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# Chapter 2. Traffic Flow: Theory and Analysis

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1. Chapter objectives: By the end of this chapter the student will be able to: Chapter 2. Traffic Flow: Theory and Analysis Design and conduct activities to measure traffic stream characteristics Analyze field data and estimate values for traffic stream parameters Build a standard traffic flow model to fit observed traffic data Apply a random model in the analysis of traffic flow when appropriate Evaluate the performance of a roadway by using standard procedures Chapter 2

2. 2.1 Measuring Traffic Flow and Spacing • Objectives of 2.1 • Measure vehicle headways and calculate average headway • Distinguish between volume and flow rate, and explain how to convert from one to the other • Display and quantify the variability of traffic flow • List and classify techniques and technologies for measuring traffic flow characteristics • Convert traffic flow measures into data for roadway design By the end of this section, the student will be able to... Chapter 2

3. www.utahcommuterlink.com Chapter 2

4. 2.1 Measuring Traffic Flow and Spacing 2.1.1 Headway calculations • Headway = the separation between a given point on adjacent vehicles. (Example 2.1) 2.1.2 Definition of traffic flow and volumes Volume (counts) = the number of vehicles that pass a point during a specified time period Flow rate = usually in units of vehicles per hour, converted from volume counts taken over different lengths of time Chapter 2

5. Relationship between flow rate and average headway (Example 2.2) Vehicle classification = the process of breaking down the total number of vehicles observed into categories or classes  used for many purposes, e.g., user cost estimation, pavement design, economic analysis, safety analysis, taxation, capacity and level of service analysis, signal timing, etc. Chapter 2

6. Peak 15 minutes within a peak hour 2.1.3 Accounting for uneven flows Some analyses of traffic engineering require the largest of the four 15-minute flow rates during the hour being studied. If one hour volumes are given, it is necessary to convert it to the largest 15-minute flow rate to do the computation, which reflect the distribution of traffic within an hour. Often, volume counts are given as hourly volumes. Highway Capacity Manual analyses require peak 15-minute flow rates. (The topic of Chapter 3) Chapter 2 (Example 2.3)

7. 2.1.4 Traffic flow data collection techniques and technologies Use of traffic count data: • Documenting locations of congestion • Establishing trends in traffic growth as a basis for future investment to increase the capacity of a roadway • Determining whether a stop sign or traffic signal is needed at an intersection • Supporting the design and evaluation of new or improved traffic signal-timing plans • Meeting the data reporting requirements of the federal government or other agencies Chapter 2

8. Techniques & technologies Temporary counting station Permanent counting station Manual counts and machine counts. (see the figures in the textbook.) Machine counts: • Sensor tube – based counters • Nu-Metrics Hi-Star Magnetic Counter • Image analysis • Microwave-based Manual counts: • Pen and paper • Jamar counter Chapter 2

9. Volume count by microwave sensors RTMS Box Multiple-lane sensing by one microwave sensor Firing example Smart Sensor by Wavetronix (look up at 500 W and Bulldog Blvd) Chapter 2

10. 2.1.5 Turning counts into design data Two typical goals of traffic count programs: • Estimate as accurate as possible annual average daily traffic (AADT) • Estimate design hourly volume (DHV) from AADT • You need to have a permanent count station to get accurate AADT. But having a permanent count station on every location of your interest is costly and not necessary.  Use the permanent count station data to develop daily, weekly, monthly and seasonal adjustment factors and multiply these factors to “coverage” count data (short-time count data). Chapter 2

11. DHV is derived from AADT. Often used to determine how many lanes are needed for a new highway. • Often the 30th highest hourly volume in a year is used as the design hourly volume, where the hourly volume trend tapers as shown in the figure below. (There are 8760 hours in a year.) How to get DHV: • Carrying out a “special count” for one year on the road segment in question and then ranking the 1-hour volumes so recorded • Using relationships established by observing other roadway segments with similar characteristics (for designing a future highway, this is the method) Chapter 2 Remember this figure is for rural highways. (Example 2.4)

12. Use of K-factor (not PHF!) = The proportion of daily traffic at a site that occurs during the peak period (K-Factors can be determined from Figure 2.12.) Example of K-factors in Table 2.4 Urbanized K = 0.091 Urban K = 0.093 Transitioning/urban K = 0.093 Rural developed K = 0.095 Rural undeveloped K = 0.1 Typically K-factors for urban roads are smaller than those for suburban roads where peaking is prominent. Why? (Example 2.5) Chapter 2

13. 2.2 Measuring Traffic Speeds and Densities • Objectives of 2.2 • Explain how vehicle speeds are measured and used • Design an experiment to determine traffic density • Distinguish between density and occupancy, and explain how to convert from one to the other • Explain how loop detectors work By the end of this section, the student will be able to... Chapter 2

14. 2.2 Measuring Traffic Speeds and Densities 2.2.1 Speed calculations • To monitor the quality of traffic flow (slow speed?  congestion) • To monitor speeds with respect to the speed limit or with respect to driving conditions • To establish the basis for a speed limit Why do we collect speed data? Two ways to compute average speeds: • Time mean speed (TMS): the average speed of all vehicles passing a point on a highway over some specified time period – spot speed studies • Space mean speed (SMS): the average speed of all vehicles moving over a given section (i.e., distance involved) of a highway over some specified time period – travel time studies Chapter 2

15. TMS vs. SMS TMS = Arithmetic mean of the spot speeds SMS = Harmonic mean of the spot speeds or distance divided by the average travel time OR Chapter 2 (Example 2.6)

16. 2.2.2 Monitoring speeds on roadways We must determine the sample size because we need to do a statistical sampling to determine the average speed of a highway The following formulas are based on a statistical theory of sampling. In this class , just memorize which formula must be used for estimating the mean (or median) speed or the 85th percentile speed. Sample sizes: 85th percentile speed Mean (or median) speed U = 1.04 for 85th percentile speedU = 0 for mean speed See Table 2.6 & 2.7. Chapter 2 (Example 2.7 & 2.8)

17. 2.2.3 Density Density is a good measure of the current amount of travel on a highway besides flow rate; this is a value that drivers can relate to estimate the level of congestion (How do you typically measure congestion while you are driving? Units: • Vehicles per mile • Vehicles per mile per lane Alas! Aerial photos are costly! Observation from a vehicle in the traffic stream is difficult also. So, what should we do then? Chapter 2 (Example 2.9)

18. Quick estimate of the “physical capacity” of a freeway lane Assumptions: • Time gap (the gap is NOT the headway) of 2 seconds is maintained for safety (a decent estimate, sometimes about 1.8 seconds) When speed = 60 mph = 88 f/s At a 2-second interval (gap), = 2 sec x 88 f/s = 176 feet Add 15 feet for the length of the typical vehicle The resulting density is (5280 ft/mi)/(176 + 15 ft/veh) = 27.64 vpmpl (los = D, close to C) Gap Chapter 2

19. 2.2.4 Detector loops and occupancy calculations • Density is difficult to measure. So, we use “occupancy” as a surrogate measure for density. This can be obtained by traffic detectors of any kind. • Occupancy: the percent of the roadway (in terms of time) that is covered (occupied) by vehicles. Apparent occupancy Actual occupancy This is the occupancy measured at a point. EL = Effective (sensing) length Chapter 2 (Example 2.10)

20. Flow rate, speed and occupancy are given; estimate density Typically occupancies given by the detectors are apparent occupancies. OR Chapter 2

21. Derivation of the Density-Occupancy Relationship • Estimate SMS using detector data • Compute total time occupied (not occupancy) by N vehicles detected in time period T • Solve the first equation for average time occupied by each vehicle • Plug in the 3rd eq into 2nd eq • Compute the occupancy Oapp. N/T turned out to be flow rate, q. Also q/SMS is density by definition. Now the relation between occupancy, Oapp, and density, D, was established. • Solve for D. Voilà, you get Eq. 2.11 Chapter 2

22. 2.3 Traffic Models for Continuous Flow By the end of this section, the student will be able to... • Objectives of 2.3 • Derive and explain the relationships involving speed, flow and density. • Fit traffic data to one proposed model of traffic flow Chapter 2

23. Goal of Modeling: Finding relationship among: Flow rate, Speed (SMS), and Density Categories: Chapter 2

24. 2.3.1 Speed-density relationship Sf Dj This linear model is the famous and useful Greenshields linear model. Regardless of the speed-density relationship: Flow = speed * Density or q = S * Dholds. Chapter 2

25. 2.3.2 Speed-flow relationship Chapter 2 Eq. 2.16 – 2.18)

26. 2.3.3 Speed-density relationship Eq. 2.9 Table 2.9 data Chapter 2

27. 2.3.4 The Greenshields model The Greenshields model has neat characteristics: • The optimal density that occurs when the flow rate is maximum is ½ of the jam density and the optimal speed is ½ of the free-flow speed. • The model can reflect the reality for the entire range of density and flow. Flow-density: Flow-speed: Seed-density: Take a derivative dq/dD and dq/dS and equate to zero. This is a typical minimum-maximum problem. Chapter 2

28. The interrelationships of the fundamental traffic flow relationships Chapter 2

29. 2.4 The Poisson Model for Continuous Random Traffic Flow By the end of this section, the student will be able to... • Objectives of 2.4 • Pronounce “Poisson” correctly • List the assumptions that underlie the Poisson distribution • Apply Poisson and negative exponential equations to analyze traffic flow Chapter 2

30. 2.4.1 The Poisson process The Poisson model can answer the question: “How many events will likely be taking place during a time interval of a specified length?” -- e.g. How many left turn vehicles are arriving during a signal cycle length? Or, how many gaps are available to cross a street safely? Conditions: • The events are random. • The event rate remains constant for the duration of the analysis. • The probability that a single event will occur during a short time interval is proportional to the length of the time interval. t = duration of time interval of your interest (sec) P(n) = probability that exactly n vehicles arrive in time interval t λ = average flow rate or arrival rate (veh/sec) Chapter 2 (Discuss Example 2.16)

31. 2.4.2 The time between vehicle arrivals The Poisson process deals with discrete values (1 car, 2 cars, etc.). Suppose you are interested in time in between two adjacent vehicles arriving. How can we model this? By definition, the time between two consecutive vehicles has no other vehicles arriving. Let x be the time to the first event after the previous event. The probability that the length of time to the first event will exceed x is the same as the probability that no events will occur in x. The complement of this (headway smaller than x: Chapter 2

32. Plot of negative exponential distribution fitted to a data set (A negative exponential curve) Table 2.11 data (Discuss Example 2.20, 2.21, 2.22) Chapter 2

33. 2.5 Measuring Roadway Performance By the end of this section, the student will be able to... • Objectives of 2.5 • Define “capacity” as it applies to a road segment. • Explain the concept of “level of service.” • Apply the standard classification schemes for roads and vehicles Chapter 2

34. 2.5.1 Capacity and level of service Capacity as defined by Highway Capacity Manual: “the maximumhourly rate at which persons or vehicles can be reasonably expected to traverse a point or uniform segment of a lane or roadway during a given time periodunder prevailing conditions.” This assumes that all vehicles are passenger cars. If you have trucks mixed in traffic flow, we need to know the percentage of trucks in traffic flow and their passenger car equivalency values. Chapter 2

35. Speed/Travel Time Delay Density Level of Service: “A level of service is a letter designation that describes a range of operating conditions on a particular type of facility.” LOS A (best) LOS F (worst or system breakdown) Urban Arterials Freeway and rural arterials LOS is defined by a single measure of effectiveness (MOE). Intersections Chapter 2

36. Level of service example LOS B LOS C or D LOS A LOS E or F Chapter 2 http://www.utahcommuterlink.com

37. Level of Service and Fundamental Diagram of Traffic Flow Stable flow SFE Unstable flow E F Flow D C SFA B A Density Chapter 2

38. 2.5.2 Highway performance monitoring The Highway Performance Monitoring System (HPMS) is a federally mandated traffic data reporting program. The HPMS attempts to estimate information (traffic volume, annual vehicle classification, and truck weights) for all public roads in the US by having each state DOT use a specified statistical sampling program. • Volume or flow data • Road functional classification • Vehicle classification (13 classes, see table 2.13 and figures 2.30 and 2.31.) – Many technologies are available. Their software programs have a routine to classify vehicles. But this does not mean these programs are perfect. Chapter 2