Problem $1[20$ points$]$

An intersection has a four$-$phase signal with the movements allowed in each phase and corresponding analysis and saturation flow rates as shown in the following table:

$\backslash$table$[[\backslash$table$[[$Phase$],[$Movements$]],1,2,3,4],[$EB L$,$WB L$,$EB T$/$R$,$WB T$/$R$,$SB L$,$SB T$/$R$,$NB L$,$NB T$/$R$],[v[$veh$/$hr$],245,230,975,1030,255,235,225,215],[s[$veh$/$hr$],1750,1725,3350,3400,1725,1750,1700,1750]]$

The lost time is $4$ seconds per phase. Assume that the signal is isolated and it operates under pre$-$timed signal control. In addition, assume that the traffic flow accounts for the peak $15$min period, and the average delay per vehicle due to initial queue at start of the analysis time period, $d_3,$ is zero.

Calculate the following:

a$)$ The sum of the flow ratios for the critical lane groups

b$)$ The minimum cycle length if a critical intersection $\frac{v}{c}$ of $0\mathrm{.}95$ is desired

c$)$ The optimal cycle length

d$)$ The effective green time for each phase given the optimal cycle length such that the lane group $\frac{v}{c}$ ratios are equalized

e$)$ The EB average approach delays $($including both delay due to uniform arrivals and delay due to random arrivals$)$ and level of service $($LOS$)$

f$)$ Assuming that the WB average approach delay is $43\mathrm{.}67$ sec $,$ the SB average approach delay is $63\mathrm{.}28$ sec $,$ the NB average approach delay is equal to $67\mathrm{.}24$ sec $,$ and the EB is as found in part e$,$ determine the overall intersection average delay and level of service. Show all work