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DELAYS AT SIGNALIZED INTERSECTIONS WITH EXHAUSTIVE TRAFFIC CONTROL*

Published online by Cambridge University Press:  08 June 2012

M.A.A. Boon ,
I.J.B.F. Adan ,
E.M.M. Winands  and
D.G. Down
M.A.A. Boon
Affiliation:
EURANDOM and Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlands E-mail: marko@win.tue.nl, iadan@win.tue.nl
I.J.B.F. Adan
Affiliation:
EURANDOM and Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlands E-mail: marko@win.tue.nl, iadan@win.tue.nl
E.M.M. Winands
Affiliation:
FNWIKorteweg de Vries Instituut voor Wiskunde, Universiteit van Amsterdam, P.O. Box 942481090, GE AMSTERDAM, The Netherlands E-mail: emm.winands@few.vu.nl
D.G. Down
Affiliation:
McMaster University, 1280 Main Street West, Hamilton, ON, CanadaL8S 4L7 E-mail: downd@mcmaster.ca
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Abstract

In this paper, we study a traffic intersection with vehicle-actuated traffic signal control. Traffic lights stay green until all lanes within a group are emptied. Assuming general renewal arrival processes, we derive exact limiting distributions of the delays under heavy traffic (HT) conditions. Furthermore, we derive the light traffic (LT) limit of the mean delays for intersections with Poisson arrivals, and develop a heuristic adaptation of this limit to capture the LT behavior for other interarrival-time distributions. We combine the LT and HT results to develop closed-form approximations for the mean delays of vehicles in each lane. These closed-form approximations are quite accurate, very insightful, and simple to implement.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 26 , Issue 3 , July 2012 , pp. 337 - 373
Copyright
Copyright © Cambridge University Press 2012

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References

1.Boon, M.A.A., Adan, I.J.B.F., & Boxma, O.J. (2010). A two-queue polling model with two priority levels in the first queue. Discrete Event Dynamic Systems, 20(4): 511–536.CrossRefGoogle Scholar
2.Boon, M.A.A., Adan, I.J.B.F., & Boxma, O.J. (2010). A polling model with multiple priority levels. Performance Evaluation, 67: 468–484.CrossRefGoogle Scholar
3.Boon, M.A.A., van der Mei, R.D., & Winands, E.M.M. (2011). Applications of polling systems. Surveys in Operations Research and Management Science, 16: 67–82.CrossRefGoogle Scholar
4.Boon, M.A.A., Winands, E.M.M., Adan, I.J.B.F., & van Wijk, A.C.C. (2011). Closed-form waiting time approximations for polling systems. Performance Evaluation, 68: 290–306.CrossRefGoogle Scholar
5.Coffman, E.G. Jr., Puhalskii, A.A., & Reiman, M.I. (1995). Polling systems with zero switchover times: A heavy-traffic averaging principle. The Annals of Applied Probability 5(3): 681–719.CrossRefGoogle Scholar
6.Coffman, E.G. Jr., Puhalskii, A.A., & Reiman, M.I. (1998). Polling systems in heavy-traffic: A Bessel process limit. Mathematics of Operations Research 23: 257–304.CrossRefGoogle Scholar
7.Cohen, J.W. (1982). The single server queue, revised edition, Amsterdam North-Holland.Google Scholar
8.CROW. (2006) Handboek verkeerslichtenregelingen. Publicatie 213. CROW kenniscentrum voor verkeer, vervoer en infrastructuur, Ede, ISBN 90 6628 444 7.Google Scholar
9.Dai, J.G. (1995). On positive Harris recurrence of multiclass queueing networks: A unified queue via fluid limit models. Annals of Applied Probability 5: 49–77.CrossRefGoogle Scholar
10.Dai, J.G. (1999). Stability of fluid and stochastic processing networks. Miscellanea Publication, No. 9. Aarhus, Denmark. Centre for Mathematical Physics and Stochastics, http://www.maphysto.dk/.Google Scholar
11.Dai, J.G., & Meyn, S.P. (1995). Stability and convergence of moments for multiclass queueing networks via fluid models. IEEE Transactions on Automatic Control, 40: 1889–1904.CrossRefGoogle Scholar
12.Darroch, J.N., Newell, G.F., & Morris, R.W.J. (1964). Queues for a vehicle-actuated traffic light. Operations Research 12(6): 882–895.CrossRefGoogle Scholar
13.Dorsman, J.L., van der Mei, R.D., & Winands, E.M.M. (2011). A new method for deriving waiting-time approximations in polling systems with renewal arrivals. Stochastic Models 27(2): 318–332.CrossRefGoogle Scholar
14.Fuhrmann, S.W. (1981). Performance analysis of a class of cyclic schedules. Technical memorandum 81-59531-1, Bell Laboratories, March.Google Scholar
15.Fuhrmann, S.W. & Cooper, R.B. (1985). Stochastic decompositions in the M/G/1 queue with generalized vacations. Operations Research 33(5): 1117–1129.CrossRefGoogle Scholar
16.Greenberg, B.S., Leachman, R.C., & Wolff, R.W. (1988). Predicting dispatching delays on a low speed, single track railroad. Transportation Science 22: 31–38.CrossRefGoogle Scholar
17.Haijema, R. & van der Wal, J. (2007). An MDP decomposition approach for traffic control at isolated signalized intersections. Probability in the Engineering and Informational Sciences 22: 587–602.CrossRefGoogle Scholar
18.Heidemann, D. (1994). Queue length and delay distributions at traffic signals. Transportation Research Part B 28(5): 377–389.CrossRefGoogle Scholar
19.Lehoczky, J.P. (1972). Traffic intersection control and zero-switch queues under conditions of Markov chain dependence input. Journal of Applied Probability 9(2): 382–395.CrossRefGoogle Scholar
20.Levy, H. & Sidi, M. (1990). Polling systems: applications, modeling, and optimization. IEEE Transactions on Communications, 38: 1750–1760.CrossRefGoogle Scholar
21.Levy, H. & Sidi, M. (1991). Polling systems with simultaneous arrivals. IEEE Transactions on Communications 39: 823–827.CrossRefGoogle Scholar
22.Meyn, S.P. & Down, D. (1994). Stability of generalized jackson networks. Annals of Applied Probability, 4: 124–148.CrossRefGoogle Scholar
23.Miller, A.J. (1963). Settings for fixed-cycle traffic signals. Operational Research Quarterly 14: 373–386.CrossRefGoogle Scholar
24.Newell, G.F. (1965). Approximation methods for queues with application to the fixed-cycle traffic light. SIAM Review, 7(2): 223–240.CrossRefGoogle Scholar
25.Newell, G.F. (1969). Properties of vehicle-actuated signals: I. one-way streets. Transportation Science, 3(2): 31–52.Google Scholar
26.Newell, G.F. (1998). The rolling horizon scheme of traffic signal control. Transportation Research Part A 32(1): 39–44.Google Scholar
27.Newell, G.F. & Osuna, E.E. (1969). Properties of vehicle-actuated signals: II. two-way streets. Transportation Science 3(2): 99–125.CrossRefGoogle Scholar
28.Olsen, T.L. & van der Mei, R.D. (2003). Polling systems with periodic server routeing in heavy traffic: distribution of the delay. Journal of Applied Probability 40: 305–326.CrossRefGoogle Scholar
29.Olsen, T.L. & van der Mei, R.D. (2005). Polling systems with periodic server routing in heavy traffic: renewal arrivals. Operations Research Letters 33: 17–25.CrossRefGoogle Scholar
30.Resing, J.A.C. (1993). Polling systems and multitype branching processes. Queueing Systems 13: 409–426.CrossRefGoogle Scholar
31.Takagi, H. (1988). Queuing analysis of polling models. ACM Computing Surveys (CSUR) 20: 5–28CrossRefGoogle Scholar
32.Tijms, H.C. (1994). Stochastic models: an algorithmic approach. Chichester: Wiley.Google Scholar
33.Transportation Research Board. (2000). Capacity Manual 2000. Washington, DC: Transportation Research Board.Google Scholar
34.van den Broek, M.S., van Leeuwaarden, J.S.H., Adan, I.J.B.F., & Boxma, O.J. (2006). Bounds and approximations for the fixed-cycle traffic-light queue. Transportation Science 40(4): 484–496.CrossRefGoogle Scholar
35.van der Mei, R.D. (2001). Polling systems with simultaneous batch arrivals. Stochastic Models 17(3): 271–292.CrossRefGoogle Scholar
36.van der Mei, R.D. & Winands, E.M.M. (2008). A note on polling models with renewal arrivals and nonzero switch-over times. Operations Research Letters 36: 500–505.CrossRefGoogle Scholar
37.van Leeuwaarden, J.S.H. (2006). Delay analysis for the fixed-cycle traffic-light queue. Transportation Science 40(2): 189–199.CrossRefGoogle Scholar
38.Vishnevskii, V.M. & Semenova, O.V. (2006). Mathematical methods to study the polling systems. Automation and Remote Control 67(2): 173–220.CrossRefGoogle Scholar
39.Vlasiou, M. & Yechiali, U. (2008). M/G/∞ polling systems with random visit times. Probability in the Engineering and Informational Sciences 22: 81–106.CrossRefGoogle Scholar
40.Webster, F.V. (1958). Traffic signal settings. Technical Paper 39, Road Research Laboratory.Google Scholar
41.Whitt, W. (1989). An interpolation approximation for the mean workload in a GI/G/1 queue. Operations Research 37(6): 936–952.CrossRefGoogle Scholar
42.Yamashita, H., Ishizuka, Y., & Suzuki, S. (2006). Mean and variance of waiting time and their optimization for alternating traffic control systems. Mathematical Programming 108(2–3): 419–433.CrossRefGoogle Scholar