Even in Mythaca County, congestion is increasing. Furthermore, maintenance, repair, and reconstruction projects are causing us to close traffic lanes with increasing frequency. We would like to know how qualified you are to help us predict how traffic will flow under a variety of circumstances.

In any of the problems below, you are encouraged to use spreadsheets to assist in the preparation and presentation of your analysis. If you do so, however, give clear explanations and (as appropriate) "hardcopy" examples of any equations or other operations embedded in your spreadsheet. If you use any equations, etc. in your analysis, cite your sources.

1. Speed-density-flow relationships. You will be sent by ECN e-mail an ASCII file containing observations for 30 consecutive time periods on [3-lane] Freeway 96, which is about 1 hour east of Mythaca. Note that, besides speed and flow rate, values of "occupancy" are given. Occupancy is the percentage of time that a loop detector senses the presence of a vehicle. Somebody on the Mythaca County staff thinks that occupancy is a good proxy for density.
   
   • (15 points) Plot the relationships shown in MK Figures 5.1, 5.2, and 5.3 for the Fwy 96 data, using both occupancy and density. (Hint: MK equation 5.8) Do the plots resemble the MK Figures? Does occupancy appear to be a good proxy for density? Explain.
   • (15 points) Propose equations to describe the relationships shown in each of the figures you plotted. Use the basic equation forms given in MK Section 5.3.
   
   
2. SR361 speed-density relationship. (12 points) The speed-density relationship for a lane on SR361 is believed to be:
   u + 2.6 = 0.001*(k-240)^2,
   where u has units mph and k has units vpm. Find (a) the free-flow speed, (b) the jam density, (c) the lane capacity, and (d) the speed at capacity.
   
   
3. Back on I-80 near Salt Lake City. Let us say that, at one point on the 3-lane section of EB I-80, I counted 40 other vehicles between my rented car and an overpass ahead. It turned out to be 0.78 mile to the overpass from the point at which I started my vehicle count. EB I-80 had three lanes open at that point and the vehicles were moving at approximately 52 mph at the time.
   
   • (7 points) Calculate the vehicle density (including my vehicle) on that stretch of EB I-80.
   • (7 points) Assuming the traffic was distributed evenly over the three EB lanes, what was the flow rate?
   
   
4. Poisson calculations. The SR361 bridge across the Mythaca River between Middleville and Shoridan is a toll facility. The table below shows the SEB bridge volume counts by 20-minute periods for a recent weekday morning.
   
   

   Time Period:	7:00	7:20	7:40	8:00	8:20	8:40	9:00
   Vehicle Arrivals:	55	133	202	193	129	104	76

   
   
   • (7 points) If the arrivals between 7:00 AM and 7:20 AM are assumed to follow a Poisson Distribution, what is the probability that at least two vehicles will arrive during any given 30-second period?
   • (7 points) Adopting a M/M/1 queueing regime (treating the parallel toll booths as a composite service channel[with "mu" = 650 vph]), calculate the average length of queue, average waiting time (seconds) in the queue, and average time (seconds) spent in the system for drivers arriving between 7:00 AM and 7:40 AM.
   
   
5. I-96 Incident and queueing analysis. A truck overturned at 11:57 AM near milepost 138 on NB I-96, completely blocking that highway. Fortunately, the incident site is just beyond an overpass, between an offramp and an onramp. (See diagram at right.) This means that most vehicles will see the blockage and exit I-96 at the offramp, avoiding a long backup and long delay. This also makes the detour of through vehicles simply a matter of using these ramps to go around the incident site. For the first ten minutes after the truck's mishap, the ramp capacities were governed by the stop sign at the end of the offramp and the priority given to cross traffic, which did have a stop sign. The ramps' service rate for detouring traffic was approximately 325 vph. After ten minutes, state police began controlling traffic at the end of the offramp, increasing the ramps' service rate for detouring traffic to 650 vph. At exactly 1:00 PM, NB I-96 (capacity = 3600 vph) was reopened to through traffic. If the NB I-96 flow rate at this time of day is 1550 vph:
   
   • (10 points) Draw a queueing diagram that shows the buildup and dissipation of the queue. You are reminded to label clearly each important element of (and event in) your queueing diagram. Request: Label the horizontal axis in minutes, starting at zero, not at 11:57 AM.
   • (10 points) At approximately what time (to the nearest 2 minutes) does the queue dissipate? Show this event on your diagram.
   • (10 points) What was the longest vehicle queue and the longest vehicle delay? Show these "events" on your diagram.

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