D deterministic arrivals or departures Dt total vehicle delay
h vehicle time headway k traffic density kj traffic jam density kcap traffic density at capacity l roadway length
M exponentially distributed arrivals or departures n number of vehicles
nc departure channel number
N total number of departure channels q traffic flow
qcap traffic flow at capacity (maximum traffic flow) Q length of queue
Q average length of queue
Qmax maximum length of queue s vehicle spacing t time
t average time spent in the system u space-mean speed (also denoted us) ui spot speed of vehicle i
uf free-flow speed ucap speed at capacity
us space-mean speed (also denoted simply as u) ut time-mean speed
w average time waiting in the queue λ arrival rate
μ departure rate ρ traffic intensity
REFERENCES
Drew, D. R. “Deterministic Aspects of Freeway Operations and Control.” Highway Research Record, 99, 1965.
Lord, D., and F. Mannering. “The Statistical Analysis of Crash-Frequency Data: A Review and Assessment of Methodological Alternatives,” Transportation Research Part A, vol. 44, no. 5, 2010.
Pipes, L. A. “Car Following Models and the Fundamental Diagram of Road Traffic.”
Transportation Research, vol. 1, no. 1, 1967.
Poch, M., and F. Mannering. “Negative Binomial Analysis of Intersection-Accident Frequencies,”
Journal of Transportation Engineering, vol. 122, no.
2, March/April 1996.
Transportation Research Board. Traffic Flow Theory: A Monograph. Special Report 165. Washington, DC:
National Research Council, 1975.
Transportation Research Board. Highway Capacity Manual. Washington, DC: National Research Council, 2010.
PROBLEMS
Traffic Stream Parameters and Basic Traffic Stream Models (Sections 5.2–5.3)
5.1 Assume you are observing traffic in a single lane of a highway at a specific location. You measure the average headway and average spacing of passing vehicles as 3.2 seconds and 165 ft, respectively.
Calculate the flow, average speed, and density of the traffic stream in this lane.
5.2 Assume you are an observer standing at a point
along a three-lane roadway. All vehicles in lane 1 are traveling at 30 mi/h, all vehicles in lane 2 are traveling at 45 mi/h, and all vehicles in lane 3 are traveling at 60 mi/h. There is also a constant spacing of 0.5 mile between vehicles. If you collect spot speed data for all vehicles as they cross your observation point, for 30
minutes, what will be the time-mean speed and space- mean speed for this traffic stream?
5.3 Four race cars are traveling on a 2.5-mile tri-oval track. The four cars are traveling at constant speeds of 195 mi/h, 190 mi/h, 185 mi/h, and 180 mi/h, respectively. Assume you are an observer standing at a point on the track for a period of 30 minutes and are recording the instantaneous speed of each vehicle as it crosses your point. What is the time-mean speed and space-mean speed for these vehicles for this time period? (Note: Be careful with rounding.)
5.4 For Problem 5.3, calculate the space-mean speed
assuming you were given only an aerial photo of the circling race cars and the constant travel speed of each of the vehicles.
Problems
169
5.5 On a specific westbound section of highway,
studies show that the speed-density relationship is
3.5 f 1
j
u u k k ª § · º
« »
= −¨ ¸
¨ ¸
« © ¹ »
¬ ¼
It is known that the capacity is 4200 veh/h and the jam density is 210 veh/mi. What is the space-mean speed of the traffic at capacity, and what is the free-flow speed?
5.6 A section of highway has a speed-flow relationship of the form
q = au2 + bu
It is known that at capacity (which is 3100 veh/h) the space-mean speed of traffic is 28 mi/h. Determine the speed when the flow is 1500 veh/h and the free-flow speed.
5.7 A section of highway has the following flow-
density relationship:
q = 50k − 0.156k2
What is the capacity of the highway section, the speed at capacity, and the density when the highway is at one- quarter of its capacity?
Models of Traffic Flow (Section 5.4)
5.8 An observer has determined that the time headways between successive vehicles on a section of highway are exponentially distributed and that 65% of the headways between vehicles are 9 seconds or greater. If the observer decides to count traffic in 30-second time intervals, estimate the probability of the observer counting exactly four vehicles in an interval.
5.9 At a specified point on a highway, vehicles are
known to arrive according to a Poisson process.
Vehicles are counted in 20-second intervals, and vehicle counts are taken in 120 of these time intervals. It is noted that no cars arrive in 18 of these 120 intervals.
Approximate the number of these 120 intervals in which exactly three cars arrive.
5.10 For the data collected in Problem 5.9, estimate the percentage of time headways that will be 10 seconds or greater and those that will be less than 6 seconds.
5.11 A vehicle pulls out onto a single-lane highway that has a flow rate of 300 veh/h (Poisson distributed). The driver of the vehicle does not look for oncoming traffic.
Road conditions and vehicle speeds on the highway are such that it takes 1.7 seconds for an oncoming vehicle to stop once the brakes are applied. Assuming a standard
driver reaction time of 2.5 seconds, what is the probability that the vehicle pulling out will get in an accident with oncoming traffic?
5.12 Consider the conditions in Problem 5.11. How
short would the driver reaction times of oncoming vehicles have to be for the probability of an accident to equal 0.20?
Queuing Theory and Traffic Flow Analysis (Section 5.5)
5.13 Vehicles arrive at a single toll booth beginning at 8:00 A.M. They arrive and depart according to a uniform deterministic distribution. However, the toll booth does not open until 8:10 A.M. The average arrival rate is 8 veh/min and the average departure rate is 10 veh/min.
Assuming D/D/1 queuing, when does the initial queue clear and what are the total delay, the average delay per vehicle, longest queue length (in vehicles), and the wait time of the 100th vehicle to arrive (assuming first-in- first-out)?
5.14 Vehicles begin to arrive at a park entrance at 7:45
A.M. at a constant rate of six per minute and at a constant rate of four vehicles per minute from 8:00 A.M. on. The park opens at 8:00 A.M. and the manager wants to set the departure rate so that the average delay per vehicle is no greater than 9 minutes (measured from the time of the first arrival until the total queue clears).
Assuming D/D/1 queuing, what is the minimum departure rate needed to achieve this?
5.15 A toll booth on a turnpike is open from 8:00 A.M. to 12 midnight. Vehicles start arriving at 7:45 A.M. at a uniform deterministic rate of six per minute until 8:15
A.M. and from then on at two per minute. If vehicles are processed at a uniform deterministic rate of six per minute, determine when the queue will dissipate, the total delay, the maximum queue length (in vehicles), the longest vehicle delay under FIFO, and the longest vehicle delay under LIFO.
5.16 Vehicles begin to arrive at a parking lot at 6:00
A.M. at a rate of eight per minute. Due to an accident on the access highway, no vehicles arrive from 6:20 to 6:30
A.M. From 6:30 A.M. on, vehicles arrive at a rate of two per minute. The parking lot attendant processes incoming vehicles (collects parking fees) at a rate of four per minute throughout the day. Assuming D/D/1 queuing, determine total vehicle delay.
5.17 Vehicles begin to arrive at a toll booth at eight vehicles per minute from 9 A.M. to 10 A.M. The booth opens at 9:10 A.M. and services at a rate of ten vehicles per minute until 9:40 A.M. From 9:40 A.M. until 10 A.M. the service rate is six vehicles per minute. Assuming
D/D/1 queuing, what is the total vehicle delay from 9
A.M. to 10 A.M. assuming D/D/1 queuing?
5.18 The arrival rate at a parking lot is 6 veh/min.
Vehicles start arriving at 6:00 P.M., and when the queue reaches 36 vehicles, service begins. If company policy is that total vehicle delay should be equal to 500 veh- min, what is the departure rate? (Assume D/D/1 queuing and a constant service rate.)
5.19 At 8:00 A.M. there are 10 vehicles in a queue at a toll booth and vehicles are arriving at a rate of Ȝ(t) = 6.9
− 0.2t. Beginning at 8 A.M., vehicles are being serviced at a rate of ȝ(t) = 2.1 + 0.3t (Ȝ(t) and ȝ(t) are in vehicles per minute and t is in minutes after 8:00 A.M.).
Assuming D/D/1 queuing, what is the maximum queue length, and what would the total delay be from 8:00
A.M. until the queue clears?
5.20 At the end of a sporting event, vehicles begin
leaving a parking lot at Ȝ(t) = 12 − 0.25t and vehicles are processed at ȝ(t) = 2.5 + 0.5t (t is in minutes and Ȝ(t) and ȝ(t) are in vehicles per minute). Assuming D/D/1 queuing, determine the total vehicle delay, longest queue, and the wait time of the 50th vehicle to arrive.
5.21 Vehicles arrive at a single park-entrance booth
where a brochure is distributed. At 8 A.M. there are 20 vehicles in the queue and vehicles continue to arrive at the deterministic rate of Ȝ(t) = 4.2 − 0.1t, where Ȝ(t) is in vehicles per minute and t is in minutes after 8:00 A.M. From 8 A.M. until 8:10 A.M., vehicles are served at a constant deterministic rate of three per minute. Starting at 8:10 A.M., another brochure-distributing person is added and the brochure-service rate increases to six per minute (still at a single booth). Assuming D/D/1 queuing, determine the longest queue, the total delay from 8 A.M. until the queue dissipates; and the wait time of the 40th vehicle to arrive.
5.22 Vehicles arrive at a single toll booth beginning at 7:00 A.M. at a rate of 8 veh/min. Service also starts at 7:00 A.M. at a rate of ȝ(t) = 6 + 0.2t where ȝ(t) is in vehicles per minute and t is in minutes after 7:00 A.M. Assuming D/D/1 queuing, determine when the queue will clear, the total delay, and the maximum queue length in vehicles.
5.23 Vehicles begin arriving at a single toll-road booth at 8:00am at a time-dependent deterministic rate of Ȝ(t)
= 2 + 0.1t (with Ȝ(t) in veh/min and t in minutes). At 8:07 A.M. the toll booth opens and vehicles are serviced at a constant deterministic rate of 6 veh/min. Assuming D/D/1 queuing, what is the average delay per vehicle from 8:00 A.M. until the initial queue clears and what is the delay of the 20th vehicle to arrive?
5.24 Vehicles begin to arrive at a toll booth at 8:50 A.M. with an arrival rate of λ(t) = 4.1 + 0.01t [with t in minutes and λ(t) in vehicles per minute]. The toll booth opens at 9:00 A.M. and processes vehicles at a rate of 12 per minute throughout the day. Assuming D/D/1 queuing, when will the queue dissipate and what will be the total vehicle delay?
5.25 Vehicles begin to arrive at a toll booth at 7:50 A.M. with an arrival rate of λ(t) = 5.2 − 0.01t (with t in minutes after 7:50 A.M. and λ in vehicles per minute).
The toll booth opens at 8:00 A.M. and serves vehicles at a rate of μ(t) = 3.3 + 2.4t (with t in minutes after 8:00
A.M. and μ in vehicles per minute). Once the service rate reaches 10 veh/min, it stays at that level for the rest of the day. If queuing is D/D/1, when will the queue that formed at 7:50 A.M. be cleared?
5.26 Vehicles arrive at a freeway on-ramp meter at a
constant rate of six per minute starting at 6:00 A.M. Service begins at 6:00 A.M. such that μ(t) = 2 + 0.5t, where μ(t) is in veh/min and t is in minutes after 6:00
A.M. What is the total delay and the maximum queue length (in vehicles)?
5.27 Vehicles arrive at a toll booth according to the function λ(t) = 5.2 − 0.20t, where λ(t) is in vehicles per minute and t is in minutes. The toll booth operator processes one vehicle every 20 seconds. Determine total delay, maximum queue length, and the time that the 20th vehicle to arrive waits from its arrival to its departure.
5.28 There are 10 vehicles in a queue when an
attendant opens a toll booth. Vehicles arrive at the booth at a rate of 4 per minute. The attendant opens the booth and improves the service rate over time following the function μ(t) = 1.1 + 0.30t, where μ(t) is in vehicles per minute and t is in minutes. When will the queue clear, what is the total delay, and what is the maximum queue length?
5.29 Vehicles begin to arrive at a parking lot at 6:00
A.M. with an arrival rate function (in vehicles per minute) of λ(t) = 1.2 + 0.3t, where t is in minutes. At 6:10 A.M. the parking lot opens and processes vehicles at a rate of 12 per minute. What is the total delay and the maximum queue length?
5.30 At a parking lot, vehicles arrive according to a Poisson process and are processed (parking fee collected) at a uniform deterministic rate at a single station. The mean arrival rate is 4.2 veh/min and the processing rate is 5 veh/min. Determine the average length of queue, the average time spent in the system, and the average waiting time in the queue.
Problems
171
5.31 Consider the parking lot and conditions described in Problem 5.30. If the rate at which vehicles are processed became exponentially distributed (instead of deterministic) with a mean processing rate of 5 veh/min, what would be the average length of queue, the average time spent in the system, and the average waiting time in the queue?
5.32 Vehicles arrive at a toll booth with a mean arrival rate of 3 veh/min (the time between arrivals is exponentially distributed). The toll booth operator processes vehicles (collects tolls) at a uniform deterministic rate of one every 15 seconds. What is the average length of queue, the average time spent in the system, and the average waiting time in the queue?
5.33 A business owner decides to pass out free
transistor radios (along with a promotional brochure) at a booth in a parking lot. The owner begins giving the radios away at 9:15 A.M. and continues until 10:00 A.M. Vehicles start arriving for the radios at 8:45 A.M. at a uniform deterministic rate of 4 per minute and continue to arrive at this rate until 9:15 A.M. From 9:15 to 10:00
A.M. the arrival rate becomes 8 per minute. The radios and brochures are distributed at a uniform deterministic rate of 11 cars per minute over the 45-minute time period. Determine total delay, maximum queue length, and longest vehicle delay assuming FIFO and LIFO.
5.34 Consider the conditions described in Problem
5.33. Suppose the owner decides to accelerate the radio- brochure distribution rate (in veh/min) so that the queue that forms will be cleared by 9:45 A.M. What would this new distribution rate be?
5.35 A ferryboat queuing lane holds 40 vehicles. If
vehicles are processed (tolls collected) at a uniform deterministic rate of 5 vehicles per minute and processing begins when the lane reaches capacity, what is the uniform deterministic arrival rate if the vehicle queue is cleared 35 minutes after vehicles begin to arrive?
5.36 At a toll booth, vehicles arrive and are processed (tolls collected) at uniform deterministic rates λ and μ, respectively. The arrival rate is 3 veh/min. Processing begins 15 minutes after the arrival of the first vehicle, and the queue dissipates t minutes after the arrival of the first vehicle. Letting the number of vehicles that must actually wait in a queue be x, develop an expression for determining processing rates in terms of x.
5.37 Vehicles arrive at a recreational park booth at a uniform deterministic rate of 5 veh/min. If uniform deterministic processing of vehicles (collecting of fees) begins 20 minutes after the first arrival and the total
delay is 3200 veh-min, how long after the arrival of the first vehicle will it take for the queue to be cleared?
5.38 Trucks begin to arrive at a truck weigh station
(with a single scale) at 6:00 A.M. at a deterministic but time-varying rate of λ(t) = 4.3 − 0.22t [λ(t) is in veh/min and t is in minutes]. The departure rate is a constant 2 veh/min (time to weigh a truck is 30 seconds). When will the queue that forms be cleared, what will be the total delay, and what will be the maximum queue length?
5.39 Commercial trucks begin to arrive at a seaport
entry plaza at 7:50 a.m., at the rate of λ(t) = 6.3 – 0.25t veh/min, with t in minutes. The plaza opens at 8:00 a.m.
For the first 10 minutes, one processing booth is open.
After the first 10 minutes until the queue clears, two processing booths are open. Each booth processes trucks at a uniform rate of 2 per minute. What is the average delay per vehicle, the maximum queue length, and the average queue length?
5.40 Vehicles begin to arrive at a remote parking lot after the start of a major sporting event. They are arriving at a deterministic but time-varying rate of λ(t) = 3.3 − 0.1t [λ(t) is in veh/min and t is in minutes]. The parking lot attendant processes vehicles (assigns spaces and collects fees) at a deterministic rate at a single station. A queue exceeding four vehicles will back up onto a congested street, and is to be avoided. How many vehicles per minute must the attendant process to ensure that the queue does not exceed four vehicles?
5.41 A truck weighing station has a single scale. The time between truck arrivals at the station is exponentially distributed with a mean arrival rate of 1.6 veh/min. The time it takes vehicles to be weighed is exponentially distributed with a mean rate of 2.1 veh/min. When more than 5 trucks are in the system, the queue backs up onto the highway and interferes with through traffic. What is the probability that the number of trucks in the system will exceed 5?
5.42 Consider the convenience store described in
Example 5.14.The owner is concerned about customers not finding an available parking space when they arrive during the busiest hour. How many spaces must be provided for there to be less than a 1% chance of an arriving customer not finding an open parking space?
5.43 Vehicles arrive at a toll bridge at a rate of 420 veh/h (the time between arrivals is exponentially distributed). Two toll booths are open and each can process arrivals (collect tolls) at a mean rate of 12 seconds per vehicle (the processing time is also
exponentially distributed). What is the total time spent in the system by all vehicles in a 1-hour period?
5.44 Vehicles leave an airport parking facility (arrive at parking fee collection booths) at a rate of 500 veh/h (the time between arrivals is exponentially distributed). The parking facility has a policy that the average time a patron spends in a queue waiting to pay for parking is not to exceed 5 seconds. If the time required to pay for parking is exponentially distributed with a mean of 15 seconds, what is the smallest number of payment processing booths that must be open to keep the average time spent in a queue below 5 seconds?
Traffic Analysis at Highway Bottlenecks (Section 5.6)
5.45 A freeway with two northbound lanes is shut
down because of an accident. At the time of the accident, the traffic flow rate is 1200 vehicles per hour per lane and the flow remains at this level. The capacity of the freeway is 2200 vehicles per hour per lane when not impacted by an accident. The freeway is shut down completely for 20 minutes after the accident and then one lane is open for 20 minutes and finally both lanes are opened (40 minutes after the accident). What is the average delay per vehicle resulting from the accident (assuming D/D/1 queuing)?
5.46 A four-lane highway has a normal capacity of
1800 vehicles per hour per lane. In the southbound direction, a vehicle disablement on the roadway shoulder occurs at 4:30 p.m. Due to rubbernecking, the capacity in the southbound direction is reduced to 1200 veh/h/lane at this time. At 4:45 p.m., the disabled vehicle is removed from the shoulder and the capacity increases to 1500 veh/h/lane. At 5:00 p.m. the roadway capacity returns to its full value of 1800 veh/h/lane.
From 4:30 p.m. until the queue clears the traffic flow rate in the southbound direction is 1600 veh/h/lane.
What is the average delay per vehicle, the maximum queue length, and the average queue length in the southbound direction resulting from the incident (assuming D/D/1 queuing)?
Multiple Choice Problems (Multiple Sections) 5.47 Five minivans and three trucks are traveling on a 3.0 mile circular track and complete a full lap in 98.0, 108.0, 113.0, 108.0, 102.0, 101.0, 85.0, and 95 seconds, respectively. Assuming all the vehicles are traveling at constant speeds, what is the time-mean speed of the minivans? Pay attention to rounding.
a) 102.332 mi/h b) 107.417 mi/h
c) 102.079 mi/h d) 102.400 mi/h
5.48 Vehicles arrive at an intersection at a rate of 400 veh/h according to a Poisson distribution. What is the probability that more than five vehicles will arrive in a one-minute interval?
a) 0.7944 b) 0.6560 c) 0.6547 d) 0.1552
5.49 In studying of traffic flow at a highway toll booth over a course of 60 minutes, it is determined that the arrival and departure rates are deterministic, but not uniform. The arrival rate is found to vary according to the function Ȝ(t) = 1.8 + 0.25t – 0.0030t2. The departure rate function is ȝ(t) = 1.4 + 0.11t. In both of these functions, t is in minutes after the beginning of the observation and Ȝ(t) and ȝ(t) are in vehicles per minute.
At what time does the maximum queue length occur?
a) 49.4 min b) 2.7 min c) 19.4 min d) 60.0 min
5.50 A theme park has a single entrance gate where
visitors must stop and pay for parking. The average arrival rate during the peak hour is 150 veh/h and is Poisson distributed. It takes, on average, 20 seconds per vehicle (exponentially distributed) to pay for parking.
What is the average waiting time for this queuing system?
a) 4.167 min/veh b) 2.0 min/veh c) 1.667 min/veh d) 0.833 min/veh
5.51 At an impaired driver checkpoint, the time
required to conduct the impairment test varies (according to an exponential distribution) depending on the compliance of the driver, but takes 60 seconds on average. If an average of 30 vehicles per hour arrive (according to a Poisson distribution) at the checkpoint, determine the average time spent in the system.
a) 0.033 min/veh b) 1.5 min/veh c) 1.0 min/veh d) 2.0 min/veh