Problem $1.$ You have collected speed and density data for a roadway segment, shown in the attached fileProblem $1.$ You have collected speed and density data for a roadway segment, shown in the attached file

$($speed$\_$density.csv$).$ In this data set, density is in vehicles per kilometer, and speed is in kilometers per

hour.

$($Greenshields model.$)$ Create a plot of the data along with a linear trendline, with speed on the vertical

axis and density on the horizontal axis. $($You can use Excel or any other spreadsheet or statistical tool

with these capabilities.$)$

Use the slope and intercept from your linear trendline to estimate the free$-$flow speed $u_f$ and jam

density $k_j$ for this roadway segment.

One criticism of the Greenshields model is that it oversimplifies traffic flow by assuming a linear

relationship between speed and density. An alternative is the Greenberg model which is derived from

fluid dynamics principles and assumes a logarithmic relation $u=u_{0}ln(\frac{k_j}{k}).$ Use the same data set

to estimate the parameters $u_0$ and $k_j$ for the Greenberg model. $($Hint: the Greenberg speed$-$density

relationship can be written as $u=u_{0}ln{k}_j-u_{0}lnk,$ so you can do a linear regression between $u$ and

$lnk.)$

Yet another speed$-$density relationship is the Underwood model with $u=u_f$exp$(-\frac{k}{k_0}).$ Use the same

data set to estimate the parameters $u_f$ and $k_0$ for the Underwood model. $($Hint: write the model as

$lnu=ln{u}_f-\frac{k}{k_0},$ so you can do a linear regression between $lnu$ and $k.)$

What density does each model predict on the roadway for a speed of $30kph?$

Compare the three models based on their fit to the measured data. Offer brief arguments to support

your statements.

The Greenshields model is criticized because the linear assumption is too simple. Yet no model is

perfect. Offer critiques of the Greenberg and Underwood models. $($You may want to consider what

happens to these models at very high or low densities or speeds.$)$

$($speed$\_$density.csv$).$ In this data set, density is in vehicles per kilometer, and speed is in kilometers per

hour.

$($Greenshields model.$)$ Create a plot of the data along with a linear trendline, with speed on the vertical

axis and density on the horizontal axis. $($You can use Excel or any other spreadsheet or statistical tool

with these capabilities.$)$

Use the slope and intercept from your linear trendline to estimate the free$-$flow speed $u_f$ and jam

density $k_j$ for this roadway segment.

One criticism of the Greenshields model is that it oversimplifies traffic flow by assuming a linear

relationship between speed and density. An alternative is the Greenberg model which is derived from

fluid dynamics principles and assumes a logarithmic relation $u=u_{0}ln(\frac{k_j}{k}).$ Use the same data set

to estimate the parameters $u_0$ and $k_j$ for the Greenberg model. $($Hint: the Greenberg speed$-$density

relationship can be written as $u=u_{0}ln{k}_j-u_{0}lnk,$ so you can do a linear regression between $u$ and

$lnk.)$

Yet another speed$-$density relationship is the Underwood model with $u=u_f$exp$(-\frac{k}{k_0}).$ Use the same

data set to estimate the parameters $u_f$ and $k_0$ for the Underwood model. $($Hint: write the model as

$lnu=ln{u}_f-\frac{k}{k_0},$ so you can do a linear regression between $lnu$ and $k.)$

What density does each model predict on the roadway for a speed of $30kph?$

Compare the three models based on their fit to the measured data. Offer brief arguments to support

your statements.

The Greenshields model is criticized because the linear assumption is too simple. Yet no model is

perfect. Offer critiques of the Greenberg and Underwood models. $($You may want to consider what

happens to these models at very high or low densities or speeds.$)$