# Dynamic User Optimal Signal Design at Isolated Intersections

### Abstract

This paper presents a proposal of novel signal design problem at isolated intersections, which assumes that the effective green times assigned to each signal phase follow dynamic user optimal (DUO) principle. At the DUO state, the average delays of vehicles using the signal phases with positive additional green times (the assigned effective green times minus the minimum effective green times) are equal and maximum. The proposed signal design problem is formulated as a variational inequality (VI) problem. The point queue (PQ) model is applied to represent traffic dynamics and to generate the cumulative traffic flows, which is further used to estimate the average delay of each signal phase. The existence of the solution of the proposed VI problem is proved and a solution algorithm based on the method of successive averages (MSA) is developed to solve the proposed signal design problem. Finally, a sample intersection is used to illustrate the application of the proposed model and the solution algorithm.### References

Allsop, R.E.: Some possibilities of using traffic control to influence trip distribution and route choice, In: Transportation and Traffic Theory (Proceedings of the Sixth International Symposium on Transportation and Traffic Theory). Elsevier, Amsterdam, 1974, pp. 345-374

Gao, Z.Y., Song, Y.F.: A reserve capacity model of optimal signal control with user-equilibrium route choice, Transportation Research Part B, Vol. 36, 2002, pp. 313-323

Ying, J.Q., Lu, H.P., Shi, J.: An algorithm for local continuous optimization of traffic signals, European Journal of Operational Research, Vol. 181, 2007, pp. 1189-1197

Veljanovska, K., Bombol, K.M., Maher, T.: Reinforcement learning technique in multiple motorway access control strategy design, Promet-Traffic & Transportation, Vol. 22, 2010, pp. 117-123

Chang, T.H., Sun, G.Y.: Modeling and optimization of an oversaturated signalized network, Transportation Research Part B, Vol. 38, 2004, pp. 687-707

Ukkusuri, S.V., Ramadurai, G., Patil, G.: A robust transportation signal control problem accounting for traffic dynamics, Computers & Operations Research, Vol. 37, 2010, pp. 869-879

Jiang, Y., Li, S., Shamo, D.E.: A platoon-based traffic signal timing algorithm for major–minor intersectiontypes, Transportation Research Part B, Vol. 40, 2006, pp. 543–562

Ran, B., Boyce, D.E.: Modelling Dynamic Transportation Networks, Springer-Verlag, Heidelberg, Germany, 1996

Huang, H.J., Lam, W.H.K.: Modeling and solving the dynamic user equilibrium route and departure time choice problem in network with queues, Transportation Research B, Vol. 36, 2002, pp. 253-273

Lo, H.K., Szeto, W.Y.: A cell-based variational inequality formulation of the dynamic user optimal assignment problem, Transportation Research Part B, Vol. 36, 2002, pp. 421-443

Zhong, R.X., Sumalee, A., Friesz, T.L., Lam, W.H.K.: Dynamic user equilibrium with side constraints for a traffic network: theoretical development and numerical solution algorithm, Transportation Research Part B, Vol. 45, 2011, pp.

-1061

Mun, J.S.: Traffic performance models for dynamic traffic assignment: an assessment of existing models, Transport Reviews, Vol. 27, No. 2, 2007, pp. 231-249

Nie, X.J., Zhang, H.M.: A comparative study of some macroscopic link models used in dynamic traffic assignment, Networks and Spatial Economics, Vol. 5, No. 1, 2005, pp. 89-115

Daganzo, C.F.: The cell-transmission model: a simple dynamic representation of highway traffic, Transportation Research Part B, Vol. 28, 1994, pp. 269-287

Daganzo, C.F.: The cell-transmission model, Part II: Network traffic, Transportation Research Part B, Vol. 29, 1995, pp. 79-93

Yperman, I.: The link transmission model for dynamic network loading, Ph D. thesis, Katholieke Universiteit Leuven, Leuven, Belgium, 2007

Long, J.C., Gao, Z.Y., Szeto, W.Y.: Discretised link travel time models based on cumulative flows: formulations and properties, Transportation Research Part B, Vol. 45, 2011, pp. 232-254

Li, S.G., Zhou, Q.H., Ju, Y.F.: A mixed integer linear program for the single destination system optimum dynamic traffic assignment problem with physical queue, Promet-Traffic & Transportation, Vol. 22, 2010, pp. 245-249

Lo, H.K.: A cell-based traffic control formulation: strategies and benefits of dynamic timing plans, Transportation Science, Vol. 35, No. 2, 2001, pp. 148-164

Chen, H.K., Hsueh, C.F.: A model and an algorithm for the dynamic user-optimal route choice problem, Transportation Research Part B, Vol. 32, 1998, pp. 219-234

He, B.S., Zhou, J.: A modified alternating direction method for convex minimization problems, Applied Mathematics Letters, Vol. 13, 2000, pp. 123-130

Tong, C.O., Wong, S.C.: A predictive dynamic traffic assignment model in congested capacity-constrained road networks, Transportation Research Part B, Vol. 34, 2000, pp. 625-644

Mounce, R., Carey, M.: Route swapping in dynamic traffic networks, Transportation Research Part B, Vol. 45, 2011, pp. 102-111

Webster, F.V.: Traffic signal settings, Road Research Technical Paper No. 39, Road Research Laboratory, 1958

Authors who publish with this journal agree to the following terms:

- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).