Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T16:20:58.796Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence to equilibrium in a traffic model with restricted passing

Published online by Cambridge University Press:  14 July 2016

Hans Dieter Unkelbach
Get access Rights & Permissions [Opens in a new window]

Abstract

A road traffic model with restricted passing, formulated by Newell (1966), is described by conditional cluster point processes and analytically handled by generating functionals of point processes.

The traffic distributions in either space or time are in equilibrium, if the fast cars form a Poisson process with constant intensity combined with Poisson-distributed queues behind the slow cars (Brill (1971)). It is shown that this state of equilibrium is stable, which means that this state will be reached asymptotically for general initial traffic distributions. Furthermore the queues behind the slow cars dissolve asymptotically like independent Poisson processes with diminishing rate, also independent of the process of non-queuing cars. To get these results limit theorems for conditional cluster point processes are formulated.

Keywords

POISSON TRAFFICROAD TRAFFIC THEORYTRAFFIC MODELSRESTRICTED PASSINGPOINT PROCESSESCLUSTER POINT PROCESSESTRAJECTORY PROCESSESGEOMETRIC PROBABILITYGENERATING FUNCTIONALSLIMIT THEOREMS FOR STOCHASTIC PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 4 , December 1979 , pp. 881 - 889
Copyright
Copyright © Applied Probability Trust 1979 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Breiman, L. (1963) The Poisson tendency in traffic distribution. Ann. Math. Statist. 34, 308–311.Google Scholar
Brill, E. A. (1971) Point processes arising in vehicular traffic flow. J. Appl. Prob. 8, 809–814.CrossRefGoogle Scholar
Brown, M. (1969) Some results on a traffic model of Rényi. J. Appl. Prob. 6, 293–300.CrossRefGoogle Scholar
Brown, M. (1972) Low density traffic streams. Adv. Appl. Prob. 4, 177–192.Google Scholar
Erlander, S. (1967) A mathematical model for traffic on a two lane road. In Vehicular Traffic Science , ed. Edie, C. L. et al. Elsevir, New York, 153–167.Google Scholar
Haight, F. A. (1963) Mathematical Theories of Traffic Flow. Academic Press, New York.Google Scholar
Kallenberg, O. (1978) On the asymptotic behavior of line processes and systems of non-interacting particles. Z. Wahrscheinlichkeitsth. 43, 65–95.CrossRefGoogle Scholar
Newell, G. F. (1966) Equilibrium probability distributions for low density highway traffic. J. Appl. Prob. 3, 247–260.Google Scholar
Rényi, A. (1964) On two mathematical models of the traffic on a divided highway. J. Appl. Prob. 1, 311–320.Google Scholar
Tanner, J. C. (1951) The delay to pedestrians crossing a road. Biometrika 38, 383–392.Google Scholar
Thedéen, T. (1964) A note on the Poisson tendency in traffic distribution. Ann. Math. Statist. 35, 1823–1824.Google Scholar
Thedéen, T. (1967) Convergence and invariance questions for point systems in R1 under random motion. Ark. Mat. 7, 211–239.CrossRefGoogle Scholar
Unkelbach, H. D. (1974) Poissontendenz im Straßenverkehr. Dissertation, Darmstadt.Google Scholar
Unkelbach, H. D. (1975) Poissonprozesse als Grenzprozesse in zwei Straßenverkehrsmodellen. In Operations Research Verfahren XXI, ed. Henn, R. et al. Meisenheim am Glan, 239–261.Google Scholar
Vere-Jones, D. (1968) Some applications of probability generating functionals to the study of input–ouput streams. J. R. Statist. Soc. B 30, 321–333.Google Scholar