CHAOS CONTROL AND SCHEDULE OF SHUTTLE BUSES Hyunwoong Chang Yiling - PowerPoint PPT Presentation

CHAOS CONTROL AND SCHEDULE OF SHUTTLE BUSES Hyunwoong Chang Yiling Ding Zichao Guo Ranfei Jiang Mingjun Zhou

TRAFFIC FLOW Traffic flow is the study of interactions between travelers and infrastructure. The aim of studying it is understanding and developing an optimal transport network to create the effective traffic system and solve the minimal traffic congestion problems.

SCHEDULE OF SHUTTLE BUSES • Chaos control and schedule of shuttle buses , by Takashi Nagatani, 15 February 2006 • Why we choose this article? • - close to real life • - Bus is a main transportation

INTRODUCTION • The bus schedule is closely related to the dynamic motion of the bus. The bus shuttle schedule is influenced by complex factors • Influencing factors: • - Weather • - Traffic jam • - Passengers • - …..

SET UP BUS 1 • The motion of a shuttle bus depends on the inflow rate of passengers and the delayed speedup control. The delayed speedup control has an important effect on the dynamic motion of the bus. In this article, we simplify the • influencing factors, we only consider BUS 2 the loading parameter ( Γ ) and speedup parameter (S). Bus 2 takes more time on boarding passengers

• In order to particularly discuss the problem of bus shuttle schedule, we further simplify it into a circular system.

Basic Concept • Headway : the time interval between the arriving time of previous bus and the arriving time of ‘bus i’ for m-th trip. • Example: Suppose you are given a table below . Find the headway of bus 2 for 4-th trip. # trip 1 2 3 4 5 6 Bus 1 0.1 0.2 0.3 0.4 0.5 0.6 2 0.2 0.23 0.34 0.41 0.56 1 3 0.05 0.1 0.2 0.33 0.35 0.405

Basic Concept • Headway : the time interval between the time of previous arriving bus and ‘bus i’. • Example: Suppose you are given a table below . Find the headway of bus 2 for 4-th trip. H 2 (4) = T 2 (4) - T 3 (6) = 0.41 - 0.405 = 0.005 # trip 1 2 3 4 5 6 Bus 1 0.1 0.2 0.3 0.4 0.5 0.6 2 0.2 0.23 0.34 0.41 0.56 1 3 0.05 0.1 0.2 0.33 0.35 0.405

Basic Concept • Headway :the time interval between the arriving time of previous bus and the arriving time of ‘bus i’ for m-th trip. • Role of Headway : headway determines how many people are waiting for the ‘bus i’. → The longer the headway of ‘bus i’ is, The more people the bus should take. In turn, the bus take longer trip. → The shuttle bus company would like to control the headway.

Basic Concept • Tour time : the time spent of ‘bus i‘ for m-th trip. # trip 1 2 3 4 5 6 Bus 1 0.1 0.2 0.3 0.4 0.5 0.6 2 0.2 0.23 0.34 0.41 0.56 1 3 0.05 0.1 0.2 0.33 0.35 0.405

Explanation of main equation M shuttle buses on operation. 2L Main objective : to mark arriving time back to the origin for every trip of each bus.

Explanation of main equation We can intuitively think that there are two components consisting of tour time for each • bus. Moving time on the road Boarding and taking off time ϒ : one passenger’s boarding time. Vi(m) : average speed of ‘bus i’ ɳ : one passenger’s getting off time. at m-th trip. Bi(m) : the number of passengers boarding ‘bus i’ at m-th trip.

Explanation of main equation We are able to modify the equation a little bit. •

Explanation of main equation We finally get the equation by reducing the number of parameters •

Reproduce Detail ● 4 groups of graphs ○ 12 graphs for the first two groups ○ 8 graphs for the latter two groups ○ 2 million ● MATLAB ● Difference: diverge limit (1 instead of 2) ○ parameter ○ running environment ● Format

Detail of graphs Time headway of bus 1 against loading parameter Γ from trip m=900-1000

Detail of graphs E Enlargement for 0 < Γ < 0.5

Detail of graphs Time headway of bus 1 against loading parameter Γ from trip m=900-1000

Detail of graphs Enlargement for 0 < Γ < 0.5

Detail of graphs Time headway of bus 1 against loading parameter Γ from trip m=900-1000

Detail of graphs Enlargement for 0< Γ <0.5

Detail of graphs Time headway of bus 1 against loading parameter Γ from trip m=900-1000 E

Detail of graphs 3 dynamical 2 1 Enlargement for 0 < Γ < 0.5

Detail of graphs Tour times of bus 1 against loading parameter from trip m=900-1000 with S 1 =0.5, S 2 =0.2

Detail of graphs Enlargement of 0 < Γ < 0.5

Detail of graphs Tour times of bus 2 against loading parameter from trip m=900-1000

Detail of graphs Enlargement of 0 < Γ < 5

Detail of graphs

Detail of graphs H1(m+1) against H1(m) from m = 1000 to 2000

Detail of graphs H1(m+1) against H1(m) from m = 1000 to 2000

Detail of graphs H1(m+1) against H1(m) from m = 1000 to 2000

Detail of graphs H1(m+1) against H1(m) from m = 1000 to 2000

Definition ● Mean (arithmetic) ○ the sum of the sampled values divided by the number of items in the sample ○ ● Root-mean square (rms) ○ the square root of the arithmetic mean of the squares of the values ○

Detail of graphs Mean headways H1a, H2a and tour times DT1a, DT2a

Detail of graphs Enlargement for 0 < Γ < 0.5

Detail of graphs Mean headways H1a, H2a and tour times DT1a, DT2a

Detail of graphs Enlargement for 0 < Γ < 0.5

Detail of graphs Root-mean square’s of headways H1a, H2a and tour times DT1a, DT2a

Detail of graphs Enlargement for 0 < Γ < 0.5

Detail of graphs Phase diagram (region map) for the regular and periodic (or chaotic) motions

Summary: Based on the simulation result, we conclude that: • the variation of arrival time is due to the chaos and chaotic motion is dependent on both parameters. • the dynamical transition occurs (from regular to periodic or chaotic motion) when loading passengers are more than the threshold, which is dependent on the speedup parameter.

Future Work: We are going to continue working on this topic and our goal is to find a way to make bus schedule regular rather than chaotic. • As the dynamical transition occurs when loading parameter is higher than the threshold, we expect the bus schedule would be regular if each bus has limited capacity (lower than the threshold). • We also plan to change the loading condition, from one bus loading up every passenger whenever the next bus arrives to two buses loading up at the same time if the previous one is still loading when the next one arrives.

Papers we’ve read: 1. Schedule and complex motion of shuttle bus induced by periodic inflow of passengers 2. Dynamics and schedule of shuttle bus controlled by traffic signal 3. The night driving behavior in a car-following model 4. A new stochastic cellular automata model for traffic flow simulation with drivers’ behavior prediction 5. Multiple car-following model of traffic flow and numerical simulation 6. Transitions to chaos of a shuttle bus induced by continuous speedup 7. Complex motions of shuttle buses by speed control 8. Jam formation in traffic flow on a highway with some slowdown sections 9. Theory and simulation for jamming transitions induced by a slow vehicle in traffic flow 10. Traffic jam and discontinuity induced by slowdown in two-stage optimal-velocity model 11. Effect of speedup delay on shuttle bus schedule

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