Anticipative control of switched queueing systems

  • S. LämmerEmail author
  • R. Donner
  • D. Helbing
Topical issue dedicated to ECCS2007 - Dresden


The relevant dynamics of a queueing process can be anticipated by taking future arrivals into account. If the transport from one queue to another is associated with transportation delays, as it is typical for traffic or productions networks, future arrivals to a queue are known over some time horizon and, thus, can be used for an anticipative control of the corresponding flows. A queue is controlled by switching its outflow between “on” and “off” similar to green and red traffic lights, where switching to “on” requires a non-zero setup time. Due to the presence of both continuous and discrete state variables, the queueing process is described as a hybrid dynamical system. From this formulation, we derive one observable of fundamental importance: the green time required to clear the queue. This quantity allows to detect switching time points for serving platoons without delay, i.e., in a “green wave” manner. Moreover, we quantify the cost of delaying the start of a service period or its termination in terms of additional waiting time. Our findings may serve as a basis for strategic control decisions.


02.30.Yy Control theory 02.30.Ks Delay and functional equations 89.75.-k Complex systems 89.40.-a Transportation 


  1. T. Nagatani, Rep. Progr. Phys. 65, 1331 (2002) CrossRefADSGoogle Scholar
  2. D. Helbing, Rev. Mod. Phys. 73, 1067 (2001) CrossRefADSGoogle Scholar
  3. J. Perkins, P.R. Kumar, IEEE Trans. Automat. Control 34, 139 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  4. D. Helbing, S. Lämmer, in Networks of Interacting Machines: Production Organization in Complex Industrial Systems and Biological Cells, edited by D. Armbruster, A. Mikhailov, K. Kaneko (World Scientific, Singapore, 2005), pp. 33–66 Google Scholar
  5. D. Helbing, S. Lämmer, J.P. Lebacque, in Optimal Control and Dynamic Games, edited by C. Deissenberg, R.F. Hartl (Springer, Dortrecht, 2005), pp. 239–274 Google Scholar
  6. S. Lämmer, Ph.D. thesis, University of Technology, Dresden, 2007 Google Scholar
  7. T. van Woensel, N. Vandaele, Asia-Pacific J. Oper. Res. (accepted) (2006) Google Scholar
  8. G.F. Newell, Applications of queueing theory (Chapman and Hall, 1971) Google Scholar
  9. S.K. Bose, An Introduction to Queueing Systems (Springer, 2001) Google Scholar
  10. R.J. Troutbeck, Transportation Sci. 20, 272 (1986) Google Scholar
  11. B. de Schutter, B. de Moor, in Hybrid Systems V, edited by P. Antsaklis, W. Kohn, M. Lemmon, A. Nerode, S. Sastry (Springer-Verlag, Berlin, 1999), Vol. 1567 of Lecture Notes in Computer Science, pp. 70–85 Google Scholar
  12. B. de Schutter, Eur. J. Oper. Res. 139, 400 (2002) zbMATHCrossRefGoogle Scholar
  13. G.F. Newell, Oper. Res. 7, 589 (1959) MathSciNetCrossRefGoogle Scholar
  14. P.G. Michalopoulos, V. Pisharody, Transportation Sci. 14, 365 (1980) MathSciNetGoogle Scholar
  15. M. McDonald, N.B. Hounsell, in Concise Encyclopedia of Traffic & Transportation Systems, edited by M. Papageorgiou (Pergamon, Exeter, 1991), Advances in Systems Control and Information Engineering, pp. 400–408 Google Scholar
  16. G.F. Newell, The effect of queues on the traffic assignment to freeways, in Proc. 7th Int. Symposium on Transportation and Traffic Theory, edited by T. Sasaki, T. Yamaoka (Institute of Systems Science Research, Kyoto, Japan, 1977), pp. 311–340 Google Scholar
  17. C.F. Daganzo, Transportation Sci. 32, 3 (1998) zbMATHGoogle Scholar
  18. D. Helbing, J. Siegmeier, S. Lämmer, Networks and Heterogeneous Media 2, 193 (2007) zbMATHMathSciNetGoogle Scholar
  19. A.V. Savkin, Controllability of complex switched server queueing networks modelled as hybrid dynamical systems, in Proc. 37th IEEE Conf. Decis. & Control, Vol. 4, 4289 (1998) Google Scholar
  20. A.V. Savkin, R.J. Evans, Hybrid Dynamical Systems (Birkhäuser, Boston, 2002) Google Scholar
  21. B. de Schutter, SIAM J. Control Optim. 39, 835 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  22. G.B. Arfken, H.J. Weber, Mathematical Methods for Physicists, 4th edn. (Academic Press, 1995) Google Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007

Authors and Affiliations

  1. 1.Dresden University of TechnologyDresdenGermany
  2. 2.Swiss Federal Institute of TechnologyZurichSwitzerland
  3. 3.Collegium Budapest – Institute for Advanced StudyBudapestHungary

Personalised recommendations