CE 353 Laboratory Week 2: Transportation Analysis

The objectives of this lab are to introduce the student to several areas of analysis within transportation. Aspects of planning, design, and operation are included within this lab. The student will have an opportunity to use basic concepts from prerequisite courses of physics, surveying, and statistics. In addition, this laboratory will serve to evaluate strengths and weaknesses in preparing acceptable lab reports for the remainder of the semester.

Initial Instructions

1. Get a new partner (i.e., one that you haven't had before) - for this lab this shouldn't be a problem!
2. Work in teams of 2
3. Submit only one report for the entire team

Problems

1. Studies

Three basic components examined when studying traffic flow on highways are speed (miles per hour), traffic density (vehicles per mile), and traffic volume (vehicles per hour). The basic relationship among these variables is:

Volume = Speed * Density (Eq. 1)

One researcher has identified a linear relationship between speed and density on a particular roadway to be:

Speed = 54 - 0.30 (Density) (Eq. 2)

Tasks:

1. Verify that the units on Eq. 1 are correct.
2. Establish an equation to show the relationship between speed and volume for the special roadway under study and draw a figure showing this equation. Place volume on the ordinate.
3. At what speed would we be able to achieve the maximum volume on this road? Prove this mathematically.
4. What are the maximum volume and density on this road?

2. Stopping Distance

When a driver is faced with an object on a roadway he/she needs time to perceive a problem, make a decision and react. During this time the car travels at full speed. For design situations we use a perception-reaction time of 2.5 seconds. If the driver panics and slams on the brakes the only force available to reduce the kinetic energy on a level roadway is the frictional force between the tires and pavement. Under these conditions, braking distance is:

D = V²/(30 f), where:

D = distance in feet
V = velocity in mph
f = coefficient of friction

Task:

If the initial speed is 60 mph, determine the total distance needed to see an object, react to it and bring the vehicle to a stop without hitting the object. An allowable f value for this speed is 0.30.

3. Design

When we have vertical curves, the roadway itself can restrict visibility to an object on the road. Consider a situation on a rural county route where a 1200 foot symmetrical crest vertical curve is being considered for joining a +4% grade and a -5% grade. If a 6" object is laying on the road 120 feet beyond the high point of the curve, would a driver going 60 mph (as in problem 2) have sufficient distance to stop? To answer this you must also know that the driver's eye height is taken to be 3.50 feet.

Reference material is provided in the SURVEYING TEXT BOOK.

HINT: AFTER reading the whole problem, use, in order, equations 13-22, 13-30, 13-28, equation of a line, 2 equations 2 unknowns (2/2), and the quadratic equation.

NOTE: Rather than using the equation of a line, 2/2, and quadratic, you could make use of the hint below and assume the line of sight is nearly horizontal; your choice.

Tasks:

1. Draw a sketch of this curve assuming that the PI is Station 100+00.00 and has an elevation of 85.00 feet. Under these conditions, find the station of the crest of curve, the elevation of the crest, and the elevation of the top of the object.
2. Determine the station at which a driver going up the 4% grade would first be able to look over the crest and just see the top of the object. (HINT: You should find for this special case that the line-of-sight is very nearly horizontal).
3. Consider the stopping distance for 60 mph found in Problem 2. Does that value suggest we would have a problem with sight distance here?
4. Identify one or more factors that would cause less distance to be needed here than in the 60 mph Problem 2 case. Discuss possible design changes to provide adequate sight distance.

4. Statistical Analysis

A traffic signal designer promised to retime the traffic signals along an arterial street and increase the average speed on the street by at least 3 mph from its current average of 19 mph. After the job was completed the city officials did speed runs through the system and found the following speeds on the runs.

Travel Speeds in MPH:

23.3, 25.3, 20.3, 20.6, 23.7, 20.5, 22.9, 21.2, 22.8

Tasks:

1. Based on this sample, would you be able to reject the claim that the average speed through this area has reached at least 22 mph with 95% confidence? See reading from STATISTICS MANUAL (use t-test).

   (Stat. Manual citation: Statistics Manual. Crow, Edwin L., Frances A. Davis, Margaret W. Maxfield. Dover Publications, Inc., New York, 1960.)

2. What can the city do to gain more confidence in their assessment about the true mean speed on this arterial? (i.e. from a statistical point of view?)

   5. Queuing Theory

   Part 1: Single Server Queues

   Transportation problems often deal with queueing situations. A simple queueing model involves vehicles arriving and being served by a single server, such as a toll booth. Using a model based upon Poisson arrivals and exponentially distributed service times, some simple values can be derived for this model.

   Single Server Model:

   

   P(n), probablity of n vehicles in the system, including vehicle at server =	(q/Q)n(1-q/Q)
   E(n), expected vehicles in the system =	(q/(Q-q))
   E(w), expected wait in the queue =	(q/Q(Q-q))
   P{w<=t}, probablity of spending t or less in the queue =	1-(q/Q)e-(1-q/Q)qt

   Vehicles arrive at a toll booth at a rate of 375 vph. The toll booth serves at rate of 500 vph.

   Tasks:

   1. What percent of the time is the toll booth not serving any vehicles?
   2. What are the expected number of vehicles in the system?
   3. What is the probablity of spending 5 minutes or less in the queue?
   4. Plot the expected number of vehicles in the system vs the ratio of arrival to service rate. What can be said about E(n) when the ratio approaches 1? What does this mean?

   Part 2:Queueing with interrupted flow

   For queuing situations with interrupted flow, like traffic signals, it useful to create a queueing diagram. Interpreting this diagram can provide parameters about the queue.

   

   Items of interest:

   maximum queue length
   maximum waiting time/delay
   average queue length
   average waiting time
   total delay
   

   For this class, assume first into the queue, the first out of the queue.

   Definitions:

   Wj = total time spent in queue by the jth arrival
   Q(t) = queue length at time t
   A(t)= cummulative arrival function
   D(t)= departure function
   Q(t) = A(t) - D(t), initial queue length = 0 (conservation principle)
   A(t) and D(t) can never cross because the queue length (or # in storage) can't be negative
   

   

   

   Equation explanations:

   Eq 1= Average waiting time for 'n' arrivals = Area / n
   Eq 2= Total time spent in queue (by all arrivals) = area between 2 curves
   Eq 3= Average length of queue
   Eq 4= Average length of queue
   

   Geometric interpretation:

   TT = area between A(t) and D(t) and the 2 vertical lines at t=0 and t=T

   If the queue disappears at T (queue length = 0), when the nth object leaves,n/T = average number of arrivals per unit time for any service order and between points where storage is empty Average queue length = (Average waiting time) * (Average arrival rate)

   Summarizing, 3 things must be remembered when drawing D(t):

   1. The departure rate cannot exceed the service rate of the server. It may be less.
   2. Cumulative departures can never exceed cumulative arrivals. D(t) can never be above A(t) on the diagram.
   3. When a queue is present, there are more complicated situations such as storage areas with limited capacity, servers which turn themselves off and on periodically and servers whose service rate changes with the queue length. However, the 3 items above always apply.

   Cars arrive at a traffic signal at a rate of 360 vph. The green time is 30 seconds, and the cycle time is 60 seconds. During the green time, vehicles can depart from the queue at a rate of 1200 vph.

   Tasks:

   1. Plot the queuing diagram for one cycle.
   2. Determine total queuing delay during 1 cycle and average delay per vehicle.
   3. Determine maximum delay.
   4. Determine maximum queue length.