Time AND Reliabilty

How The Grinch  Stole Christmas

It came without ribbons, it came without tags.
It came without packages, boxes or bags.
Maybe Christmas doesn't come from a store.
Maybe Christmas (he thought) means a little bit more.

What if congested time in Travel Demand Models was a little bit more

Travel Demand Models that utilize the User Equilibrium assignment method proposed by Sparrow and Dafermos, use the Frank-Wolfe, F-W, algorithm to iteratively solve for this equilibrium. This requires the use of an impedance function.  The purpose of this impedance function is to recompute the impedance of links on paths between F-W iterations.  As implemented in Travel Demand Models, this impedance function is typically called the Volume Delay Function, VDF.  This function expresses impedance in units of time, but this time is more properly a measure of impedance, not observable times. The impedance function must have the volume for which equilibrium is being solved, as one of its independent variables.  The other variables may NOT change in value during F-W iterations. 

VDFs are typically expressed as  a function of volume to capacity ratios, v/c.  The Bureau of Public Roads, BPR, proposed an empirical impedance function which is based on v/c ratios, but the empirical observations were only for v/c ratios less than 1.2 and it was NEVER proposed that this equation be used for conditions with v/c ratios greater than 1.2.  Nonetheless, the BPR equation is the basis of the VDF that is recommended by the Travel Model Improvement Program, and is used for ALL v/c ratios.

Before proposing a theoretical impedance function, it is useful to identify those variables where the impedance will increase with increasing volumes.  The impedance function used in TDMs does not have to include all impedance choice variables, only those choice variables that cause a change in impedance with a change in volume.  If the impedance does not change with a choice variable, that variable  need not be included in the “impedance” function.  More properly what is needed is NOT an impedance function, but a CHANGE in impedance function.  This function need only consider how impedance changes in response to changes in volume.  For example, if the number of traffic control devices on a route is an impedance choice variable, the impedance associated with that choice variable does not change with a change in volumes, and that need not be considered in the UE.

The impedance as a function of volume DOES change as the reliability changes and as the travel time changes.  It is suggested that the “delay” in the BPR is more properly time, and reliability expressed in units of time, compared to the time and reliability with zero volume. If reliability and travel time are functions of volume, then a proposed equation of reliability, expressed in units of time, and a proposed equation of mean travel time, that change with increasing volume, is needed.  The function of reliability is a function of the mean time, but since the mean time is a function of volume, then reliability is, de facto, a function of volume.

It is suggested that the empirical BPR  already must include both mean time, and reliability expressed in units of time. Since the mean travel time is proposed to be a discontinuous function, any impedance function derived using mean travel time thus will also be a discontinuous function.

Azizi and Beagan  showed that route choice, was better correlated with the Planning Index, PI, than with the mean Travel Time Index, TTI.  It is proposed that the impedance function should be a function of PI. It was also shown that PI resulted in fewer assignment, F-W,  iterations before reaching essentially the same equilibrium volumes  It is proposed that the VDF should be PI and that it would include both time and reliability, e.g. be a Volume Delay AND Reliability Function, a VD(R)F.  The proposed theoretical VD(R)F,  based on the equations of travel time and reliability, is proposed to be

If v/c>=1  then TTI = FFS/TS*exp((v/c-1)/γ_T)) else TTI = 1/(γ_L*ln(1-v/c)/FFS+1))

VD(R)F =PI=TTI*(1+0.645*(1-exp(-(TTI-1)/.110171)))

where FFS, TS are link attributes similar to capacity (FFS is already coded on links),  and γ_T and γ_L are functions of FFS and TS, i.e. γ_T=( FFS-TS)/5.8 and γ_L =TS/5.8. The coefficients of reliability, δ and ε,  are shown using their theoretical values which are true for every link. 

This VD(R)F is very well correlated with the existing BPR equation for v/c ratios less than 1.2.  It is proposed that the existing validation of the VDF using the BPR formatted curve be used in this domain.  Figure 1 shows the traditional BPR curve with α =0.5 and β=4, (the original formulation of the BPR has α=0.15 but it also uses the practical capacity in the ratio.  The α =0.5 is the value consistent with the current convention of coding the physical capacity on links), and PI for a Free Flow Speed of 70 MPH, a Transition Speed of 60 MPH, and capacity of 2000 veh./hr./ln . The proposed interim BPR is the validated BPR for v/c less than 1.2,  and a simple queuing equation rotated by an angle θ,,where α=sin(θ), after a transition at v/c=1.2. The queuing equation is also coordinate translated to move the origin from (0,0) to (1.2, 1+α*1.2^β). The congestion portion of the BPR equation, α *(v/c)^β,  is itself coordinate transformed from an origin of  (0,0) to an origin of  (0,1). The proposed interim BPR is

1+α *(v/c)^β                               for v/c ≤ 1.2

and

1+α*(v/c-1.2)+α*1.2^β                for v/c > 1.2

Figure 1 Various Impedance Functions