Investigation the Stochastic behaviour of the Traffic Flow: A Case Study of a Section of a Road




The stochastic behavior is one of the key for the current state of vehicles flow for the real time traffic behavior. This paper describe the study to investigate the stochastic behavior of real time traffic flow for a section of road using probability distribution fit over the section of road, the traffic data was collected for a week from 7:00 to 19:00 at the location Nawabshah Pakistan. The different distribution such as Normal, Lognormal, Weibull, Gamma, Exponential distribution was fit using MATLAB distribution fit by probability plot of traffic flow data. The same distribution was used for the goodness-of-fit tests by considering Kolmogorov-Smirnov, Kolmogorov-Smirnov modified, Anderson-Darling were used with p-values at 95% of confidence level and justification to accept the hypothesis test are accepted or rejects. The hypothesis accept for Normal, Weibull and Gamma distribution which accept the all hypothesis test and among these three accepted fit distribution the Normal probability distribution fit is most fitted distribution using the rank by p-value of the hypothesis tests.

Keywords: Traffic flow, Goodness-of-fit, Probability Distributions, Nawabshah


Abul-Magd, A. [2007], ‘Modeling highway-traffic headway distributions using superstatistics’, Physical Review E 76(5), 057101.

Alasmar, M., Clegg, R., Zakhleniuk, N. and Parisis, G. [2021], ‘Internet traffic volumes are not gaussian—they are log-normal: An 18-year longitudinal study with implications for modelling and prediction’, IEEE/ACM Transactions on Networking 29(3), 1266–1279.

Ali, Y., Sharma, A., Haque, M. M., Zheng, Z. and Saifuzzaman, M. [2020], ‘The impact of the connected environment on driving behavior and safety: A driving simulator study’, Accident Analysis & Prevention 144, 105643.

Alperovich, T. and Sopasakis, A. [2008], ‘Stochastic description of traffic flow’, Journal of Statistical Physics 133, 1083–1105.

Antoniou, I., Ivanov, V. V., Ivanov, V. V. and Zrelov, P. [2002], ‘On the log-normal distribution of network traffic’, Physica D: Nonlinear Phenomena 167(1-2), 72–85.

Berry, D. S. and Belmont, D. M. [1951], Distribution of vehicle speeds and travel times, in ‘Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability’, Vol. 2, University of California Press, pp. 589–603.

Bramich, D. M., Menendez, M. and Ambühl, L. [2022], ‘Fitting empirical fundamental diagrams of road traffic: A comprehensive review and comparison of models using an extensive data set’, IEEE Transactions on Intelligent Transportation Systems 23(9), 14104–14127.

Chakravarti, I. M., Laha, R. G. and Roy, J. [1967], ‘Handbook of methods of applied statistics’, Wiley Series in Probability and Mathematical Statistics (USA) eng .

Cheng, Q., Lin, Y., Zhou, X. S. and Liu, Z. [2024], ‘Analytical formulation for explaining the variations in traffic states: A fundamental diagram modeling perspective with stochastic parameters’, European Journal of Operational Research 312(1), 182–197.

Chiabaut, N. and Faitout, R. [2021], ‘Traffic congestion and travel time prediction based on historical congestion maps and identification of consensual days’, Transportation Research Part C: Emerging Technologies 124, 102920.

Das, K. R. and Imon, A. [2016], ‘A brief review of tests for normality’, American Journal of Theoretical and Applied Statistics 5(1), 5–12.

Dubey, S. K., Ponnu, B. and Arkatkar, S. S. [2013], ‘Time gap modeling using mixture distributions under mixed traffic conditions’, Journal of Transportation Systems Engineering and Information Technology 13(3), 91–98.

Fang, Z., Zhu, S., Fu, X., Liu, F., Huang, H. and Tang, J. [2022], ‘Multivariate analysis of traffic flow using copula-based model at an isolated road intersection’, Physica A: Statistical Mechanics and its Applications 599, 127431.

Fras, M., Mohorko, J. and Čučej, Ž. [2012], ‘Modeling of statistical data sources based on measured network traffic’, Simulation 88(10), 1216–1232.

Ghosh, S., Roy, M. K. and Biswas, S. C. [2016], ‘Determination of the best fit probability distribution for monthly rainfall data in bangladesh’, American Journal of Mathematics and Statistics 6(4), 170–174.

Guessous, Y., Aron, M., Bhouri, N. and Cohen, S. [2014], ‘Estimating travel time distribution under different traffic conditions’, Transportation Research Procedia 3, 339–348.

Handscombe, J. and Yu, H. Q. [2019], ‘Low-cost and data anonymised city traffic flow data collection to support intelligent traffic system’, Sensors 19(2), 347.

Khakifirooz, M., Tercero-Gómez, V. and Woodall, W. H. [2021], ‘The role of the normal distribution in statistical process monitoring’, Quality Engineering 33(3), 497–510.

Kosun, C. and Ozdemir, S. [2016], ‘A superstatistical model of vehicular traffic flow’, Physica A: statistical mechanics and its applications 444, 466–475.

Laurikkala, M. [2009], ‘Goodness-of-fit tests and heavy-tailed distributions in network traffic data analysis’.

Leduc, G. et al. [2008], ‘Road traffic data: Collection methods and applications’, Working Papers on Energy, Transport and Climate Change 1(55), 1–55.

Li, L., Jiang, R., He, Z., Chen, X. M. and Zhou, X. [2020], ‘Trajectory data-based traffic flow studies: A revisit’, Transportation Research Part C: Emerging Technologies 114, 225–240.

Li, M., Fang, L., Jia, W. and Guo, J. [2022], ‘Traffic condition uncertainty quantification under nonnormal distributions’, Journal of transportation engineering, Part A: Systems 148(10), 04022086.

Mondal, S. and Gupta, A. [2021], ‘Speed distribution for interrupted flow facility under mixed traffic’, Physica A: Statistical Mechanics and Its Applications 570, 125798.

Monti, G. S., Mateu Figueras, G., Ortego Martínez, M. I., Pawlowsky Glahn, V. and Egozcue Rubí, J. J. [2017], Modified kolmogorov-smirnov test of goodness of fit, in ‘The 7th International Workshop on Compositional Data Analysis Proceedings book’, pp. 152–158.

Pun, L., Zhao, P. and Liu, X. [2019], ‘A multiple regression approach for traffic flow estimation’, IEEE access 7, 35998–36009.

Sánchez, E., González-Navarrete, M. and Caamaño-Carrillo, C. [2021], ‘Bivariate superstatistics: An application to statistical plasma physics’, The European Physical Journal B 94, 1–7.

Stephens, M. A. [1974], ‘Edf statistics for goodness of fit and some comparisons’, Journal of the American statistical Association 69(347), 730–737.

Ximenes, P. d. S. M. P., Silva, A. S. A. d., Ashkar, F. and Stosic, T. [2021], ‘Best-fit probability distribution models for monthly rainfall of northeastern brazil’, Water Science and Technology 84(6), 1541–1556.

Yang, Y., Fan, Y. and Royset, J. O. [2019], ‘Estimating probability distributions of travel demand on a congested network’, Transportation Research Part B: Methodological 122, 265–286.

Zang, Z., Xu, X., Qu, K., Chen, R. and Chen, A. [2022], ‘Travel time reliability in transportation networks: A review of methodological developments’, Transportation Research Part C: Emerging Technologies 143, 103866.

Zefreh, M. M. and Török, A. [2020], ‘Distribution of traffic speed in different traffic conditions: An empirical study in budapest’, Transport 35(1), 68–86.




How to Cite

Mehboob Ali Jatoi, Shakeel Ahmed Kamboh, Oshaque Ali Abro, Saeed Ahmed Rajput, & Liaquat Ali Zardari. (2024). Investigation the Stochastic behaviour of the Traffic Flow: A Case Study of a Section of a Road. VFAST Transactions on Mathematics, 12(1), 189–201.