Modeling and Performance Evaluation of Complex Traffic

Transcript Modeling and Performance Evaluation of Complex Traffic

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

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

1. Introduction

Traffic Control System

Characterized by

:    shared resources  resource conflicts a tendency to deadlock and overflow, requirement of well-planned synchronization, scheduling and control 4

The State of Arts of Performance Analysis of Traffic Control Systems

  Queuing Theory Models Simulation  Petri Net Models 5

We present:

 a compositional method for modeling and analyzing complex traffic control systems.  a typical STPN model of traffic networks using the compositional method.

2. Traffic Control of Networks

Isolated Intersection

Urban Traffic Control Systems

3    n-2 1 2 n-1 n 1 2 3 … … m-2 m-1 m Closed Network 8

Two Phases of Regular Intersections

Phase A1 Phase A2 9

Four Phases of High Type Intersections

Phase A1 Phase A2 Phase B1 Phase B2 10

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.

3. STPN Model of Intersection Traffic Control

Petri Nets

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

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

STPN Model of An Intersection -- Control Model (I)

ea_A1 GA_A1 A_A1 eg_A1 G_A1 G_A2 eg_A2 A_A2 GA_A2 ea_A2 The control model of 2-phase intersection

PLACE:

G_A1: A_A1: GA_A1: Green signal for direction EW Amber signal for direction EW Green or amber signal for direction EW

TRANSITION:

eg_A1: Green signal ends ea_A1: Amber signal ends 15

STPN Model of An Intersection -- Control Model (II)

G_B1 ea_A1 eg_B1 GA_A1 A_A1 G_A1 G_A2 eg_A1 eg_A2 A_A2 GA_A2 ea_A2 eg_B2 G_B2 The control model of 4-phase intersection 16

STPN Model of An Intersection -- Traffic Flow Model (I)

GA_A1 Control Part IN nrto RSL ent RDY arr rto INT nlto MF dep OUT R lto LOUT lti rti ROUT LIN RIN alt The traffic flow model of 2-phase intersection

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 17

STPN Model of An Intersection -- Traffic Flow Model (II)

RDY GA_A1 Control Part IN sf RSL dep arr rto G_B2 ROUT lto LTO ent tlo INT G_B1 lti LOT alt LIN OUT rti RIN LAR The traffic flow model of 4-phase intersection.

4. Modeling and Performance Evaluation of Traffic Network

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. 20

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 21

STPN Model of 2-phase Intersection

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

STPN Model of 4-phase Intersection

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

Compositional Analysis

i+j=2 i+j=3 i+j=4 i 1 2 3 … … m-2 m-1 m j 3    n-2 n-1 n 1 2

Analyze intersections along the increasing i+j line

Example

Intersection (0,0)(4p) (0,1)(2p) (1,0)(4p) (0,2)(2p) (1,1)(4p) (2,0)(2p) (0,3)(4p) (1,2)(2p) (2,1)(4p) (3,0)(2p) (1,3)(4p) (2,2)(2p) (3,1)(4p) (2,3)(2p) (3,2)(4p) (3,3)(2p) Offset 0 60 60 0 0 0 60 60 60 60 0 0 0 60 60 0 AQL 10.5575

10.808

13.7656

9.87869

7.93539

5.54243

7.33246

4.95944

18.0724

5.03002

14.7134

15.1647

13.8706

8.97933

5.99334

7.10573

ATD 35.433

37.9069

47.0608

33.9058

26.9096

19.0341

24.0968

17.8978

57.2082

16.7118

53.6546

48.6528

44.8415

30.5812

18.6261

25.482

Offset 0 30 30 60 60 60 90 90 90 90 0 0 0 30 30 60 AQL 10.5575

9.80259

10.7584

3.95742

10.7768

8.39731

9.24824

8.93898

13.8584

9.83201

10.2563

12.8156

12.4933

9.95824

16.4617

12.1873

ATD 35.433

34.217

36.273

13.4698

35.9995

29.021

30.4988

31.7011

43.9798

32.6142

37.0925

41.1378

39.9578

34.2142

50.5948

43.395

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

reachable states for each intersection in average) Our model: 16 

; Global models:

16 .

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. 27