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    TRAFFIC CHARACTERISTICS: Speed, Volume, Density (Definitions and Formulas)

    TRAFFIC CHARACTERISTICS: Speed, Volume, Density (Definitions and Formulas)






    It is very important that highway engineers have a good understanding of transport planning, design and construction of road facilities, transport economic analysis and evaluation of existing transport facilities.
    In this treatise, I will discuss some core elements of the aforementioned.
    These traffic characteristics which include traffic speed, volume, and density, are used to describe the quality and quantity of traffic in a stream.

    Speed:

    Traffic Speed is the ratio of the distance travelled with time. It is quite often measured in meters/second (m/s) or kilometers/hour (km/h) in metric system.
    Traffic speed can be measured using different types of instruments such as stop watches, radars or speed guns, speed cameras, pneumatic tubes, induction loops, etc.

    Speed is determined by various characteristics such as driver characteristics, type of vehicle, time of the day, traffic composition, weather or surrounding environment, type of the area, etc.

    Types of speed include:
    1.     Journey Speed
    2.     Running Speed
    3.     Time Mean Speed
    4.     Space Mean Speed

    Journey Speed:

    Journey speed could be defined as the ratio of total distance travelled to total journey time, which includes delays due to traffic congestion, or other factors.

    Mathematically journey speed is expressed as:


    \[\mathop \mu \limits^ - = \frac{{Total\;Dis\tan ce\;Travelled\;}}{{Total\;Journey\;Time}}\]


    Uses of Journey Speed:

    1.     It is used for determining the adequacy of existing road network
    2.     Used for assigning trips from origin-destination (O-D) survey or trip distribution model to various routes in a road network.

    Running Speed:

    Running speed could be seen as the ratio of total distance travelled to running time. Consequently, running time, is the total journey time minus delays, simply put, it is the time that the vehicle is actually in motion.
    Journey and running speeds are functions of travel time and delays, used for evaluating road performance.
    Mathematically,

    $Running{\rm{ speed = }}\frac{{Total{\rm{ distance}}}}{{{\rm{Running time}}}}$


    Uses of Running Speed:

    1.     Running speed is used for examining speed-flow relationship for economic assessment of road improvement schemes

    Time Mean Speed (Spot Speed):

    Spot speed or time mean speed, is also referred to as instantaneous speed.
    It is the arithmetic mean of observed speeds. Its expression is given as:



    \[{\mathop \mu \limits^ - _t} = \mathop {{\mu _{spot}}}\limits^ - = \frac{1}{N}\sum\limits_{i = 1}^N {{\mu _i}} \]


    Where:
    N = total number of observations
     = observed speed of the ith vehicle

    Uses of Time Mean Speed

    1.     Used for examining the operations of traffic control devices
    2.     Used for determining the size of letters for road signs
    3.     Used for measuring the trends in vehicle speeds over a given period of time
    4.     Used for determining the enforceable speed limits, which is the 85th percentile
    5.     Used for the purpose of highway design, for which, 98th percentile is employed
    6.     Used for monitoring flow characteristics, in traffic surveillance and control scheme
    7.     50th percentile of observations, is useful for research purposes

    Methods of Measuring Time Mean Speed (Spot Speed)

    1.     Stop watch (and enoscope): This is a manual method of measuring the time mean speed. It involves establishing two separate points along the lane, with enumerators observing time and identifying vehicles as they pass the observation points, then spot speeds are estimated using time lapse and distance apart.
    2.     Pneumatic Tube or Pressure Contact Strip Method: In this method, two pneumatic tubes separated by 1000mm are fixed on the pavement surface, and counting box is attached to the tubes for counting vehicle speeds (and classification). The pneumatic tube or pressure contact strip, estimates speed of vehicles by making use of the lapse in time between bursts which it receives, due to compression pressure caused by passing vehicles over the tubes. It also classifies vehicles based on the distance between axles with respect to time.
    3.     Radar Speed meter or Speed gun: Radar speed meter, requires the use of speed gun device by a traffic enumerator assisted by another person for recording collected data. Both enumerators stand by the road side for as long as it behooves them to collect the data.
    4.     Time Lapse Photography: This is a computerized process that requires capturing or videoing vehicle images as they travel along sections of the road on the field. The video is thereafter played in the laboratory for analysis, in order to generate data using on screen marks or pavement markings.
    5.     Induction Loops: This is a loop of turned wire imbedded into the pavement. It is made up of an oscillator and a cable which sends signals to a traffic counting device when vehicles pass over a set of two loops imbedded on the pavement in series, such that vehicle lengths and speeds are estimated on the basis of time lapse whenever there are changes in the magnetic field due to the passing of vehicles over the magnetic field.

    Space Mean Speed (Harmonic Mean Speed)

    Space mean speed, could be defined as the mean travel speed of vehicles traversing a road segment of a known distance. Space mean speed, is always lower than spot speed, and has more usefulness to traffic engineering, than spot speed (time mean speed, instantaneous speed).
    Assuming that the road segment traversed by vehicles is denoted by the letter D, average travel time, is given as:

    \[\bar t = \frac{1}{N}\sum\nolimits_{i = 1}^N {\frac{D}{{{u_i}}}} \]


    \[{\bar \mu _s} = \frac{D}{{\bar t}} = \frac{D}{{\frac{1}{N}\sum\limits_{i = 1}^N {\frac{D}{{{\mu _i}}}} }} = \frac{1}{{\frac{1}{N}\sum\limits_{i = 1}^N {\frac{1}{{{\mu _i}}}} }}\]


    Wardrop first recognized the relationship between mean speeds as:



    ${{\bar \mu }_t} = {{\bar \mu }_s} + \frac{{{\sigma ^2}_s}}{{{{\bar \mu }_s}}}$


    Where:



    ${{\sigma ^2}_s}$= Variance about the space mean speed.


    Note:
    Speed, and travel pace, are entirely different.
    Travel pace could be defined as the 10 mph increment in speed in which the highest percentage of drivers are observed. It is used for predicting traffic crash.

    Traffic Density:

    Traffic density (k), or concentration could be defined as the average number of vehicles occupying a unit length of roadway at a given time. Its unit of measurement is vehicles per kilometer (veh/km).
    Mathematically,

    Traffic density could be expressed as:


    ${\rm{Density, k = }}\frac{{Average{\rm{ number of vehicles travelling over D}}}}{D}$


    Where:
    D = length of road segment.
    Due to challenges associated with traffic density estimation, it is quite easier to estimate the parameter using the following expression:



    ${\rm{Density, k = }}\frac{q}{{{{\bar \mu }_s}}}$

    Where:


    $q$ = flow rate (veh/h), and


    ${{{\bar \mu }_s}}$ = space mean speed (km/h)


    There are two major types of densities viz:
    Critical Density: Critical density could be defined as the maximum density attainable under free flow condition.

    Jam Density: Jam density could be defined, as the minimum density achieved under congestion.

    Speed-Density Curve





    \[\mu = {\mu _f} = (1 - \frac{k}{{{k_j}}})\]



    The inverse of density is known as spacing, (s), which is the distance between two successive vehicles measured from bumper to bumper.



    Spacing Measurement
                             
    Mathematically,
    Spacing is express as:

    S = 1/k (veh/km)

    Volume:

    Traffic volume (v) could be defined as the number of vehicles which pass through a reference point on a highway, or through a given lane or direction of a highway, within a specified unit of time, usually one hour.
    Traffic volume is expressed in vehicles per hour, or vehicles per hour per lane.
    When the period of count is less than one hour, this parameter is no longer referred to as volume, but rather as flow rate.
    Therefore, flow rate could be defined as the number of vehicles passing through a reference point on a highway, or through a given lane or direction of a highway, within a specified unit of time, less than an hour.

    Flow rate, q = Number of vehicles x 60/period of count.

    In other words, flow rate, is the product of density and volume
    I.e., q = kv

    The unit of flow rate is Vehicle per hour (veh/h).

    The inverse of flow rate (q) is headway (h).
    Headway is defined as the time lapse between two successive vehicles (ith and I + ith) passing a reference point in space.
    Mathematically, headway is expressed as:
    Headway, h = 1/q (hours/veh.)
    Headway is also defined in terms of space as the space between two successive vehicles in a traffic stream. In essence, headway is the time between the front of a lead vehicle passing a point, and the front of a following vehicle passing the same point in seconds.

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