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Two routes connect an origin and a destination. Routes 1 and 2 have performance functions t1 = 2 + X1 and t2 = 1 + X2, where the t's are in minutes and the x's are in thousands of vehicles per hour. The travel times on the routes are known to be in user equilibrium. If an observation for route 1 finds that the gaps between 30% of the vehicles are less than 6 seconds. Estimate the volume and average travel times for the two routes

Respuesta :

Solution :

Given

[tex]$t_1=2+x_1$[/tex]

[tex]$t_2=1+x_2$[/tex]

Now,

[tex]$P(h<5)=1-P(h \geq5)$[/tex]

[tex]$0.4=1-P(h \geq5)$[/tex]

[tex]$0.6=P(h \geq5)$[/tex]

[tex]$0.6= e^{\frac{-x_1 5}{3600}}$[/tex]

Therefore,   [tex]$x_1=368 \ veh/h$[/tex]

                        [tex]$=\frac{368}{1000} = 0.368$[/tex]

Given,   [tex]$t_1=2+x_1$[/tex]

                 = 2 + 0.368

                 = 2.368 min

At user equilibrium, [tex]$t_2=t_1$[/tex]

∴  [tex]$t_2$[/tex] = 2.368 min

[tex]$t_2=1+x_2$[/tex]

[tex]$2.368=1+x_2$[/tex]

[tex]$x_2 = 1.368$[/tex]

[tex]$x_2 = 1.368 \times 1000$[/tex]

    = 1368 veh/h

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