Traffic Flow

 

Row, Row, RowYour Boat

Row, row, row your boat
Gently down the stream
Merrily merrily, merrily, merrily
Life is but a dream

Do vehicles in traffic behave like water in a stream?

This topic was rejected by a national publication, in part because one of the reviewers disagreed that people in traffic behave  like drops of water in a stream with laminar and turbulent flow. Oh, yeah?

Polynomial equations of speed/time have been proposed and are widely used. These equations describe the behavior of speed/time in two domains.  They require a Free Flow Speed, FFS, the speed at zero volume; the capacity, which separates the uncongested and congested domains; and either an explicit or implicit Transition Speed, TS. The two domains are:

·       Uncongested/ laminar flow, where the volume is less than the capacity, and

·       Congested/ turbulent flow, where the volume is greater than the capacity.

The polynomial equations adequately explain the speed/time in the uncongested domain.  Their use in the congested domain is always not necessary for Traffic Simulation models, but is necessary for Travel Demand Models.  The use of these polynomial equations creates some inconsistencies when they are used in the congested domain.  Polynomial  equations are smooth, i.e. have no discontinuities.  But the use of these equations creates a discontinuity at a v/c of 2.0, and requires the allowance of negative or imaginary, not real and positive, capacities. It is proposed that exponential equations of speed/time which are different in the uncongested and congested domains may be preferable. They will have a discontinuity at the transition between the two domains, but create no additional discontinuities and require only positive capacities.  These equations are:

Uncongested domain, v/c ≤1.

Vol/Cap = 1-exp (-1/γ_L *(FFS-Speed)))

Speed = f(Vol/Cap) = FFS*(γ_L/FFS*ln(1-Vol/Cap)+1)

where γ_L is a range ( transient) coefficient of the laminar equation.  If 99.97%, 3 Sigma, of the transition occurs during the range which is the absolute value of (FFS-TS), it has the value of

γ_L = -|FFS -TS|/ln(1-0.9997) = (FFS-TS)/5.8

This is a horizontal mirroring of the traditional exponential association of speed and a translation of the origin from (0,0) to an origin of (FFS,0), restated with v/c as a function of speed.

Congested domain, v/c >1

Vol/Cap = (1-γ_T*ln(1+(Speed-TS)/TS))

Speed=f(Vol/Cap) = TS*((1-exp((-(Vol/Cap-1))/γ_T ))+1)

where γ_T is a range ( transient) coefficient of the turbulent equation.  If 99.97%, 3 Sigma, of the transition occurs during the absolute value of the range (TS-0), it has the value of

γ_T = -|TS-0|/ln(1-0.997) =  TS/5.8

This is the traditional exponential association of transition and a translation of the origin from (0,0) to an origin of (TS,1), restated with v/c as a function of speed.