Performance Analysis of Traffic Networks Based on St ochastic Timed Petri Net Models

1. Performance Analysis of Traffic Networks Based on Stochastic Timed Petri Net Models Jiacun Wang, Chun Jin and Yi Deng Center for Advanced Distributed Systems Engineering School of Computer Science Florida International University Miami, FL 33199

2. Contents 1. Introduction 2. Traffic Control of Networks 3. STPN Model of Intersection Traffic Control 4. Modeling and Performance Evaluation of Traffic Networks 5. Conclusion

3. 1. Introduction

4. Traffic Control System Characterized by:  shared resources  resource conflicts  a tendency to deadlock and overflow,  requirement of well-planned synchronization, scheduling and control

5. The State of Arts of Performance Analysis of Traffic Control Systems • Queuing Theory Models • Simulation • Petri Net Models

6. We present: • a compositional method for modeling and analyzing complex traffic control systems. • a typical STPN model of traffic networks using the compositional method.

7. 2. Traffic Control of Networks

8. Urban Traffic Control Systems Isolated Intersection Closed Network

9. Two Phases of Regular Intersections Phase A1 Phase A2

10. Phase B1 Phase A2 Phase B2 Phase A1 Four Phases of High Type Intersections

11. Urban Traffic Control Systems-- Timing Plan Issues • Cycle length: The time period of a complete sequence of signal indications. • Split: A division of the cycle length allocated to each of the various phases. • Offset: The time relationship determined by the difference between a defined interval portion of the coordinated phase green and a system reference point. • Phase: A portion of signal cycle during which an assignment of right of way is made to a given traffic movements.

12. 3. STPN Model of Intersection Traffic Control

13. p t p t 5 3 3 5 2 p p 4 1 2 p 6 t t t p 1 2 4 2 p 7 p t p t 5 3 3 5 2 p p 4 1 2 p 6 t t t p 1 2 4 2 p 7 Petri Nets t1 fires

14. Stochastic Timed Petri Nets • When "time" is assigned to transitions (or places) of Petri nets, they are called Timed Petri Nets. • If the "time" is random in timed Petri nets, they are called Stochastic Timed Petri Nets.

15. ea_A1 A_A1 G_A2 eg_A2 eg_A1 GA_A1 GA_A2 G_A1 A_A2 ea_A2 The control model of 2-phase intersection STPN Model of An Intersection -- Control Model (I) PLACE: G_A1: Green signal for direction EW A_A1: Amber signal for direction EW GA_A1: Green or amber signal for direction EW TRANSITION: eg_A1: Green signal ends ea_A1: Amber signal ends

16. G_B1 eg_B1 ea_A1 A_A1 G_A2 eg_A1 eg_A2 GA_A1 GA_A2 G_A1 A_A2 ea_A2 eg_B2 G_B2 The control model of 4-phase intersection STPN Model of An Intersection -- Control Model (II)

17. RDY arr ROUT rto IN nrto RSL ent Control Part GA_A1 INT lto nlto LOUT MF alt R lti dep LIN rti OUT RIN The traffic flow model of 2-phase intersection STPN Model of An Intersection -- Traffic Flow Model (I) PLACE: RDY_A1 Incoming vehicles IN_A1 Vehicles ready to enter intersection RSL_A1 Ready for going straight or turning left ROUT_A1 Right-turn-out vehicles INT_A1 Vehicles entering intersection MF_A1 Vehicles moveing forward OUT_A1 Vehicles out RIN_A1 Right-turn-in vehicles LOUT_A1 Vehicles turn left out LAR_A2 Left-turn-in incoming vehicles LIN_A2 Left-turn-in incoming vehicles to enter intersection TRANSITION: arr_A1 Vehicles arrive rto_A1 Vehicles right-turn out Nrto_A1 Vehicles not right-turn out lto_A1 Vehicles left-turn out Nlto_A1 Vehicles not left-turn out ent_A1 Vehicles enter intersection Dep_A1 Vehicles depart intersection lti_A1 Vehicles left-turn in rti_A1 Vehicles right-trun in alt_A1 Left-turn-in vehicles arrive

18. RDY arr ROUT IN rto sf G_B2 lto RSL LTO ent tlo GA_A1 Control Part LOT INT G_B1 alt lti dep LAR LIN rti OUT RIN The traffic flow model of 4-phase intersection. STPN Model of An Intersection -- Traffic Flow Model (II)

19. 4. Modeling and Performance Evaluation of Traffic Network

20. Compositional Modeling • Traffic and traffic control of an intersection: STPN model • Traffic of an road segment: Random motion model • Compositional model of a traffic system: STPN + random motion model • Interactions between different directions are partially approximately by statistical models.

21. Compositional Analysis • Based on individual intersection model • One direction of traffic along a two-way street is considered separately from the other, • Incrementally evaluate system’s performance by analyzing intersections one by one according to a carefully selected order

22. RDY_A2 RDY_A1 arr_A2 arr_A1 rto_A2 rto_A1 IN_A2 ROUT_A1 IN_A1 ROUT_A2 nrto_A2 nrto_A1 ea_A1 RSL_A2 RSL_A1 ent_A2 ent_A1 lto_A1 A_A1 G_A2 lto_A2 LOUT_A2 INT_A2 eg_A2 INT_A1 eg_A1 LOUT_A1 nlto_A2 nlto_A1 alt_A2 GA_A1 GA_B MF_A2 GA_A2 G_B2 MF_A1 lti_A lti_A1 G_A1 A_A2 dep_A2 dep_A1 LOUT_A2 LAR_A2 LIN_A2 ea_A2 OUT_A1 rti_A2 rti_A1 OUT_A2 ROUT_A1 RIN_A1 STPN Model of 2-phase Intersection West East North South

23. RDY_A2 RDY_A1 arr_A2 arr_A1 rto_A2 rto_A1 RIN_A1 IN_A2 RIN_A2 IN_A1 G_B1 lto_A1 lto_A2 G_B1 sf_A2 G_B2 sf_A1 eg_B1 ea_A1 RSL_A2 RSL_A1 LTO_A1 LTO_A1 OUT_A1 ent_A2 ent_A1 tlo_A2 tlo_A1 A_A1 G_A2 LOUT_A1 alt_A2 MF_A11 MF_A2 eg_A1 G_B2 GA_B1 lti_A2 lti_A1 eg_A2 GA_A1 GA_A2 LOUT_A2 dep_A2 dep_A1 LAR_A2 G_A1 LIN_A2 A_A2 rti_A2 rti_A1 OUT_A2 OUT_A1 ea_A2 RIN_A1 eg_B2 RIN_A1 G_B2 STPN Model of 4-phase Intersection West East North South

24. Compositional Analysis i+j=2 i+j=4 i+j=3 i j Analyze intersections along the increasing i+j line

25. Example

26. Discussion • Dimension of State Space (number of places used in PN model) Our model: 27; Global models:  3316 = 528. • Number of Reachable States(suppose that there are M reachable states for each intersection in average) Our model: 16M; Global models: M16.

27. Conclusion • A compositional method for modeling and performance evaluation of complex traffic control systems is presented; • The method is based on individual intersection models; • It dramatically reduces the computing complexity.