Fundamentals of Transportation/Traffic Flow

Traffic Flow

We aim to develop Mathematical relationships between flow, density, and speed. This helps in planning, design, and operations of roadway facilities (# lanes, length of turnbays, ramp metering policies, etc.).

Traffic flow elements such as time-space diagram tell us where on the roadway is the vehicle.

File:TimeSpace.png

Traffic flow theory

Flow and Density

Flow (q) = the rate at which vehicles pass a fixed point (vehicles per hour) $q=\frac{3600N}{t_{measured}}$

Density (Concentration) (k) = number of vehicles (N) over a stretch of roadway (L) (vehicles per kilometer) $k=\frac{N}{L}$

where:

• N = number of vehicles passing a point in the roadway in $t_{m}easured$ sec
• q = equivalent hourly flow
• L = length of roadway
• k = density

Speed

Measuring speed of traffic is not as obvious as it may seem, we can average the measurement of the speeds of individual vehicles over time or over space, and each produces slightly different results.

Time mean speed

Time mean speed ($\overline{v_s}$) = arithmetic mean of speeds of vehicles passing a point

$\overline{v_t}=\frac{1}{N}\sum _{n=1}^N{v}_n$

Space mean speed

Space mean speed ($\overline{v_s}$) is defined as the harmonic mean of speeds passing a point during a period of time. It also equals the average speeds over a length of roadway

$\overline{v_s}=\frac{N}{\sum _{n=1}^N\frac{1}{v_n}}$

Headways

Time headway

Time headway ($h_t$) = difference between the time when the front of a vehicle arrives at a point on the highway and the time the front of the next vehicle arrives at the same point (in seconds)

Average Time Headway ($\overline{h_t}$) = Average Travel Time per Unit Distance * Average Space Headway

$\overline{h_t}=\bar{t}*\overline{h_s}$

Space headway

Space headway ($h_s$) = difference in position between the front of a vehicle and the front of the next vehicle (in meters)

Average Space Headway ($\overline{h_s}$)= Space Mean Speed * Average Time Headway

$\overline{h_s}=\overline{v_s}*\overline{h_t}$

Note that density and space headway are related:

$k=\frac{1}{\overline{h_s}}$

Fundamental Diagram of Traffic Flow

The variables of flow, density, and space mean speed are related definitionally as:

$q=k\overline{v_s}$

Tradition Model (Parabolic)

Properties of the traditional fundamental diagram.

• When density on the highway is zero, the flow is also zero because there are no vehicles on the highway
• As density increases, flow increases
• When the density reaches a maximum jam density ($k_j$), flow must be zero because vehicles will line up end to end
• Flow will also increase to a maximum value ($q_m$), increases in density beyond that point result in reductions of flow.
• Speed is space mean speed.
• At density = 0, speed is freeflow ($v_f$). The upper half of the flow curve is uncongested, the lower half is congested.
• The slope of the flow density curve gives speed. Rise/Run = Flow/Density = Vehicles per hour/ Vehicles per km = km / hour.

Observation (Triangular or Truncated Triangular)

Actual traffic data is often much noisier than idealized models suggest. However, what we tend to see is that as density rises, speed is unchanged to a point (capacity), and then begins to drop. The relationship between flow and density is thus more triangular than parabolic.

Microscopic and Macroscopic Models

Models describing traffic flow can be classed into two categories, microscopic and macroscopic.

Microscopic Models

Microscopic models predict the following behavior of cars (their change in speed and position) as a function of the behavior of the leading vehicle.

Macroscropic Models

Macroscopic models study traffic as if it were a stream. The most widely used model is the Greenshields model which posited that the relationships between speed and density is linear. These were most appropriate before the advent of high-powered computers enabled the use of microscopic models. Macroscopic properties like flow and density are the product of individual (microscopic) decisions. Yet those microscopic decision-makers are affected by the environment around them, i.e. the macroscopic properties of traffic.

Traffic Phases

Queueing Input-Output Diagram with Phases

Application

• Problem (Solution)

Key Terms

• Time-space diagram
• Flow, speed, density
• Headway (space and time)
• Space mean speed, time mean speed
• Microscopic, Macroscopic

Variables

• $d_n$= distance of n^th vehicle
• $t_n$= travel time of n^th vehicle
• $v_n$= speed (velocity) of n^th vehicle
• $h_{t,nm}$= time headway between vehicles $n$ and $m$
• $h_{s,nm}$ = space (distance) headway between vehicles $n$ and $m$
• $q$ = flow past a fixed point (vehicles per hour)
• $N$ = number of vehicles
• $t_{measured}$ = time over which measurement takes place (number of seconds)
• $t$= travel time
• $k$= density (vehicles per km)
• $L$= length of roadway section (km)
• $v_t$= time mean speed
• $v_s$= space mean speed
• $v_f$= freeflow (uncongested speed)
• $k_j$= jam density
• $q_m$= maximum flow

Supplementary Reading

• Revised Monograph on Traffic Flow Theory

End Notes

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