graphical analysis ii

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Wu Graphical analysis II Cumulative diagrams Cathy Wu 1.041/1.200/11.544 Transportation: Foundations and Methods 2021-09-15

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Page 1: Graphical analysis II

Wu

Graphical analysis IICumulative diagrams

Cathy Wu

1.041/1.200/11.544 Transportation: Foundations and Methods

2021-09-15

Page 2: Graphical analysis II

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References1. Some slides adapted from Profs. Carolina Osorio and Dan

Work.

2. Chap 2. of Prof. Carlos Daganzoā€™s book Fundamentals of Transportation and Traffic Operations (2007)

3. Electronic version available via MIT Librairies Catalog: Barton

4. Prof. Nikolas Geroliminisā€™ lecture Fundamentals of Traffic Operations and Control, Spring 2010 EPFL

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Outline1. Introduction to cumulative diagrams, queues, and delays

2. Cumulative diagrams vs. time-space diagrams

3. Working with cumulative diagrams

4. Analysis & interpretation

5. Littleā€™s Law

6. Application: Signalized intersections

7. Reconstruction

8. Ramp metering problem

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Ā§ Cumulative count curve is started from a basic technique known as ā€˜mass curve analysisā€™ in hydrologic synthesis

Ā§ Introduced to transportation by Moskowitz (1954)[2] and Gazis and Potts (1965)[3]; Gordon Newell (1971,1982,1993)[1,4,5] demonstrated their full potential.

Ā§ The following diagram represents the cumulative number (count) ofarrivals at a fixed location

Ā§ Cumulative diagram also calledcumulative count curve, N-curve, queueing diagram

Cumulative diagram

Slide adapted from Prof. Zhengbing He

1. Newell GF. Applications of queueing theory (2nd edition). Chapman and Hall London, 1982.2. Moskowitz, K. Waiting for a gap in a traffic stream. Proc. Highway Res. Board, 33, 385-395, 1954.3. Gazis, D.C. and R.B. Potts (1965). The over-saturated intersection. Proc. 2nd Int. Symp. on the Theory of Road Traffic Flow.4. Newell, G.F. Applications of queueing theory, Chapman Hall, London, 1971.5. Newell, G.F. A simplified theory of kinematic waves in highway traffic, I general theory, II queuing at freeway bottlenecks,

III multi-destination flows. Transportation Research Part B, 1993.

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Ā§ We can use two observers:ā€¢ Observer A looks at the arrivals to the systemā€¢ Observer D looks at the departures from the system

Ā§ The following diagram represents the cumulative number of arrivals and departures

Cumulative diagram: illustrating system delaysObserver D Observer A

0,ā€¦

Ā§ š‘„ š‘” = number of customers in queue at time š‘”

Ā§ š‘¤ š‘› = waiting time of person š‘›Ā§ š“' = time of arrival of the š‘›()

customerĀ§ š·' = time of departure of the š‘›()

customerĀ§ Area = total wait time in the system

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Cumulative diagrams - ApplicationsĀ§ Applications of this technique appear in all transportation modes.Ā§ Are mainly used to evaluate waiting time and queue lengths.Ā§ Examples:ā€¢ Predict queues caused by scheduled maintenance or accidentsā€¢ Design a suitable ā€œwaiting areaā€: e.g. length of an on-rampā€¢ Control purposes: e.g. determine suitable traffic signal plans

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Outline1. Introduction to cumulative diagrams, queues, and delays

2. Cumulative diagrams vs. time-space diagrams

3. Working with cumulative diagrams

4. Analysis & interpretation

5. Littleā€™s Law

6. Application: Signalized intersections

7. Reconstruction

8. Ramp metering problem

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Cumulative diagram vs. time space diagramā€¢ Cumulative diagram can be obtained from the

time-space diagramā€¢ Time space diagram has complete infoā€¢ Cumulative diagram has a subset of the infoā€¢ Note:

Slope of cumulative diagram = flow

ā€¢ Use cases:ā€¢ Time-space diagram: when depicting

more than one vehicle trajectory;ā€¢ Cumulative diagram: when depicting

aggregations of multiple vehicles at fixed points (bottlenecks), and in particular the ā€˜arrivalā€™ and ā€˜departureā€™ curves at a single bottleneck.

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Ā§ Flow at š‘„: š‘ž š‘„ = ./ 0,(.(

Ā§ Density at š‘”: š‘˜ š‘” = ./ 0,(.0

Slide adapted from Prof. Zhengbing He

Three-dimensional N-curve: N(x, t)

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Outline1. Introduction to cumulative diagrams, queues, and delays

2. Cumulative diagrams vs. time-space diagrams

3. Working with cumulative diagrams

4. Analysis & interpretation

5. Littleā€™s Law

6. Application: Signalized intersections

7. Reconstruction

8. Ramp metering problem

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Cumulative diagram ā€“ discrete Ć  continuousĀ§ The main advantage for analysis is that differential calculus can be used:

ā€¢ š‘ž š‘” ā‰ˆ ! "# $!$

Ā§ In most transportation applications cumulative count is made up of discrete objects (e.g. passengers, buses, cars, etc... ), so š‘ š‘” is a step function

Ā§ However, in applications dealing with many moving objects, where predicting the exact number to the nearest integer is not important, it is preferable to work with a smooth approximation of š‘ š‘” , "š‘ š‘” , e.g.an interpolation

Ā§ Typical applications of cumulative plots involve long observation periods where many observations are collected. In these cases, the individual steps of the curve cannot be seen.

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Ā§ Seconds or minutes: queue (a)

Ā§ Hours:peak demand or rush hours (b,c);

Ā§ Days:different patterns on different days (d);

Scales of counts and time

Slide adapted from Prof. Zhengbing He

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Ā§ In freeways accumulation of arrival counts is very fast since the flow is large. The detail is hard to observe in the regular cumulative diagram.

Ā§ An oblique coordinate system is used to show arrival changes, through a linear transformation, which keeps the relative quantity (See, for example, [1, 2]):

š‘, š‘” āŸ¹ š‘ āˆ’ š‘ž+š‘”, š‘”where š‘ž+ is a coefficient.

Oblique cumulative diagram

Slide adapted from Prof. Zhengbing He

1 Cassidy M. Bivariate relations in nearly stationary highway traffic. Transportation Research Part B. 1998.2 Cassidy M. Some traffic features at freeway bottlenecks. Transportation Research Part B. 1999;33:25-42.

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Ā§ A 3-lane section of the North-bound M42 freeway near Birmingham international airport, England

Ā§ Covered by detectors every 50-100mĀ§ Data: 7 selected days in 2008, counts aggregated every one minute from

3:10pm to 6:20pm[1]

Ā§ Locations š‘„- = 10.9, š‘„3 = 11.646 and š‘„6 = 12.179 km.Raw oblique cumulative diag. Corrected oblique cumulative diag.

Cumulative diagrams in practice

Slide adapted from Prof. Zhengbing He1. Laval, J. A., Z. He, and F. Castrillon. "Stochastic extension of Newell's three-detector method." Transportation Research Record 2315 (2012).

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Outline1. Introduction to cumulative diagrams, queues, and delays

2. Cumulative diagrams vs. time-space diagrams

3. Working with cumulative diagrams

4. Analysis & interpretation

5. Littleā€™s Law

6. Application: Signalized intersections

7. Reconstruction

8. Ramp metering problem

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Ā§ Arrivals at š‘„3; departures at š‘„4Ā§ Starting counting with a reference

customer / vehicle;ā€¢ Let š“(š‘”) denote the arrivalsā€¢ Let š·(š‘”) denote the departuresā€¢ Let šœ‡ denote the service rate

(customers / time)

Continuous cumulative arrivals and departures

n0

t0

Slide adapted from Prof. Zhengbing He

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Analysis ā€“ Virtual departuresĀ§ Total time in the system is composed of

ā€¢ ā€œService timeā€: time to go through the system independently of traffic conditionsĀ§ E.g. travel time along a link in uncongested conditions: free flow travel time (fftt)

ā€¢ Delay: additional time in system due to congestionĀ§ To better analyze and interpret the queueing system, we shift š“(š‘”) right by

fftt, and obtain the virtual arrival count curve š‘‰(š‘”) = š“(š‘” āˆ’ fftt).Ā§ š‘‰(š‘”): represents the departure time under no delay (congestion/queueing)

Slide adapted from Prof. Zhengbing He

Ā§ The delay of vehicle š‘›:š‘¤ š‘›Ā§ Queue at š‘”D ā‰ˆ excess vehicle

accumulation: š‘„ š‘”DĀ§ Total Delay:

š‘‡š· = H$!

$"š‘‰ š‘” āˆ’ š· š‘” š‘‘š‘”

= āˆ«$!$" š‘„ š‘” š‘‘š‘”

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Analysis ā€“ Littleā€™s Law (deterministic case)

Ā§ The delay of vehicle š‘›:š‘¤ š‘›Ā§ Queue at š‘”D ā‰ˆ excess vehicle

accumulation: š‘„ š‘”DĀ§ Total Delay:

š‘‡š· = H$!

$"š‘‰ š‘” āˆ’ š· š‘” š‘‘š‘”

= āˆ«$!$" š‘„ š‘” š‘‘š‘”

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Littleā€™s LawĀ§ Beauty is in its simplicity:

Kš‘„ = šœ†Mš‘¤Ā§ Previous slide: deterministic proof.

Ā§ Also holds for probabilistic settings.Weā€™ll revisit in Unit 2.

Ā§ Assumptions:ā€¢ System stabilityā€¢ No pre-emption

Ā§ Preview: Littleā€™s Law is independent of:ā€¢ Arrival process distributionā€¢ Service distributionā€¢ Service orderā€¢ Structure of queue(s)ā€¢ Etc.

Ā§ Applications:ā€¢ Highway traffic, transit, airportsā€¢ Shopsā€¢ Manufacturing plantsā€¢ Bank tellersā€¢ Emergency roomsā€¢ Operations managementā€¢ Computer architecture (webserver requests,

CPU, DRAM, RAM, HDD)

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Outline1. Introduction to cumulative diagrams, queues, and delays

2. Cumulative diagrams vs. time-space diagrams

3. Working with cumulative diagrams

4. Analysis & interpretation

5. Littleā€™s Law

6. Application: Signalized intersections

7. Reconstruction

8. Ramp metering problem

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Signalized intersectionsĀ§ Some definitions ā€¢ Cycle: time needed for one complete color change sequence; it is made up of

phases.ā€¢ Phase: (e.g. red, green, yellow, left turn only) the part of a cycle that is

allocated to one or more movements. ā€¢ Effective green: the time when vehicles can enter the intersection (green +

part of the yellow)ā€¢ Lost time: the time in one cycle when no one can enter the intersection from

any approach (e.g. every time phases change there is some lost time, all-red phase)ā€¢ Saturation flow: service rate (traffic flow) during green

approach 2 (E-W)

approach 1 (N-S)Signal timing (two approaches)

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Some typical timing plansĀ§ Simple two phase timing plan

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Some typical timing plansĀ§ Three phase timing plan (dual left on N-S)

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Some typical timing plansĀ§ Four phase timing plan (lead-lag on E-W)

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Queuing diagram applicationĀ§ Example: On-off service (traffic signal basics)ā€¢ Given: Arrival flow (rate) at a one way signalized intersection is constant šœ†;

we have information about the green/red timing plan of the signal (server), and the service rate during green šœ‡ā€¢ Find: The departure curve, total delay, average delay

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Deterministic intersection delay

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Outline1. Introduction to cumulative diagrams, queues, and delays

2. Cumulative diagrams vs. time-space diagrams

3. Working with cumulative diagrams

4. Analysis & interpretation

5. Littleā€™s Law

6. Application: Signalized intersections

7. Reconstruction

8. Ramp metering problem

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Cumulative diagrams - Reconstructing departuresĀ§ Example: Constructing departure curve ā€¢ We often have incomplete info. We might have š‘‰(š‘”) or š“(š‘”) and the

operating features of the server (e.g., constant service rate šœ‡), but we need š·(š‘”)ā€¢ Rule:

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Ā§ We may have incomplete information:ā€¢ E.g. we know š“(š‘”) (or š‘‰(š‘”)) and the

operating characteristics of the server/system (e.g. service rate Āµ(š‘”)), but we need to find š·(š‘”)

Ā§ If š· š‘” < š“ š‘” , then #$ %#%

= šœ‡ š‘”

Ā§ Otherwise, #$ %#%

= #& %#%

Cumulative diagrams - Reconstructing departures (soln)

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Outline1. Introduction to cumulative diagrams, queues, and delays

2. Cumulative diagrams vs. time-space diagrams

3. Working with cumulative diagrams

4. Analysis & interpretation

5. Littleā€™s Law

6. Application: Signalized intersections

7. Reconstruction

8. Ramp metering problem

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Ramp metering problemĀ§ A ramp meter is being considered at an entrance to a freeway.Ā§ Currently, rush hour traffic arrives at the on-ramp at a rate š‘žO from time š‘” =0 to time š‘” = š‘”āˆ—. After š‘” = š‘”āˆ—, vehicles arrive at a (lower) rate š‘žD.

1. Assuming that drivers will not change their trips, draw and label a cumulative plot showing a metering (i.e. departure) rate of šœ‡(š‘žD < šœ‡ <š‘žO).ā€¢ Label the maximum delay experienced by any vehicle, š‘¤RST .ā€¢ What is š‘¤RST as a function of š‘žO, š‘žD, šœ‡, and š‘”āˆ—?

2. If an alternate route is available to drivers, and it is known that they will take this route if their expected delay at the ramp meter is greater than U'()D

, add this new scenario to your diagram. Now, show graphically the following:

a) The number of vehicles which will divert.b) How much earlier the queue will dissipate (compared to part 1)?

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Ramp metering problemN

t0

Ā§ A ramp meter is being considered at an entrance to a freeway.

Ā§ Currently, rush hour traffic arrives at the on-ramp at a rate š‘ž+ from time š‘” = 0 to time š‘” =š‘”āˆ—. After š‘” = š‘”āˆ—, vehicles arrive at a (lower) rate š‘ž0.

1. Assuming that drivers will not change their trips, draw and label a cumulative plot showing a metering (i.e. departure) rate of šœ‡(š‘ž0 < šœ‡ < š‘ž+).ā€¢ Label the maximum delay experienced by any

vehicle, š‘¤780 .ā€¢ What is š‘¤780 as a function of š‘ž4 , š‘ž9 , šœ‡, and š‘”āˆ—?

2. If an alternate route is available to drivers, and it is known that they will take this route if their expected delay at the ramp meter is greater than 5;<=

0, add this new scenario to

your diagram. Now, show graphically the following:

a) The number of vehicles which will divert.b) How much earlier the queue will dissipate

(compared to part 1)?