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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

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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

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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.

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2. Traffic Control of Networks

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Urban Traffic Control Systems

Isolated Intersection

1 2 3 … … m-2 m-1 m

1

2

3

n-2

n-1

n

Closed Network

9

Two Phases of Regular Intersections

Phase A1 Phase A2

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Four Phases of High Type Intersections

Phase B1

Phase A2 Phase B2

Phase A1

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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.

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3.

STPN Model of

Intersection Traffic Control

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Petri Nets

p 1

t2

p6

p 5

p 4

p 3

p 2

p7

t1

t 3 t 5

t4

2

2

p 1

t 2

p6

p 5

p4

p 3

p 2

p 7

t1

t3 t 5

t4

2

2t1 fires

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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.

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

G_A2

ea_A2

GA_A2

eg_A2

G_A1

eg_A1

A_A1

A_A2

ea_A1

GA_A1

The control model of 2-phase intersection

PLACE:G_A1: Green signal for direction EW A_A1: Amber signal for direction EWGA_A1: Green or amber signal for direction

EW

TRANSITION:eg_A1: Green signal endsea_A1: Amber signal ends

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

G_B1

G_A2

ea_A2

GA_A2

G_A1

ea_A1

eg_A1

A_A1

A_A2

GA_A1

eg_B2

eg_B1

G_B2

eg_A2

The control model of 4-phase intersection

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

IN

LOUT

ROUT

RDY

arr

rto

lto

ent

INT

OUT

LIN

RIN

Rdep

lti alt

rti

RSL

nrto

nlto

MFControl Part

GA_A1

The traffic flow model of 2-phase intersection

PLACE:RDY_A1 Incoming vehiclesIN_A1 Vehicles ready to enter intersectionRSL_A1 Ready for going straight or turning leftROUT_A1 Right-turn-out vehiclesINT_A1 Vehicles entering intersectionMF_A1 Vehicles moveing forwardOUT_A1 Vehicles outRIN_A1 Right-turn-in vehiclesLOUT_A1 Vehicles turn left outLAR_A2 Left-turn-in incoming vehicles LIN_A2 Left-turn-in incoming vehicles to enter intersection

TRANSITION:arr_A1 Vehicles arriverto_A1 Vehicles right-turn outNrto_A1 Vehicles not right-turn outlto_A1 Vehicles left-turn outNlto_A1 Vehicles not left-turn outent_A1 Vehicles enter intersectionDep_A1 Vehicles depart intersectionlti_A1 Vehicles left-turn inrti_A1 Vehicles right-trun inalt_A1 Left-turn-in vehicles arrive

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

ROUT

arr

rto

ent

INT

IN

RDY

RSL

sf

G_B2

OUT

LIN

RIN

G_B1

LARdep

lti alt

rti

lto

tlo

LTO

LOTGA_A1

Control Part

The traffic flow model of 4-phase intersection.

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4. Modeling and Performance

Evaluation of Traffic Network

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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.

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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

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STPN Model of 2-phase Intersection

IN_A2

LOUT_A2

ROUT_A2

G_A2

ea_A2

GA_B

RDY_A2

arr_A2

rto_A2

lto_A2

ent_A2

INT_A2

OUT_A2

LIN_A2

ROUT_A1

G_B2

LAR_A2

dep_A2lti_A1

alt_A2

rti_A2

eg_A2

RSL_A2

nrto_A2

nlto_A2

MF_A2G_A1

ea_A1

eg_A1

A_A1

A_A2

GA_A1

IN_A1

ent_A1

INT_A1

OUT_A1

dep_A1

RSL_A1

nrto_A1

nlto_A1

MF_A1

rto_A1

ROUT_A1

RIN_A1

LOUT_A2

rti_A1

GA_A2lti_A

arr_A1

RDY_A1

LOUT_A1

lto_A1

West East North South

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STPN Model of 4-phase Intersection

IN_A2 RIN_A2

G_A2

ea_A2

GA_A2

RDY_A2

arr_A2

rto_A2

lto_A2

ent_A2

OUT_A2

LIN_A2

RIN_A1

G_B2

LAR_A2dep_A2

lti_A1

alt_A2

rti_A2

RSL_A2

sf_A2

MF_A2

G_A1

ea_A1

eg_A1

A_A1

A_A2

GA_A1

IN_A1

ent_A1

OUT_A1

dep_A1

RSL_A1

sf_A1

MF_A11

rto_A1RIN_A1

RIN_A1

LOUT_A2

rti_A1

GA_B1 lti_A2

arr_A1

RDY_A1

lto_A1

eg_B2

eg_B1

G_B1

G_B2

eg_A2

G_B1

OUT_A1

G_B2

LOUT_A1

LTO_A1

tlo_A1tlo_A2

LTO_A1

West East North South

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Compositional Analysis

1 2 3 … … m-2 m-1 m

1

2

3

n-2

n-1

n

j

ii+j=2 i+j=3 i+j=4

Analyze intersections along the increasing i+j line

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Example

Intersection Offset AQL ATD Offset AQL ATD(0,0)(4p) 0 10.5575 35.433 0 10.5575 35.433(0,1)(2p) 60 10.808 37.9069 30 9.80259 34.217(1,0)(4p) 60 13.7656 47.0608 30 10.7584 36.273(0,2)(2p) 0 9.87869 33.9058 60 3.95742 13.4698(1,1)(4p) 0 7.93539 26.9096 60 10.7768 35.9995(2,0)(2p) 0 5.54243 19.0341 60 8.39731 29.021(0,3)(4p) 60 7.33246 24.0968 90 9.24824 30.4988(1,2)(2p) 60 4.95944 17.8978 90 8.93898 31.7011(2,1)(4p) 60 18.0724 57.2082 90 13.8584 43.9798(3,0)(2p) 60 5.03002 16.7118 90 9.83201 32.6142(1,3)(4p) 0 14.7134 53.6546 0 10.2563 37.0925(2,2)(2p) 0 15.1647 48.6528 0 12.8156 41.1378(3,1)(4p) 0 13.8706 44.8415 0 12.4933 39.9578(2,3)(2p) 60 8.97933 30.5812 30 9.95824 34.2142(3,2)(4p) 60 5.99334 18.6261 30 16.4617 50.5948(3,3)(2p) 0 7.10573 25.482 60 12.1873 43.395

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Discussion

Dimension of State Space (number of places used in PN model)

Our model: 27;

Global models: 3316 = 528. Number of Reachable States (suppose that there are M reachable

states for each intersection in average)

Our model: 16M;

Global models: M16.

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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.