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quiz_3_cee3604_2011_solution

quiz_3_cee3604_2011_solution - Name Signature CEE 3604...

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CEE 3604: Introduction to Transportation Engineering Fall 2011 Exam 3 (Open Notes) Problem 1 (40 Points) Data collected at a weigh station near Roanoke, Virginia suggests that during the peak-hour period, 90 trucks per hour arrive randomly (i.e., Poisson) to the station. The station has one weighing scale at the site (see Figure 1). A survey reveals that a truck takes an average of 35 seconds (with a negative exponential distribution) going through the station to obtain an accurate weight measurement. a) Find the expected number of trucks at the weigh station (including those in service) during the peak-hour period of the typical day. Show your hand calculations. Solution: Identify the queue system as an M/M/1 queueing system. Infinite source, single server. The equations for the M/M/1 system are given on page 58 of the queueing system theory handout and reproduced below for completeness. Identify the queueing parameters: λ , μ and s Name _________________________________________ Signature _______________________________________ CEE 3604 Exam 3 Trani Page 1 of 11
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The number of trucks at the weigh station is (L). The equation for L (expected number of trucks in the queueing system) is: L = λ μ λ L = 1.5 trucks/min (1.7143 1.5) trucks / min L = 7 trucks b) The weigh station was built in the 1980’s with a total holding capacity of 8 trucks (see Figure 2). Find the probability that more than 8 trucks are simultaneously at the weigh station and could disrupt the traffic on the right-hand lane of the interstate highway. The probability that more than 8 trucks are at the station is calculated from: P ( x > 8) = 1 P n n = 0 8 P ( x > 8) = 1 ( P o + P 1 + ... + P 8 ) Find the probabilities from equation: P n = ρ n P o where: ρ = λ μ = 0.875 Use Matlab or hand computations to find the probabilities from zero to eight. Table 1 shows the numerical values of
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