assignment 8: intersection analysis and queueing...

CEE 3604: Introduction to Transportation Engineering Fall 2011

Assignment 8: intersection Analysis and Queueing TheorySolution Instructor: Trani

Problem 1The intersection shown in Figure 1 is to be studied for level of service characteristics. The intersection has a cycle length of 55 seconds. After consulting with the traffic engineer in town, you are told the green time for University Avenue traffic is 35 seconds. The intersection has two phases: 1) phase 1 allows green time for the traffic on University Avenue and 2) phase 2 allows green time for traffic on Elmo Street. The traffic flows recorded in a recent survey are shown in Figure 1. Assume that the D/D/1 queueing system adequately describes each lane for each approach at the intersection. In other words, in your analysis assume that each lane is independently studied as a D/D/1 queueing system. Assume no time loss in the cycle. The saturation flows for straight approaches (i.e., when cars move in straight line through the intersection) are 1,900 vehicles per hour. The saturation flows for turning movements are 1,600 veh/hr.

Figure 1. Intersection for Problem 1.

a) Estimate the average delay per vehicle for every approach and every lane. Show a set of sample calculations by hand if using a Matlab or Excel.

The nomenclature to calculate intersection delays is presented in Figure 2.

CEE 3604 A8 Trani of 8

Figure 2. Basic Queueing Diagram to Estimate Intersection Delays.

Equations to estimate total delay ( d ) and the average delay ( d ) are presented below.

Example calculation:For the approach with 800 veh/hr on University Avenue moving from West to East we know:

g = 35 secondsc = 55 secondsλ = 800 veh/hr = 0.222 veh/sµ = 1900 veh/hr = 0.528 veh/s

ρ = λµ= 0.222

0.528= 0.42

λc = (0.222 veh/s)(55 seconds) = 12.2 vehicles


µg = (0.528 veh/s)(35 seconds) = 18.5 vehicles Since λc ≤ µgUndersaturated conditions apply

Using equations 17 through 21 in the notes,Arrival flow = 800 veh/hrSaturation flow = 1900 veh/hrAverage delay (for this approach) = 6.3 secondsMaximum Queue = 4.4 vehStopped vehicles = 7.7 vehPercent stopped vehicles = 62.8 %

b) Are the green times for this intersection optimal? Explain.For the critical movement on approach from the North on Elmo Street turning left on University Avenue with 450 veh/hr we know:

Arrival flow = 0.125 veh/secondSaturation flow = 0.44 veh/secondTraffic intensity = 0.28 dimensionlessArriving vehicles per cycle = 6.88 vehiclesDeparting vehicles during green time = 15.56 vehicles Unsaturated conditionsArrival flow = 450 veh/hrSaturation flow = 1600 veh/hrAverage delay = 5.06 seconds Maximum Queue = 2.5 vehStopped vehicles = 3.48 vehPercent stopped vehicles = 50.59 %

The average delays are pretty well balanced between the two critical approaches. However use the method explained in class and in the notes (see page 42 Problem 3), requires that we estimate the delay contributions of each movement and every approach.

c) Allocate effective red and green times among approaches in such a way to minimize the total delay of all approaches at the intersection.

Table 1. Calculated Delays and Flow Rates on Each Movement.

Approach Flow RateVeh/hr

Flow rate (veh/s)

Average Delay (s)

Univ. Avenue (from West - straight) 800 0.222 6.3

Univ. Avenue (from West - right turn)

450 0.125 5.06

Univ. Avenue (from East - right lane)

700 0.194 5.8


Approach Flow RateVeh/hr

Flow rate (veh/s)

Average Delay (s)

Univ. Avenue (from West - left lane)

750 0.208 6.01

Elmo Street (from North - straight)

450 0.125 4.76

Elmo Street (from North - left turn)

450 0.125 5.06

You can setup an equation that combines all the contributions to delay (similar to example 3 in the notes). This equation will have 6 terms representing each one of the approaches that contributes to delay. For example,

TD = λU−straightrU2

2c(1− ρU−straight )+ λU−rightTurn

rU2

2c(1− ρU−rightTurn )+ ...λElmo−leftTurn

rElmo2

2c(1− ρElmo−leftTurn )

Here the red times for each approach have been labeled as rU and rElmo for the university Avenue and Elmo Street respectively. Note that three terms have been omitted. We also know that,

rU + rElmo = crU = c − rElmoSubstitute in the long equations above and take the derivative and set it to zero.

Problem 2The Port Authority of Acapulco (in México) is planning to expand its limited cruise ship port facility. Figure 2 shows the current situation with only two positions to accommodate large cruise ships of up to 70,000 metric tons. During the peak season, cruise ships arrive to the port randomly at a rate of 7.5 per week. Ships arrive to port every day of the week. The average ship docks in port 1.25 days allowing visitors to enjoy the weather and the local hospitality. Assume the service times to be negative exponentially distributed. Cruise ships wait in Acapulco Bay when the two docking positions at the cruise ship terminal are busy (see Figure 2).

a) Under the current conditions, estimate the expected number of cruise ships waiting in Acapulco Bay and unable to dock during the peak season. This is the Lq parameter.

b) Under the current conditions, estimate the average waiting time (in hours) for a cruise ship waiting in the bay.Recognize the queueing parameters for the problem. This problem can be solved using stochastic queueing theory. Use the infinite source, multiple server equations.

λ = 1.07 ships/dayµ = 0.8 ships/days = 2 servers (berths)

The problem is modeled as an infinite source, multiple server queue (M/M/2) with Poisson arrivals and neg. exponential service times. Recall the equations for the M/M/s queueing system.


Queueing Parameters System utilization (%) = 66.875 Idle probability (dim) = 0.1985 Expected No. of ships in queue (Lq) = 1.0821 Expected No. of ships in system (L) = 2.4196 Average Waiting Time in Queue (days) = 1.0113 (or 24.3 hours) Average Waiting Time in System (includes service time) (days) = 2.2613


Figure 2. Acapulco, México Cruise Ship Port Facility. Source: Google Earth.


Problem 3

A 4-lane divided freeway (2 lanes each direction) near Detroit has a free flow speed u f =120 (km/hr). During the morning peak

period (5 AM-10 AM) the volume of traffic flowing to the city center on the two inbound lanes of the highway is recorded by cameras and shown in Table 1.

Table 1. Traffic Volumes Recorded for Two Inbound Lanes.Time (hrs) Recorded Traffic Volume

(vehicles/hr)

6:00 1850

6:30 3300

7:00 3890

7:30 3600

8:00 2340

8:30 2500

9:00 2200

9:30 1900

10:00 and later 1380

a) Using the level of service values provided in the notes (i.e., Level_of_Service_Notes.pdf) in Exhibit 23-2 of the Highway Capacity Manual (HCM), determine the level of service of this freeway during the peak period spanning from 6:00 to 10:00 AM. Do your calculations of level of service for each interval recorded and shown in Table 1. Make a plot of the expected speed vs. service flow rate and indicate the level of service in your plot.

For example, for the first period, using the table above and the HCM table in the notes we know the level of service is C. This process is repeated for all other periods.

b) Find the level of service at 10:00 AM.Level of Service B

c) One morning a minor accident blocks the inbound right lane for 45 minutes until emergency crews clear the accident scene. The accident occurs at 7:30 AM.

d) Plot the demand and supply rates for the highway on day of the accident vs. time.


Figure 3. Plot of Rates and Queue Length.

e) Find the total delay for vehicles traveling on the road in the day of the accident. In this analysis employ the Maximum Service Flow Rate (measured in passenger cars per hour) provided by the HCM as the capacity of each lane.

f) Find the maximum queue length in the day of the accident. Calculate the total queue distance (km) from the accident site to the last vehicle in the queue.

From Figure 3 we know 287 vehicles is the maximum queue length at 287 vehicles. The total delay is 185 vehicle-hours. This is the area under the Queue function shown in Figure 3.


assignment 8: intersection analysis and queueing...

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