Transportation Engineering - I Traffic Flow Characteristics. Dr. Attaullah Shah
Traffic Flow • Traffic flow: No of vehicle passing per unit time at a road section q = n/t n: No of vehicles t: duration of time, usually an hour • Time head way: The time between passing of successive vehicles i.e. bumpers • t = Σ hi t; duration of time interval hi; time head way of ith vehicle. • q = n/ Σ hi = Average mean time headway • The average speed is defined in two ways: • Arithmetic mean of total speeds at a particular point also called time mean speed: Mean Ut = Σui/n • Space Mean Speed: Assumed that the average speed for all vehicles is measured at the same length of roadway. • Space mean speed in unit distance per unit time = length of strip/mean time
Example • The speed of five vehicles is measured with radar at the midpoint of 0.8Km strip (0.5 mi) The speeds are measured as 44,42,51,49 and 46 mi/h. Assuming the vehicles were travelling constant speed, determine the time mean speed. • Tim Mean speed Ut = Σui/n = (44+42+51+49+46)/5 = 46.4 mi/h • The space mean = = 5/ [1/44+1/42+1/51+1/49+1/46] = 46.17 mi/h Space mean speed is always smaller than time mean speed.
Traffic density • K = n/l The No of vehicles per unit length. • The density is also related to individual spacing of the vehicles; • The total length of roadway l = Σ si • k= 1/( Mean space) • Traffic flow = q = uk • q: flow of traffic No of vehicles/hour • Speed ( Space mean speed) • k = density units of vehicle/mile. • Example 5.2
Traffic Flow • Complex: between vehicles and drivers, among vehicles • Stochastic: variability in outcome, cannot predict with certainty • Theories and models • Macroscopic: aggregate, steady state • Microscopic: disaggregate, dynamics • Human factor: driver behavior
Speed (v) • Rate of motion • Individual speed • Average speed • Time mean speed Arithmetic mean • Space mean speed Harmonic mean
Individual Speed (1) Spot Speed
Time Mean Speed Mile post Observation Period
Space Mean Speed Observation Distance Observation Period
Volume (q) • Number of vehicles passing a point during a given time interval • Typically quantified by Rate of Flow (vehicles per hour)
Density (k) • Number of vehicles occupying a given length of roadway • Typically measured as vehicles per mile (vpm), or vehicles per mile per lane (vpmpl)
Spacing (s) • Front bumper to front bumper distance between successive vehicles S2-3 S1-2
Headway (h) • Time between successive vehicles passing a fixed point T=3sec T=0 sec h1-2=3sec
Spacing and Headway spacing headway
Spacing and Headway What are the individual headways and the average headway measured at location A during the 25 sec period? A
Spacing and Headway What are the individual headways and the average headway measured at location A during the 25 sec period? A h1-2 h2-3
Lane Occupancy • Ratio of roadway occupied by vehicles L3 L2 L1 D
Clearance (c) and Gap (g) • Front bumper to back bumper distance and time Clearance (ft) or Gap (sec) Spacing (ft) or headway (sec)
Basic Traffic Stream models 1. Speed density Model: Relationship between speed and density • Initially when there is only one vehicle, the density is very low and driver will move freely at the speed closely to the design speed of the highway. This is called free flow speed. • When more and more vehicles would come, the density of the road will increase and the speed of vehicles would reduce. The speed of the vehicle may ultimately reduce to u=0 • This high traffic density is referred as “Jam Density” • This model can be expressed as: U = Uf ( 1- k/kj) • U= Space mean speed in mi/h (km/h) • Uf = Free flow speed in mi/h • k = density in veh/mi ( veh/km) • This gives relationship between traffic flow speed and density • The speed density relationship tends to be non linear at low density and very high density. • Non linear relationship at low densities that has speed slowly declining from free flow to value • Linear relationship over the medium density region ( speed declining linearly with the density). • Non linear relationship near the traffic jam density relationship • Flow density Model: • Speed Flow Model
Speed vs. Density ufFree Flow Speed Speed (mph) kjJam Density Density (veh/mile)
Basic Traffic Stream models • Flow Density Model: • As we know that q = uk and U = Uf ( 1- k/kj) • Parabolic density model q = Uf ( k – k2/Kj ) • The max flow rate qcap: The highest flow rate • The traffic density that corresponds to this capacity flow rate is kcap and the corresponding values qcap ,Kcap ,Ucap • dq/dk = Uf (1-2k/kf )= 0 Since the free flow speed cannot be zero. • Kcap = Kj / 2 • Substituting the value: Ucap = ( k – Kj /2kj ) = Uf /2
Flow vs. Density Congested Flow Optimal flow, capacity, qm FLow (veh/hr) kmOptimal density kjJam Density Uncongested Flow Density (veh/mile)
Speed Flow Model • This is parabolic relationship • Two speeds are possible as it is quadratic equation, up to the highway capacity • Similarly two densities are possible for given flow
Speed vs. Flow ufFree Flow Speed Uncongested Flow um Speed (mph) Optimal flow, capacity, qm Congested Flow Flow (veh/hr)
Example At a section of highway in Islamabad the free flow speed is 90km/h and capacity of 5000 veh/h. During traffic census at a particular section 2500 vehicles were counted. Determine the space mean speed of the vehicle. Solution: The traffic flow is related to the space mean speed and jam density as The jam To determine the jam density From above equation of traffic flow we can rearrange the equation as Both the speeds are feasible as shown on previous slide
Traffic Flow Models-Poisson Model • The Poisson distribution: • Is a discrete (as opposed to continuous) distribution • Is commonly referred to as a ‘counting distribution’ • Represents the count distribution of random events
Poisson Distribution P(n) = probability of having n vehicles arrive in time t λ = average vehicle arrival rate in vehicles per unit time t= duration of time interval over which vehicles are counted e= base of the natural logarithm (e=2.718)
Example • An observer counts 360veh/h at a specific highway location. Assuming the arrival of vehicles at this point follows Poisson’s distribution. Estimate the probabilities of 0,1,2,3,4 and 5 vehicles arriving in 20 sec time interval. • Solution: • The average arrival rate =λ= 360 veh/h = 360/60x60=0.1 veh/sec • t=20 sec • For probability of no vehicle • P(1) = 0.271 P(2) =0.271 P(3) = 0.180 P(4) = 0.090 • For 5 or more vehicles: • P(n> 5) =1-P(n<5)= 1-(0.135+0.271+0.271+0.18+0.090 = 0.053 • Draw the histogram of the distribution
Poisson Example • Example: • Consider a 1-hour traffic volume of 120 vehicles, during which the analyst is interested in obtaining the distribution of 1-minute volume counts
Poisson Example • What is the probability of more than 6 cars arriving (in 1-min interval)?
Poisson Example • What is the probability of between 1 and 3 cars arriving (in 1-min interval)?
Poisson distribution • The assumption of Poisson distributed vehicle arrivals also implies a distribution of the time intervals between the arrivals of successive vehicles (i.e., time headway)
Negative Exponential • To demonstrate this, let the average arrival rate, , be in units of vehicles per second, so that • Substituting into Poisson equation yields (Eq. 5.25)
Negative Exponential • Note that the probability of having no vehicles arrive in a time interval of length t [i.e., P (0)] is equivalent to the probability of a vehicle headway, h, being greater than or equal to the time interval t.
Negative Exponential • So from Eq. 5.25, (Eq. 5.26) Note: This distribution of vehicle headways is known as the negative exponential distribution.
Negative Exponential Example • Assume vehicle arrivals are Poisson distributed with an hourly traffic flow of 360 veh/h.Determine the probability that the headway between successive vehicles will be less than 8 seconds.Determine the probability that the headway between successive vehicles will be between 8 and 11 seconds.
Negative Exponential Example • By definition,
Negative Exponential For q = 360 veh/hr