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Using the Input-Output Diagram to Determine the

Spatial and Temporal Extents of a Queue

Upstream of a BottleneckTim W. Lawson

David J. Lovell

Carlos F. Daganzo

University of California at Berkeley

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Outline

Background Purpose and objective

Bottleneck with constant departure rate “Conventional” (time-space) Approach Proposed (input-output) Approach

Extensions to Approach Automation, varying capacity, traffic signal

Conclusions

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Background

Concepts of “Delay” and “Time in Queue” Delay actual time free flow time Time in Queue Delay for “point” queues Time in Queue Delay for traffic queues Concepts confused in the literature

Evaluation and MOEs Value of time Energy and emissions

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Motivation

Time-Space Diagram Approach clear distinction: Delay & Time in Queue (often) well understood difficult to construct

Objective clear up some of the confusion provide a simple approach based on

familiar tools (input-output diagram)

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Assumptions

Constant free-flow speed, vf

speed is constant, regardless of flow

Congested speed, v

speed is dependent on bottleneck capacity Typical time-space diagram assumptions

e.g., instantaneous speed changes

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“Conventional” Approach

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Conventional Approach

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Lessons From t-x Diagram

w dv v Qf

1 1

dw

Qv v f

1 1

tw

Q v

v f

1

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Basic Input-Output Diagram

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Proposed Approach

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Interpretation

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Interpretation

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Other Applications

Automation on a spreadsheet required: upstream arrival times, , vf, v

provides same measures Bottleneck whose capacity changes once

simple extension to above approach Undersaturated Traffic Signal

“limiting” case

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Conclusions

Simplicity modifies widely used and understood tool much less tedious than t-x; automation

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Conclusions

Simplicity modifies widely used and understood tool much less tedious than t-x; automation

Utility estimates of wait times, etc.; impacts queue lengths; time of maximum queue

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Conclusions

Simplicity modifies widely used and understood tool much less tedious than t-x; automation

Utility estimates of wait times, etc.; impacts queue lengths; time of maximum queue

Superiority corrects significant misunderstanding