Derivation of a fundamental diagram for urban traffic flow

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12 The European Physical Journal B defined as ∫x 1 dx ′ w(x ′ )f(x ′ ) x 0 ∫x 1 . (A.1) dx ′ w(x ′ ) x 0 In case of uniform arrivals of vehicles, we have a functional relationship of the form f(x) =a + bx for the travel time, and the weigth function is constant, i.e. w(x) =w. Here, a = Ti 0, x = ΔN i,andb =1/A i =1/(u i ̂Q i ). With x 0 =0 and x 1 = ΔNi max , the formula for the average travel time becomes w[(ax 1 + bx 2 1 /2) − (ax 0 + bx 2 0 /2)] = a + b x 1 + x 0 , w(x 1 − x 0 ) 2 (A.2) wherewehaveused(x 2 1 − x 2 0 )=(x 1 − x 0 )(x 1 + x 0 ). Inserting the above parameters, we obtain the previously derived result T i = T 0 i + ΔN max i 2u i ̂Qi . (A.3) When determining the average velocity Vi av , the function to average over is of the form f(x) =c/(a + bx), where c = L i and the other parameters are as defined before. We use the relationship ∫ x 1 dx ′ wc a + bx ′ = wc ( ) ln |a + bx 1 |−ln |a + bx 0 | b x 0 = wc ∣ ∣ ∣∣∣ b ln a + bx 1 ∣∣∣ . (A.4) a + bx 0 Dividing this again by the normalization factor w(x 1 −x 0 ) and inserting the above parameters finally gives V av i = L iu i ̂Qi ΔN max i ( max i ln 1+ΔN ∣ u i ̂Qi Ti 0 ∣ ≈ L i Ti 0 1 − ΔN max i 2u i ̂Qi T 0 i ) , (A.5) wherewehaveusedln(1+x) ≤ x − x 2 /2. This formula corrects the naive formula V av i ≈ L i T i = T 0 i + L i ΔN max 1 2u i ̂Qi ≈ L i T 0 i ( 1 − ) max ΔN1 , (A.6) 2u i ̂Qi Ti 0 wherewehaveused1/(1 + x) ≈ 1 − x. Therefore, the above Taylor approximations of both formulas agree, but higher-order approximations would differ. The formulas in the main part of the paper result for Ni av = Ni max /2, which corresponds to the case δ i = 0 (i.e. f i − u i ). Generalizing the above approach to the case δ i > 0, we must split up the integrals into one over wc/(a + bx ′ ) extending from x 0 =0tox 1 = ΔNi max and another one over wc/a from x 1 = ΔNi max to x 2 =(1− u i )A i T cyc = (1−u i )u i ̂Qi T cyc , where the specifications of a, b, andc are unchanged. Taking into account Vi 0 = L i /Ti 0,thisgives V av i = wL iu i ̂Qi ln |1+ΔN max i /(u i ̂Qi T 0 i )| + Z w(1 − u i )u i ̂Qi T cyc , (A.7) where Z = wV 0 i [(1 − u i )u i ̂Qi T cyc − ΔN max i ]. (A.8) Considering equation (15), we get V av i = L i (1 − u i )T cyc ln ( 1+(1− f i ) T cyc T 0 i ) + Vi 0 f i − u i . 1 − u i (A.9) In second-order Taylor approximation, this results in [ ( 1 − Vi av ≈ Vi 0 fi 1 − (1 − f ) i)T cyc 1 − u i 2Ti 0 + f ] i − u i , 1 − u i (A.10) which can also be derived from equation (A.6), considering equation (15) and the percentage of delayed vehicles, which is given by equation (19).Thesameresultfollows from V av i = T 0 i L i + T av i ≈ V 0 i together with equation (21). References (1 − T i av ) Ti 0 (A.11) 1. D.C. Gazis, Traffic Theory (Kluwer Academic, Boston, 2002) 2. J. Esser, M. Schreckenberg, Int. J. Mod. Phys. B 8, 1025 (1997) 3. P.M. Simon, K. Nagel, Phys. Rev. E 58, 1286 (1998) 4. K. Nagel, Multi-Agent Transportation Simulations, see http://www2.tu-berlin.de/fb10/ISS/FG4/archive/ sim-archive/publications/book/ 5. M. Hilliges, W. Weidlich, Transpn. Res. B 29, 407 (1995) 6. D. Helbing, J. Siegmeier, S. Lämmer, Networks and Heterogeneous Media 2, (2007) 7. M. Cremer, J. Ludwig, Math. Comput. Simul. 28, 297ff (1986) 8. C.F. Daganzo, Transpn. Res. B 29, 79 (1995) 9. T. Nagatani, Phys. Rev. E 48, 3290 (1993) 10. D. Chowdhury, A. Schadschneider, Phys. Rev. E 59, R1311 (1999) 11. O. Biham, A.A. Middleton, D. Levine, Phys. Rev. A 46, R6124 (1992) 12. J.-F. Zheng, Z.-Y. Gao, X.-M. Zhao, Phys. Stat. Mech. Appl. 385, 700 (2007) 13. N.A. Irwin, M. Dodd, H.G. Von Cube, Highway Research Board Bulletin 347, 258 (1961) 14. R.J. Smock, Highway Research Board Bulletin 347, 60 (1962) 15. W.W. Mosher, Highway Research Record 6, 41 (1963) 16. Bureau of Public Roads, Traffic Assignment Manual (US Dept. of Commerce, Urban Planning Division, Washington, D.C., 1964) 17. T.J. Soltmann, Highway Research Record 114, 122 (1965) 18. K.B. Davidson, in Proceedings of the 3rd ARRB Conference, Part 1 (Australian Road Research Board, Melbourne, 1966), pp. 183–194
D. Helbing: Derivation of a fundamental diagram for urban traffic flow 13 19. R.J. Smeed, Traffic Engineering and Control 8, 455 (1966) 20. K.R. Overgaard, Traffic Quarterly, 197 (1967) 21. J.M. Thomson, Traffic Engineering and Control 8, 721 (1967) 22. J.G. Wardrop, Traffic Engineering and Control 9, 528 (1968) 23. Y. Zahavi, Traffic Engineering and Control 14, (1972) 24. R. Akcelik, Australian Road Research 21, 49 (1991) 25. K.M. Lum, H.S.L. Fan, S.H. Lam, P. Olszewski, J. Transpn. Eng. 124, 213 (1998) 26. H.M. Zhang, Transpn. Res. Rec. 1676, 109 (1999) 27. H. Tu, Monitoring Travel Time Reliability on Freeways, Ph.D. thesis, Delft University of Technology, (2008) 28. C.F. Daganzo, Transport. Res. B 41, 49 (2007) 29. N. Geroliminis, C.F. Daganzo, Transpn. Res. B 42, 759 (2008) 30. N. Geroliminis, C.F. Daganzo, Macroscopic modeling of traffic in cities, TRB 86th Annual Meeting, Paper #07-0413, Washington D.C. (2007) 31. C.F. Daganzo, N. Geroliminis, An analytical approximation for the macroscopic fundamental diagram of urban traffic, accepted for publication (2008) 32. J.W. Godfrey, Traffic Engineering and Control 11, 323 (1969) 33. R. Herman, I. Prigogine, Science 204, 148 (1979) 34. R. Herman, S. Ardekani, Transportation Science 18, 101 (1984) 35. M. Eichler, C.F. Daganzo, Transportation Research B 40, 731 (2006) 36. D. Helbing, Rev. Mod. Phys. 73, 1067 (2001) 37. D. Helbing, M. Treiber, A. Kesting, M. Schönhof, Theoretical vs. empirical classification and prediction of congested traffic states, Eur. Phys. J. B, accepted for publication (2009), see e-print http://arxiv.org/abs/0903.0929 38. G.B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974) 39. R. Hall, Queueing Methods for Service and Manufacturing (Prentice Hall, Upper Saddle River, NJ, 1991) 40. D. Helbing, J. Phys. A: Math. Gen 36, L593 (2003) 41. D. Helbing, A. Mazloumian, Operation regimes and sloweris-faster effect in the control of traffic intersections, submitted (2008), see e-print http://arxiv.org/abs/0903.0926 42. D. Helbing, T. Seidel, S. Lämmer, K. Peters, in Econophysics and Sociophysics, edited by B.K. Chakrabarti, A. Chakraborti, A. Chatterjee (Wiley, Weinheim, 2006), p. 552
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Derivation of a fundamental diagram for urban traffic flow

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