1 - Purdue University



CE 361 Introduction to Transportation Engineering |Out: Fri. 9 September 2005 | |

|Homework 3 (HW 3) Solutions |Due: Fri. 16 September 2005 |

HIGHWAY DESIGN FOR PERFORMANCE

• You will be permitted to submit this HW with one or two other CE361 students.

• For every problem, identify the problem by its number and name, be clear, be concise, cite your sources, attach documentation (if appropriate), and let your methodology be known.

1. Poisson models for natural events. A member of the County Highway Department staff discovers a list of Atlantic hurricanes since 1851 at . Having become an expert in the use of Poisson models to analyze traffic, he decides to look at the number of Atlantic storms listed for each year, 1985 to 2004, as a Poisson process. For example, there were 17 Atlantic storms in 2004, ranging from Tropical Depressions and Tropical Storms to five categories of hurricanes.

A. (5 points) For the years 1985-2004, what was the average “arrival rate” for Atlantic storms?

[pic]= 11.5 storms/yr. See spreadsheet table below.

B. (15 points) Given the arrival rate calculated in Part A, calculate P(n) for the range of events per year that appear in the website’s archives for 1985-2004. If you use a spreadsheet, show one P(n) calculation by hand.

Range of n is 6[pic]n[pic]19. See spreadsheet table below. Sample calculation:

(2.24) P(n) = [pic]; P(7) = [pic]= 0.053465

C. (5 points) What is the probability that the number of Atlantic storms in 2005 will exceed 21?

P(n>21) = 1 – P(n[pic]21) = 1 – 0.9962 = 0.0038

Year |TD |TS |1 |2 |3 |4 |5 |Total | |n |P(n) |sum | |2004 |2 |7 |1 |1 |2 |3 |1 |15 | |0 |0.000010 | | |2003 |5 |9 |3 |1 |1 |1 |1 |16 | |1 |0.000116 | | |2002 |2 |8 |2 |0 |1 |1 |0 |12 | |2 |0.000670 | | |2001 |2 |6 |4 |1 |2 |2 |0 |15 | |3 |0.002568 | | |2000 |4 |6 |4 |1 |1 |2 |0 |15 | |4 |0.007382 | | |1999 |4 |4 |0 |3 |0 |5 |0 |12 | |5 |0.016979 | | |1998 |0 |4 |3 |4 |1 |1 |1 |14 | |6 |0.032544 | | |1997 |1 |4 |2 |0 |1 |0 |0 |8 | |7 |0.053465 | | |1996 |0 |4 |3 |0 |4 |2 |0 |13 | |8 |0.076856 | | |1995 |2 |8 |4 |2 |2 |3 |0 |19 | |9 |0.098204 | | |1994 | | | | | | | |7 | |10 |0.112935 | | |1993 | | | | | | | |8 | |11 |0.118068 | | |1992 | | | | | | | |7 | |12 |0.113149 | | |1991 | | | | | | | |8 | |13 |0.100093 | | |1990 | | | | | | | |14 | |14 |0.082220 | | |1989 | | | | | | | |11 | |15 |0.063035 | | |1988 | | | | | | | |12 | |16 |0.045306 | | |1987 | | | | | | | |7 | |17 |0.030648 | | |1986 | | | | | | | |6 | |18 |0.019581 | | |1985 | | | | | | | |11 | |19 |0.011852 | | | | | | | | | | |230 | |20 |0.006815 | | | | | | | | | |λ’ |11.50 | |21 |0.003732 |0.9962 | |

2. (10 points) Time between events. Problem 2.48 in FTE.

Crash rate, λ = 37 collisions/ year or 37/52 = 0.711 collisions per week.

Solve for P(n>1) using Equation (2.24) and Time interval t = 1 week. Start with

[pic] = 0.491. Then Pr (n>1) = 1 – P (0) = 1 – 0.492 = 0.5093

3. (20 points) LOS on Two-Lane Highways. Problem 3.2 in FTE. Note: “PTSF-based LOS” in Part (a) should read “ATS-based LOS”.

A. In Example 3.1, ATS = FFS – 0.00776 * vp – fnp = 51.5 – 12.07 – 0.82 = 38.6 mph.

For LOS D, ATS>40 mph. Rearrange the ATS equation with ATS = 40 mph:

0.00776 * vp = = 51.5 – 40 – 0.82 = 10.68; vp must be [pic]= 1376 pcph.

In rearranged Equation 3.2, fHV = [pic]=1.06.

Because fHV > 1.0 is not possible, even PT = 0 will not make ATS = 40 mph and LOS D.

B. In Example 3.1, ATS = FFS – 0.00776 vp – fnp = 51.5 – 12.07 – 0.82 = 38.6 mph.

For LOS D, ATS>40 mph. Access point density fA appears in Equation 3.6 for FFS, which needs to be 51.5 + 1.4 = 52.9 mph. By rearranged (3.6), fA = 55.0 - 52.9 - 1.3 = 0.80. Interpolating in Exhibit 20-6, access point density = (0.80*2.5) * 10 = 3.2.

4. Left Turn Lane Analysis using Queueing Diagrams. The intersection of Coliseum Avenue and Wildcat Street near Mythaca State University has a signal with a 60-second cycle length. At the start of each cycle, a "protected left turn phase" is provided for EB traffic on Coliseum that is turning left onto NB Wildcat. This left turn (LT) phase is long enough to allow only 8 vehicles to turn left. Two conditions prevent an easy solution at the intersection: (a) Traffic near the university fluctuates during each hour, depending on the times at which classes begin or end. Between 15 and 25 minutes past each hour on MWF, an average of 11.75 drivers per minute try to enter the LT lane. During the rest of the hour, an average of 4.3 drivers per minute want to turn left. (b) Other approach volumes are too large to permit additional time to be given to the left turns made from the EB approach.

A. (15 points) Starting at 15 minutes past the hour, draw clearly a queueing diagram that shows the build-up and dissipation of the LT queue. Label the key elements in the queueing diagram. Use MK Figure 3.17 as a model.

See queueing diagram on next page.

B. Estimate the following values, showing the computations you use to support your estimates:

i) (5 points) Longest vehicle delay.

Maximum time in queue = length of horizontal dashed line from AC1/AC2 to DC. Vehicle number 10*11.75 = 117.5 arrives at t=10 minutes. The equation of DC2 is D = 8*t. Time needed to have Vehicle number 117.5 to leave queue is t = 117.5/8 = 14.69 minutes. The maximum time in queue = 14.69 – 10.0 = 4.69 minutes.

ii) (5 points) Longest vehicle queue.

Maximum length of queue = max length of vertical line from DC to AC1/AC2. At t=10 minutes, Vehicle number 117.5 has just entered the queue and Vehicle number 8*t = 8 * 10 = 80 is about to leave the queue. At t=10 minutes, the queue length is 117.5 – 80 = 37.5 vehicles.

iii) (5 points) To the nearest minute, the time at which the LT queue will have dissipated.

Queue dissipation. DC = 8*t. Starting at t=10, AC2 = 117.5 + (4.3*(t-10)). These two lines intersect at t = 20.14 minutes. The queue has cleared at that time.

[pic]

5. (15 points) Left Turn Lane Analysis using Queueing Diagrams Equations. FTE Problem 3.26, but for the intersection of Coliseum Avenue and Wildcat Street in the previous problem, not for Coliseum and Wakefield in FTE Problem 3.25.

A. Off-peak arrival rate, λ = 4.3 veh/min; Service rate, μ = 8 veh/min; ρ=0.5375

[pic]

B. [pic]

C. [pic]

Note: Unfortunately, this is not a good example of the use of the Poisson-based queueing equations, because the service process is neither random (M) nor deterministic (D). This is because the “server” (green light) is either on or off for significant continuous time intervals. All we have proved here is that you are able to find λ and μ, then use the equations for [pic], [pic], and [pic].

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