# American Institute of Mathematical Sciences

June 2018, 13(2): 241-260. doi: 10.3934/nhm.2018011

## Stability and implementation of a cycle-based max pressure controller for signalized traffic networks

 1 Dept of Civil and Environmental Engineering, Univ. of California, Berkeley, Berkeley, CA, USA 2 Dept. of Operations Research and Financial Engineering, Princeton University, Princeton, NJ, USA 3 TSS-Transport Simulation Systems, Barcelona, Spain 4 Dept of Civil and Environmental Engineering, Dept. of Electrical Engineering and Computer Science, Univ. of California, Berkeley, Berkeley, CA, USA

Received  May 2016 Revised  February 2018 Published  May 2018

Fund Project: This work was funded by the California Department of Transportation under the Connected Corridors program

Intelligent use of network capacity via responsive signal control will become increasingly essential as congestion increases on urban roadways. Existing adaptive control systems require lengthy location-specific tuning procedures or expensive central communications infrastructure. Previous theoretical work proposed the application of a max pressure controller to maximize network throughput in a distributed manner with minimal calibration. Yet this algorithm as originally formulated has unpractical hardware and safety constraints. We fundamentally alter the formulation of the max pressure controller to a setting where the actuation can only update once per multiple time steps of the modeled dynamics. This is motivated by the case of a traffic signal that can only update green splits based on observed link-counts once per "cycle time" of 60-120 seconds. Furthermore, we extend the domain of allowable actuations from a single signal phase to any convex combination of available signal phases to model intra-cycle signal changes dictated by pre-selected cycle green splits. We show that this extended max pressure controller will stabilize a vertical queueing network given restrictions on admissible demand flows that are slightly stronger than those suggested in the original formulation of max pressure. We ultimately apply our cycle-based extension of max pressure to a simulation of an existing arterial network and provide comparison to the control policy that is currently deployed at the modeled location.

Citation: Leah Anderson, Thomas Pumir, Dimitrios Triantafyllos, Alexandre M. Bayen. Stability and implementation of a cycle-based max pressure controller for signalized traffic networks. Networks & Heterogeneous Media, 2018, 13 (2) : 241-260. doi: 10.3934/nhm.2018011
##### References:
 [1] K. Aboudolas, M. Papageorgiou and E. Kosmatopoulos, Store-and-forward based methods for the signal control problem in large-scale congested urban networks, Transportation Research Part C: Emerging Technologies, 17 (2009), 163-174. doi: 10.1016/j.trc.2008.10.002. [2] R. E. Allsop, Estimating the traffic capacity of a signalized road junction, Transportation Research, 6 (1972), 245-255. doi: 10.1016/0041-1647(72)90017-2. [3] M. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, R. Vijayakumar and P. Whiting, Scheduling in a queuing system with asynchronously varying service rates, Probability in the Engineering and Informational Sciences, 18 (2004), 191-217. doi: 10.1017/S0269964804182041. [4] R. Brockett, Stabilization of motor networks, in Proceedings of the 34th IEEE Conference on Decision and Control, 1995, 1484–1488. doi: 10.1109/CDC.1995.480312. [5] J. G. Dai and W. Q. Lin, Maximum pressure policies in stochastic processing networks, Operations Research, 53 (2005), 197-218. doi: 10.1287/opre.1040.0170. [6] E. J. Davison and U. Ozguner, Decentralized control of traffic networks, IEEE Transactions on Systems, Man, and Cybernetics, 13 (1983), 476-487. doi: 10.1109/TAC.1983.1103296. [7] M. Egerstedt and Y. Wardi, Multi-process control using queuing theory, in Proceedings of the 41st IEEE Conference on Decision and Control, 2002, 1991–1996. doi: 10.1109/CDC.2002.1184820. [8] P. Giaccone, E. Leonardi and D. Shah, On the maximal throughput of networks with finite buffers and its application to buffered crossbars, in Proceedings of the 24th Annual Joint Conference of the IEEE Computer and Communications Societies, 2005,971–980. doi: 10.1109/INFCOM.2005.1498326. [9] P. B. Hunt, D. I. Robertson, R. D. Bretherton and R. I. Winton, SCOOT - a Traffic Responsive Method of Coordinating Signals, Transport and Road Research Laboratory, UK, 1981. [10] H. Ishii and B. A. Francis, Stabilizing a linear system by switching control with dwell time, IEEE Trans. Automat. Control, 47 (2002), 1962-1973. doi: 10.1109/TAC.2002.805689. [11] M. J. Neely, E. Modiano and C. E. Rohrs, Dynamic power allocation and routing for time-varying wireless networks, IEEE Journal on Selected Areas in Communications, 23 (2005), 89-103. [12] M. Pajic, S. Sundaram and G. J. Pappas, Stabilizability over Deterministic Relay Networks, in Proceedings of the 52nd IEEE Conference on Decision and Control, 2013. doi: 10.1109/CDC.2013.6760504. [13] A. G. Sims and K. W. Dobinson, The Sydney coordinated adaptive traffic (SCAT) system philosophy and benefits, IEEE Transactions on Vehicular Technology, 29 (1980), 130-137. doi: 10.1109/T-VT.1980.23833. [14] A. L. Stolyar, MaxWeight scheduling in a generalized switch: State space collapse and workload minimization in heavy traffic, Annals of Applied Probability, 14 (2004), 1-53. doi: 10.1214/aoap/1075828046. [15] L. Tassiulas, Adaptive back-pressure congestion control-based on local information, IEEE Transactions on Automatic Control, 40 (1995), 236-250. doi: 10.1109/9.341781. [16] L. Tassiulas and A. Ephremides, Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks, IEEE Transactions on Automatic Control, 37 (1992), 1936-1948. doi: 10.1109/9.182479. [17] M. van den Berg, A. Hegyi, B. De Schutter and H. Hellendoorn, A macroscopic traffic flow model for integrated control of freeway and urban traffic networks, in Proceedings of the 42nd IEEE Conference on Decision and Control, 2003, 2774–2779. [18] P. Varaiya, Max pressure control of a network of signalized intersections, Transportation Research Part C: Emerging Technologies, 36 (2013), 177-195. doi: 10.1016/j.trc.2013.08.014. [19] T. Wongpiromsarn, T. Uthaicharoenpong, Y. Wang, E. Frazzoli and D. Wang, Distributed traffic signal control for maximum network throughput, in Proceedings of the 15th International IEEE Conference on Intelligent Transportation Systems, 2012,588–595. doi: 10.1109/ITSC.2012.6338817. [20] H. M. Zhang, Y. M. Nie and Z. S. Qian, Modelling network flow with and without link interactions: The cases of point queue, spatial queue and cell transmission model, Transportmetrica B: Transport Dynamics, 1 (2013), 33-51.

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##### References:
 [1] K. Aboudolas, M. Papageorgiou and E. Kosmatopoulos, Store-and-forward based methods for the signal control problem in large-scale congested urban networks, Transportation Research Part C: Emerging Technologies, 17 (2009), 163-174. doi: 10.1016/j.trc.2008.10.002. [2] R. E. Allsop, Estimating the traffic capacity of a signalized road junction, Transportation Research, 6 (1972), 245-255. doi: 10.1016/0041-1647(72)90017-2. [3] M. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, R. Vijayakumar and P. Whiting, Scheduling in a queuing system with asynchronously varying service rates, Probability in the Engineering and Informational Sciences, 18 (2004), 191-217. doi: 10.1017/S0269964804182041. [4] R. Brockett, Stabilization of motor networks, in Proceedings of the 34th IEEE Conference on Decision and Control, 1995, 1484–1488. doi: 10.1109/CDC.1995.480312. [5] J. G. Dai and W. Q. Lin, Maximum pressure policies in stochastic processing networks, Operations Research, 53 (2005), 197-218. doi: 10.1287/opre.1040.0170. [6] E. J. Davison and U. Ozguner, Decentralized control of traffic networks, IEEE Transactions on Systems, Man, and Cybernetics, 13 (1983), 476-487. doi: 10.1109/TAC.1983.1103296. [7] M. Egerstedt and Y. Wardi, Multi-process control using queuing theory, in Proceedings of the 41st IEEE Conference on Decision and Control, 2002, 1991–1996. doi: 10.1109/CDC.2002.1184820. [8] P. Giaccone, E. Leonardi and D. Shah, On the maximal throughput of networks with finite buffers and its application to buffered crossbars, in Proceedings of the 24th Annual Joint Conference of the IEEE Computer and Communications Societies, 2005,971–980. doi: 10.1109/INFCOM.2005.1498326. [9] P. B. Hunt, D. I. Robertson, R. D. Bretherton and R. I. Winton, SCOOT - a Traffic Responsive Method of Coordinating Signals, Transport and Road Research Laboratory, UK, 1981. [10] H. Ishii and B. A. Francis, Stabilizing a linear system by switching control with dwell time, IEEE Trans. Automat. Control, 47 (2002), 1962-1973. doi: 10.1109/TAC.2002.805689. [11] M. J. Neely, E. Modiano and C. E. Rohrs, Dynamic power allocation and routing for time-varying wireless networks, IEEE Journal on Selected Areas in Communications, 23 (2005), 89-103. [12] M. Pajic, S. Sundaram and G. J. Pappas, Stabilizability over Deterministic Relay Networks, in Proceedings of the 52nd IEEE Conference on Decision and Control, 2013. doi: 10.1109/CDC.2013.6760504. [13] A. G. Sims and K. W. Dobinson, The Sydney coordinated adaptive traffic (SCAT) system philosophy and benefits, IEEE Transactions on Vehicular Technology, 29 (1980), 130-137. doi: 10.1109/T-VT.1980.23833. [14] A. L. Stolyar, MaxWeight scheduling in a generalized switch: State space collapse and workload minimization in heavy traffic, Annals of Applied Probability, 14 (2004), 1-53. doi: 10.1214/aoap/1075828046. [15] L. Tassiulas, Adaptive back-pressure congestion control-based on local information, IEEE Transactions on Automatic Control, 40 (1995), 236-250. doi: 10.1109/9.341781. [16] L. Tassiulas and A. Ephremides, Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks, IEEE Transactions on Automatic Control, 37 (1992), 1936-1948. doi: 10.1109/9.182479. [17] M. van den Berg, A. Hegyi, B. De Schutter and H. Hellendoorn, A macroscopic traffic flow model for integrated control of freeway and urban traffic networks, in Proceedings of the 42nd IEEE Conference on Decision and Control, 2003, 2774–2779. [18] P. Varaiya, Max pressure control of a network of signalized intersections, Transportation Research Part C: Emerging Technologies, 36 (2013), 177-195. doi: 10.1016/j.trc.2013.08.014. [19] T. Wongpiromsarn, T. Uthaicharoenpong, Y. Wang, E. Frazzoli and D. Wang, Distributed traffic signal control for maximum network throughput, in Proceedings of the 15th International IEEE Conference on Intelligent Transportation Systems, 2012,588–595. doi: 10.1109/ITSC.2012.6338817. [20] H. M. Zhang, Y. M. Nie and Z. S. Qian, Modelling network flow with and without link interactions: The cases of point queue, spatial queue and cell transmission model, Transportmetrica B: Transport Dynamics, 1 (2013), 33-51.
The chosen network was calibrated to represent realistic demands and physical parameters observed on a stretch of Black Mountain Road near the I-15 freeway in San Diego, California
Cb-MP demonstrated service rates that are consistent with a fully-actuated control system for similar cycle lengths
Cb-MP outperforms the actuated controller given high demand in terms of vehicle delay
While Cb-MP caused more vehicle stop events, stoppage times were similar to those observed using the actuated controller
Observed queues increase with cycle length using CbMP control
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