Problem $1$

Traffic leaving a major event enters a freeway at a major on$-$ramp located $2$ miles downstream of

the nearest off$-$ramp. The approaching freeway traffic is time independent with $q=6000vph$ and

$15\%$ of the traffic exits at the off$-$ramp $($leaving $5100vph$ of through traffic$).$ Assume that the q$-$k

relation is triangular with the following attribute: $q_{\text{m}\text{a}\text{x}}=6000vph,k_{op}=150ve\frac{h}{m}$ile, and $k_j=$

$600ve\frac{h}{m}$ile, where $q_{\text{m}\text{a}\text{x}}$ is the maximum flow, $k_{op}$ is the optimal density associated with $q_{\text{m}\text{a}\text{x}}$ and

$k_j$ is the jam density.

$($a$)$ The event traffic entering the freeway reduces the available capacity to the through vehicles as

follows:

Determine whether or not the freeway queue will back up to the off$-$ramp. To do this,

$($i$)$ Draw the fundamental diagram.

$($ii$)$ Draw a time$-$space diagram depicting the evolution of kinematic waves assuming that the

off$-$ramp does not exist, and the demand is $q=5100ve\frac{h}{h}r.$ Label clearly all traffic

states and compute all relevant wave speeds.

$($iii$)$ Compute the total distance that the queue would travel.

$($b$)$ Ramp metering is deployed at the on$-$ramp. For the traffic condition in part $($a$),$ determine the

$($constant$)$ metering rate during $0so$ that the queue does not back $upto$ the off$-$

ramp.

$(c)$ Repeat $(a)if$ the time$-$dependent available capacity $isas$ follows:

$(d)(Extra$ Credit$)$ Repeat $(a)if$ the time$-$dependent available capacity $isas$ follows: