval J (2004) Verification of a simplified car-following theory. Transp Res B Methodol 38:431–440CrossRefGoogle Scholar
- Banks JH (1991) Two-capacity phenomenon at freeway bottlenecks: A basis for ramp metering? Transp Res Rec 1320:83–90Google Scholar
- Banks JH (2006) New approach to bottleneck capacity analysis: Final report. California PATH Research Report, UCB-ITS-PRR-2006-13Google Scholar
- Bassan S, Polus A, Faghri A (2006) Time-dependent analysis of density fluctuations and breakdown thresholds on congested freeways. Transp Res Rec 1965:40–47CrossRefGoogle Scholar
- Brilon W, Geistefeldt J, Regler M (2005) Reliability of freeway traffic flow: A stochastic concept of capacity. In: Proceedings of the 16th International Symposium of Transportation and Traffic Theory, 125–144Google Scholar
- Chen X, Li L, Jiang R, Yang X (2010a) On the intrinsic concordance between the wide scattering feature of synchronized flow and the empirical spacing distributions. Chin Phys Lett 27:074501CrossRefGoogle Scholar
- Chen X, Li L, Zhang Y (2010b) A Markov model for headway/gap distribution of road traffic. IEEE Trans Intell Transp Syst 11:773–785CrossRefGoogle Scholar
- Chen X, Li Z, Li L (2012) Phase diagram analysis based on a temporal-spatial queueing model. IEEE Trans Intell Transp Syst 13:1705–1716CrossRefGoogle Scholar
- Chen X, Li Z, Li L, Shi Q (2013) Characterising scattering features in flow–density plots using a stochastic platoon model. Transportmetrica Transp Sci. doi: 10.1080/23249935.2013.822941 Google Scholar
- Chiabaut N, Buisson C, Leclercq L (2009) Fundamental diagram estimation through passing rate measurements in congestion. IEEE Trans Intell Transp Syst 10:355–359CrossRefGoogle Scholar
- Chiabaut N, Leclercq L, Buisson C (2010) From heterogeneous drivers to macroscopic patterns in congestion. Transp Res B Methodol 44:299–308CrossRefGoogle Scholar
- Coifman B, Kim S (2011) Extended bottlenecks, the fundamental relationship, and capacity drop on freeways. Transp Res A Policy Pract 45:980–991CrossRefGoogle Scholar
- Daganzo CF (2001) A simple traffic analysis procedure. Netw Spat Econ 1:77–101CrossRefGoogle Scholar
- Dong J, Mahmassani HS (2012) Stochastic modeling of traffic flow breakdown phenomenon: application to predicting travel time reliability. IEEE Trans Intell Transp Syst 13:1803–1809CrossRefGoogle Scholar
- Duret A, Buisson C, Chiabaut N (2008) Estimating individual speed-spacing relationship and assessing ability of Newell’s car-following model to reproduce trajectories. Transp Res Rec 2088:188–197CrossRefGoogle Scholar
- Elefteriadou L, Roess R, McShane W (1995) Probabilistic nature of breakdown at freeway merge junctions. Transp Res Rec 1484:80–89Google Scholar
- Elefteriadou L, Kondyli A, Washburn S, Brilon W, Lohoff J, Jacobson L, Persaud B (2011) Proactive ramp management under the threat of freeway-flow breakdown. Procedia Soc Behavioral Sci 16:4–14CrossRefGoogle Scholar
- Geistefeldt J, Brilon W (2009) A comparative assessment of stochastic capacity estimation methods. In: Proceedings of the 18th International Symposium on Traffic and Transportation Theory, (Lam W, Wong SC, Lo HK, Eds.) Springer, 583–602Google Scholar
- Habib-Mattar C, Polus A, Cohen MA (2009) Analysis of the breakdown process on congested freeways. Transp Res Rec 2124:58–66CrossRefGoogle Scholar
- Helbing D, Treiber M, Kesting A, Schönhof M (2009) Theoretical vs. empirical classification and prediction of congested traffic states. Eur Phys J B 69(4):583–598Google Scholar
- Jin X, Zhang Y, Wang F, Li L, Yao D, Su Y, Wei Z (2009) Departure headways at signalized intersections: A log-normal distribution model approach. Transp Res C: Emerg Technol 17(3):318–327Google Scholar
- Jost D, Nagel K (2003) Probabilistic traffic flow breakdown in stochastic car following models. Traffic and Granular Flow ’03, 87–103Google Scholar
- Kerner BS (2001) Complexity of synchronized flow and related problems for basic assumptions of traffic flow theories. Netw Spat Econ 1:35–76CrossRefGoogle Scholar
- Kerner BS, Klenov SL (2006) Probabilistic breakdown phenomenon at on-ramp bottlenecks in three-phase traffic theory: Congestion nucleation in spatially non-homogeneous traffic. Phys A 364:473–492CrossRefGoogle Scholar
- Kesting A, Treiber M, Helbing D (2010) Enhanced intelligent driver model to access the impact of driving strategies on traffic capacity. Phil Trans R Soc A 368:4585–4605CrossRefGoogle Scholar
- Kim T, Zhang HM (2008) A stochastic wave propagation model. Transp Res B Methodol 42:619–634CrossRefGoogle Scholar
- Kondyli A, Elefteriadou L (2011) Modeling driver behavior at freeway-ramp merges. Transp Res Rec 2249:29–37CrossRefGoogle Scholar
- Kühne R, Lüdtke A (2012) Traffic breakdowns and freeway capacity as extreme value statistics. Transp Res C Emerg Technol 27:159–168CrossRefGoogle Scholar
- Kühne R, Mahnke R, Lubashevsky I, Kaupužs J (2002) Probabilistic description of traffic breakdowns. Phys Rev E 65:066125CrossRefGoogle Scholar
- Laval JA (2011) Hysteresis in traffic flow revisited: an improved measurement method. Transp Res B Methodol 45:385–391CrossRefGoogle Scholar
- Laval JA, Leclercq L (2010) A mechanism to describe the formation and propagation of stop-and-go waves in congested freeway traffic. Phil Trans R Soc A 368:4519–4541CrossRefGoogle Scholar
- Liu HX, Wu X (2010) Stochasticity of freeway operational capacity and chance-constrained ramp metering. Transp Res C Emerg Technol 18:741–756CrossRefGoogle Scholar
- Lorenz M, Elefteriadou L (2001) Defining freeway capacity as a function of breakdown probability. Transp Res Rec 1776:43–51CrossRefGoogle Scholar
- Mahnke R, Kaupužs J (2001) Probabilistic description of traffic flow. Netw Spat Econ 1:103–136CrossRefGoogle Scholar
- Newell GF (2002) A simplified car-following theory: a lower order model. Transp Res B Methodol 36:195–205CrossRefGoogle Scholar
- NGSIM (2006) Next generation simulation. http://ngsim.fhwa.dot.gov/
- PeMS (2005) California performance measurement system. http://pems.eecs.berkeley.edu
- Persaud B, Yagar S, Tsui D, Look H (2001) Breakdown-related capacity for freeway with ramp metering. Transp Res Rec 1748:110–115CrossRefGoogle Scholar
- Polus A, Pollatschek M (2002) Stochastic nature of freeway capacity and its estimation. Can J Civ Eng 29:842–852CrossRefGoogle Scholar
- Schönhof M, Helbing D (2007) Empirical features of congested traffic states and their implications for traffic modeling. Transp Sci 41(2):135–166Google Scholar
- Shawky M, Nakamura H (2007) Characteristics of breakdown phenomenon in merging sections of urban expressways in Japan. Transp Res Rec 2012:11–19CrossRefGoogle Scholar
- Smilowitz KR, Daganzo CF (2002) Reproducible features of congested highway traffic. Math Comput Model 35:509–515CrossRefGoogle Scholar
- Son B, Kim T, Kim HJ, Lee S (2004) Probabilistic model of traffic breakdown with random propagation of disturbance for ITS application. Lect Notes Comput Sci 3215:45–51CrossRefGoogle Scholar
- Treiber M, Hennecke A, Helbing D (2000) Congested traffic states in empirical observations and microscopic simulations. Phys Rev E 62(2), 1805Google Scholar
- Wang H, Rudy K, Li J, Ni D (2010) Calculation of traffic flow breakdown probability to optimize link throughput. Appl Math Model 34:3376–3389CrossRefGoogle Scholar
- Yeo H, Skabardonis A (2009) Understanding stop-and-go traffic in view of asymmetric traffic theory. In: Proceedings of the 18th International Symposium on Traffic and Transportation Theory. Eds. Lam W, Wong SC, Lo HK. Springer, 99–116Google Scholar
- Zhang HM (2001) New perspectives on continuum traffic flow models. Netw Spat Econ 1:9–33CrossRefGoogle Scholar
- Zhang L, Levinson D (2010) Ramp metering and freeway bottleneck capacity. Transp Res A Policy Pract 44:218–235CrossRefGoogle Scholar