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A) A roadway with a capacity of 4000 vehicles per hour (vph) has 15-minute arrivals of 1000, 500, 500, and 2000 during the peak hour. What is the queue at the end of the hour most likely to be?

B) A 3-foot detector is used along a highway to help measure density. What is the density (in vehicles per mile per lane, vpmpl) if the detector is occupied 50 percent of the time? Assume a vehicle length of 28 feet.

C) If vehicles arrive at a rate of 360 vehicles per hour (vph), what is the probability of exactly 2 vehicles arriving in 20 seconds? (Round to two decimals)

D) How many vehicles have "arrived" if, for 3 consecutive 15-minute periods, the arrival rates are 240 vph, 400 vph, and 600 vph?

Answer :

B) The density in vehicles per mile per lane is 20.77 vpmpl. (C) the total number of vehicles that have arrived in the three 15-minute periods is 6240.

A) The queue at the end of the hour can be calculated by using the following formula:Lq = λ^2 / (μ (μ- λ)) where λ= traffic arrival rate = (1000+500+500+2000)/15= 2000 vphμ= roadway capacity= 4000 vphLq = (2000)^2 / (4000(4000- 2000))Lq = 1000/2Lq = 500 vehicles. Therefore, the queue at the end of the hour is most likely 500 vehicles.B) We can calculate the flow rate by using the following formula:Q = ρV where ρ = density and V = velocity.To calculate the density, we can use the following formula:ρ = N / L, where N is the number of vehicles occupying the detector and L is the length of the detector.ρ = (N / L) * 5280/60, where 5280 is the number of feet in a mile, and 60 is the number of minutes in an hour.We can calculate the number of vehicles using the following formula:

N = (0.5)(3600)(ρ)(3 / 28), where 0.5 is the occupancy rate, 3600 is the number of seconds in an hour, and 3/28 is the length of a vehicle in miles.ρ = N / L * 5280/60 = (0.5)(3600)(ρ)(3 / 28)(5280/60)L = 3 ft = 0.000568 miles.Substituting these values in the equation:N = (0.5)(3600)(ρ)(3 / 28)ρ = (N / L) * 5280/60ρ = ((0.5)(3600)(3 / 28)) / (N / L) * 5280/60ρ = 0.0118 / (N / 0.000568)ρ = 20.77N/LTherefore, the density in vehicles per mile per lane is 20.77 vpmpl.C) We can use the Poisson distribution formula to calculate the probability of exactly 2 vehicles arriving in 20 seconds:P(2) = (e^-λ λ^2) / 2!, where λ= traffic arrival rate = 360 vph / 3600 s/h = 0.1 s^-1P(2) = (e^-0.1 (0.1)^2) / 2!P(2) = 0.0045 rounded to 2 decimal places.D) We can calculate the total number of vehicles that have arrived in the three 15-minute periods as follows:N = (240 + 400 + 600) × 3N = 6240 vehicles have "arrived."

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