Fundamental Principles of Traffic Flow

Chapter 6

Dr. TALEB AL-ROUSAN

Introduction Traffic flow theory involves the development

of mathematical relationships among the primary elements of a traffic stream> Flow Density Speed

These relationships help traffic engineer in planning, designing, and evaluating the effectiveness of implementing traffic engineering measures on a highway system.

Applications of Traffic Flow Theory

Traffic flow theory is used in design to determine: Adequate lane length for storing left-turn vehicles on

separate left-turn lanes. Average delay at intersections and freeway ramp

merging areas. Changes in the level of freeway performance due to

the installation of improved vehicular control devices on ramps.

Traffic flow theory is used in simulation: Mathematical algorithms are used to study the

complex interrelationship between elements of traffic stream.

Estimate the effect of changes in traffic flow on factors such as accidents, travel time, air pollution, and gasoline consumption.

Traffic Flow Elements Time-space diagram: A graph the describes

the relationship between the location of vehicles in a traffic stream and the time as the vehicles progress along the highway.

See Figure 6.1. (X,Y) = (Time, Distance) Primary elements are: flow, density, &

speed. Another element, associated with density,

is the gap or headway between vehicles in traffic stream

Traffic Flow Elements Definitions Flow (q) : the equivalent hourly rate at which

vehicles pass a point on a highway during a time period less than 1 hour.

q = [(n x 3600)/T] ……veh/hn = number of vehicles passing a point in the roadway in

(T) seconds.

Density (k) : ,also referred to as concentration, number of vehicles traveling over a unit length (usually one mile) of highway at an instant in time.

Unit of density is (vehicle/mi)

Traffic Flow Elements Definitions Cont.

Speed (u) : the distance traveled by a vehicle during a unit of time (mi/h) or (km/h) or (ft/sec),

The speed of a vehicle at any time (t) is the slope of the time-space diagram for that vehicle at time (t).

Two types of mean speed: Time mean speed Space mean speed

Traffic Flow Elements Definitions Cont.

Time Mean Speed (ūt) :the arithmetic mean of the speeds of vehicles passing a point on a highway.

ūt = [(1/n) (Sum (ui)) ] …. i = 1 to n

n= number of vehicles passing a point on a highway.

ui = speed of the ith vehicle (ft/sec)

Traffic Flow Elements Definitions Space Mean Speed (ūs) :the harmonic mean of the speeds

of vehicles passing a point on a highway during an interval of time.

Obtained by dividing the total distance traveled by two or more vehicles on a section of a highway by the total time required by these vehicles to travel that distance.

Space mean speed is the one involved in flow-density relationship.

ūs = [n/ (Sum (1/ui)) ] …. i = 1 to n

ūs = [nL / (Sum (ti)) ] …. i = 1 to n

n= number of vehicles passing a point on a highway.ui = speed of the ith vehicle (ft/sec).ti = the time it takes the ith vehicle to travel a cross a section of

highway (sec).L= length of section of highway (ft).

Traffic Flow Elements Definitions Cont.

The time mean speed is always higher than the space mean speed.

The difference between these speeds tends to decrease as the absolute values of speeds increase.

ūt = ūs + (2/ ūs)

Traffic Flow Elements Definitions Cont.

Time Headway (h): is the difference between the time the front of a vehicle arrives at a point on the highway and the time the front of the next vehicle arrives at that same point, expressed in (seconds)..

Can be found from time-space diagram at specified distance.

Space Headway (d): is the distance between the front of a vehicle and the front of the following vehicle, expressed in (feet).

Can be found from time-space diagram at specified time (t).

See Example 6.1

Flow-Density Relationships The general equation relating flow, density,

and space mean speed is given as: Flow = density x space mean speed

q= k ūs Each of variable depends on several other

factors, including: Characteristics of the roadway. Characteristics of the vehicle Characteristics of the driver Environmental factors (e.g. weather)

Flow-Density Relationships Cont. Other relationships exist among traffic flow variables> Space mean speed = (flow) x (average space headway)

ūs = q đaverage space headway = đ = (1/k)

Density = (flow) x (average travel time for unit distance)K= q Ť

Average space headway = (space mean speed) x (average time headway)

đ = ūs Ћ Average time headway = (average travel time for unit

distance) x (average space headway)

Ћ = Ť đ

Fundamental Diagram of Traffic Flow

See Figure 6.4 Flow vs. Density:

When there are no vehicles on the highway, the density is zero and flow is also zero.

As density increase flow also increase. When density reaches max. (jam density Kj), the flow

must be zero because vehicles will tend to line up end to end.

It follows that as density increases from zero, the flow will also initially increase from zero to a max. value.

Further continuous increase in density will then result in continuous reduction of flow, which will be zero when density is equal to the jam density.

See Figure 6.4a Some controversy exist regarding the exact shape of

the curve.

Fundamental Diagram of Traffic Flow Cont.

Space Mean Speed vs. Flow: When flow is very low, there is little interaction between vehicles,

therefore drivers are free to travel at max possible speed. The absolute max speed is obtained as the flow tends to zero

(Mean Free Speed uf ). Magnitude of (uf) depends on the physical characteristics

of the highway. Continuous increase in flow will result in a continuous

decrease in speed. A point will be reached when further addition of vehicles

will result in the reduction in the actual number of vehicles that pass a point on the highway (reduction of flow).

At this point congestion is reached and eventually both speed and flow become zero.

See Figure 6.4 c.

Fundamental Diagram of Traffic Flow Cont.

Space Mean Speed vs. Density: When there are no vehicles on the highway,

the density is zero. When density is zero there will be little or no

interaction between vehicles, therefore drivers are free to travel at max possible speed.

Further continuous increase in density will then result in continuous reduction of speed, which will be zero when density is equal to the jam density

See Figure 6.4 b.

Fundamental Diagram of Traffic Flow Cont.

knowing that: [ūs = q/k ] means that slopes of lines (OB, OC, OE) in Figure 6.4a represent the space mean speeds at densities (Kb, kc, ke) respectively.

Slope of OA = the speed at density tends to zero = mean free speed (uf) = max speed that can be attained on the highway.

Slope of OE = the speed for max flow = capacity of the highway.

It is desirable for highways to operate at densities not greater than that required for maximum flow.

Mathematical Relationships Describing Traffic Flow

Macroscopic approach: considers flow-density relationship.1- Greenshields Model: used for light or dense traffic (satisfies boundary

conditions when density approach zero or jam density).

ūs = uf – ((uf /kj)k)

q = (uf k )– ((uf /kj) k2)

2- Greenberg model: used only for dense traffic (satisfies boundary conditions when density approach jam density).

ūs = c ln (kj/k)

q = ūs k = c k ln (kj/k)

Microscopic approach: (referred to as car-following theory or follow-the-leader theory) considers spacing between vehicles and speeds of individual vehicles.

Gap & Gap Acceptance Another important aspect of traffic flow is

the interaction of vehicles as they: Join a traffic stream : ramp vehicles merging

into an expressway stream. Leave a traffic stream : freeway vehicles leaving

the freeway onto frontage roads. Cross a traffic stream: changing of lanes by

vehicles on a multilane highway. the most important factor a driver consider

in making any of these maneuvers is the availability of a gap between two vehicles that, in drivers judgment, is adequate for him/her to complete the maneuver.

Important Measures In Concept of Gap Acceptance

Merging: the process by which a vehicle in one traffic stream joins another traffic stream in the same direction.

Diverging: the process by which a vehicle in a traffic stream leaves the traffic stream.

Weaving: the process by which a vehicle first merges into a stream of traffic obliquely crosses that stream, and them merges into a second stream moving in the same direction.

Important Measures In Concept of Gap Acceptance

Gap: the headway in a major stream, which is evaluated by a vehicle driver in the minor stream who wishes to merge into the major stream. Expressed in units of (time or distance)

Time Lag: the difference between the time a vehicle that merges into a main traffic stream reaches a point on the highway in the area of merge and the time a vehicle in the main stream reaches the same point.

Space lag: the difference, at an instant of time, between the distance a merging vehicle is a way from a reference point in the area of merge and the distance a vehicle in the main stream is a way from the same point.

Gap Acceptance