NEW VOLUME DELAY FUNCTION Wacław Jastrzębski

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NEW VOLUME DELAY FUNCTION Wacław Jastrzębski. The Overcapacity Problem. Volume>>Capacity. Problem. Using standard VDF functions, sometimes the forecasted demand results in volumes greater than capacity, whereas the actual capacity may in fact be sufficient .

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## NEW VOLUME DELAY FUNCTION Wacław Jastrzębski

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NEW VOLUME DELAY FUNCTIONWacław Jastrzębski

Scott Wilson Ltd – Poland Branch

The Overcapacity Problem

Volume>>Capacity

Scott Wilson Ltd – Poland Branch

Problem

Using standard VDF functions, sometimes the forecasted demand results in volumes greater than capacity, whereas the actual capacity may in fact be sufficient .

Scott Wilson Ltd – Poland Branch

Model – representation of human behaviors using the language of mathematic
• To travel or not to travel…. ?
• To the city center or closer to home ?
• By car or by transit ?
• Which route?

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Four Step Model

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Reality vs. Model Curves

capacity

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Mathematical Conditions for VDF Function

- F(x) is a strictly increasing function for the variable between 0 and + (F’(0)>0) - F(0) = T0, where T0 is the free-flow time; - F’(x) existing and is strictly increasing – that means that function is convex – this last condition is not essential but desirable;The calculation time for the new function should not use more CPU time than BPR function,

Scott Wilson Ltd – Poland Branch

Behavioral Conditions
• Time spent in traffic congestion weights much more for the traveler than the travel time at the acceptable speed;
• Within the range of 0.2-0.8 of capacity, the average speed of traffic shows little sensitivity to the volume of traffic. After reaching the capacity level the travel time increases substantially;
• Traveler chooses a path based on previous experience
• Traveler can adjust the path as new information on traffic situation is acquired.

Scott Wilson Ltd – Poland Branch

The Modeling Conditions
• The function should „force” the algorithm to seek additional paths in order to minimize the number of links with volume greater then capacity;
• The free-flow-speed is the actual average speed as determined through the surveys (regardless of legal limitations such as speed limits).
• The function takes into account that traffic lights decrease the average speed;

Scott Wilson Ltd – Poland Branch

Various Mathematical Formulas for VDF

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Various Mathematical Formulas for VDF

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Various Mathematical Formulas for VDF

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Surveys’ Results

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Surveys’ Results

t

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Surveys’ Results

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New Function

b – odd integer >1

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Mathematical Condition

b – odd integer >1 so b –1 is even

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Continuity

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New Function

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Scott Wilson Ltd – Poland Branch

Function and Surveys

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EMME Implementation

a fd27 =el1 * (1 + 1.35 * ((volau / el2) ^ 9) + .65 * volau / el2) +

.2 * (volau .gt. el2) * (volau - el2)

a fd30 =el1 * (1 + 100 * ((volau / el2 - .44) ^ 7 + .44 ^ 7) +

.45 * volau / el2) + .4 * (volau .gt. el2) * (volau - el2)

a fd31 =el1 * (1 + 90 * ((volau / el2 - .43) ^ 7 + .43 ^ 7) +

.44 * volau / el2) + .4 * (volau .gt. el2) * (volau - el2)

a fd32 =el1 * (1 + 70 * ((volau / el2 - .4) ^ 7 + .4 ^ 7) +

.3 * volau / el2) + .4 * (volau .gt. el2) * (volau - el2)

a fd33 =el1 * (1 + 28 * ((volau / el2 - .42) ^ 5 + .42 ^ 5) +

.28 * volau / el2) + .4 * (volau .gt. el2) * (volau - el2)

Scott Wilson Ltd – Poland Branch

Equilibrium Assignment

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Equilibrium Assignment

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HowDoes It Work?

- capacity 700 pcu/h

- free flow speed 70 km/h

- speed on the capacity limit 20 km/h

- practical capacity 0,65 capacity

- speed on the practical capacity limit ~ 45 km/h

Scott Wilson Ltd – Poland Branch

Various Functions

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Results for Various Functions

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Matrix Reduction to Eliminate Overcapacity

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NO FUNCTION IS PERFECT!

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ASSIGNMENT 2035

~132000 min

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Reason? No alternative paths

Scott Wilson Ltd – Poland Branch

Solution