October 1989
(Appendices 1 and 2 added August 1997)

Abstract:

The widely used BPR volume-delay functions have some inherent drawbacks. A set of conditions is developed which a ``well behaved'' volume delay function should satisfy. This leads to the definition of a new class of functions named conical volume-delay functions , due to their geometrical interpretation as hyperbolic conical sections. It is shown that these functions satisfy all conditions set forth and, thus, constitute a viable alternative to the BPR type functions.

A PDF version of this document is also available.

Contents

• Introduction
• Requirements for a Well Behaved Congestion Function
• Conical Congestion Functions
• Conclusions
• References
• Appendix 1: Marginal Cost Functions
• Appendix 2: Efficient Coding of Conical Functions in EMME/2

Introduction

In most traffic assignment methods, the effect of road capacity on travel times is specified by means of volume-delay functions t(v) which used to express the travel time (or cost) on a road link as a function of the traffic volume v. Usually these functions are expressed as the product of the free flow time multiplied by a normalized congestion function f(x)

where the argument of the delay function is the v/c ratio, c being a measure of the capacity of the road.

Many different types of volume-delay functions have been proposed and used in practice in the past (for a review article see Branston [1]). By far the most widely used volume delay functions are the BPR functions (Bureau of Public Roads [2]), which are defined as

With higher values of , the onset of congestion effects becomes more and more sudden. This can be seen in Figures 1, a and b, which show the BPR type congestion function

for exponents =2, 4, 6, 8, 10 and 12. This range of alpha values is also indicative of the wide range that is used in practice, but note that the values of are usually not restricted to integers.

The simplicity of these BPR functions is certainly one reason for their wide spread use. It is also very convenient that for any value we have , i.e. when traffic volume equals the capacity, the speed is always half the free flow speed.

Fig. 1. BPR functions for (A) small and (B) large v/c ratios

Unfortunately, these BPR functions also have some inherent drawbacks, especially when used with high values of :

a)

While for any realistic set of travel volumes, we can assume that (or at least not much larger than 1) this is usually not the case during the first few iterations of an equilibrium assignment. Values of v/c may well reach values of 3, 5 or even more. To illustrate this, the link time of a link with and a v/c ratio of 3 is increased by a factor of , which means that every minute free flow time becomes roughly one year of congested time! These aberrations slow down convergence by giving undue weight to overloaded links with high -values and can also cause numerical problems, such as overflow conditions and loss of precision.

b)

For links that are used far under their capacity, the BPR functions, especially when high values of alpha are used, yield always free flow times independent of actual traffic volume. To illustrate this, consider again a link with and a capacity of 1000. Whether the volume is 0 or 300, the volume delay function yields exactly the same numeric value (assuming single precision calculation). Therefore, the equilibrium model will locally degenerate to a all-or-nothing assignment, where the slightest change (or error) in free flow time may result in a complete shift of volume from one path to another path. Also, the solution is no longer guaranteed to be unique on the level of link flows, since the volume-delay functions are no strictly increasing functions of the volume any more.

c)

Even though the formula of the BPR function is very simple, its evaluation requires the computation of two transcendental functions, i.e. a logarithm and an exponential function to implement the power , which require a fair amount of computing resources.

Requirements for a Well Behaved Congestion Function

Are there other types of congestion functions that are not (or are less) subject to the drawbacks of the BPR functions? If yes, how would such a ``designer'' volume-delay function look like? Let us first set forth some conditions these functions need to satisfy:

1. f(x) is strictly increasing. Necessary condition for the assignment to converge to a unique solution.
2. f(0) = 1 and f(1) = 2. These conditions ensure compatibility with the well known BPR type functions. The capacity is thus still defined as the volume at which congested speed is half the free flow speed.
3. f'(x) exists and is strictly increasing. This ensures convexity of the congestion function - not a necessary, but a most desirable property.
4. . is, similar to the exponent in BPR functions, the parameter that defines how sudden the congestion effects change when the capacity is reached.
5. , where M is a positive constant. The steepness of the congestion curve is limited. This in turn limits also the values of the volume delay function not to get too high when considering v / c ratios higher than 1, avoiding the problems mentioned in a) above.
6. f'(0) > 0. This condition guarantees uniqueness of the link volumes. It also renders the assignment stable regarding small coding errors in travel time and distributes volumes on competing uncongested paths proportional to their capacity.
7. The evaluation of f(x) should not take more computing time than the evaluations of the corresponding BPR functions take.

Conditions 1 to 4 hold, of course, for the BPR function and are stated to ensure compatibility with them. Conditions 5, 6 and 7 are imposed in order to overcome the BPR functions' drawbacks a), b) and c) mentioned above.