How to Calculate Path Impedances when Reliability Is Included

If you were to assume that drivers are sensitive to the degree of reliability in travel time when they choose paths and if you wanted to include this sensitivity in your travel models, you would need a way to calculate path impedance from link impedance. You would also want a measure of variability (the inverse of reliability) for a link that at least roughly follows correct probability theory.

The most intuitively pleasing formulation of link and path variability was developed by Hyder Consulting in a series of traffic studies performed in the UK. (Full disclosure: I have no connection to Hyder, although they once gave me a very nice logoed Swiss Army knife that I still use.) In this article I will try to explain the probabilistic underpinnings and assumptions inherent in Hyder’s equations and, hopefully, explain why they are superior to some of the empirical equations that have been recently kicked around in the United States.

(Note: Hyder’s findings were published in a series of articles and reports with various combinations of co-authors. Ian Black, J. Fearon and Christina Gilliam seem to be the most prominent in that cadre.)

I will be assuming that we know the path already. Determining path choice when there exists uncertainty is a huge topic unto itself.

Reliability, or more correctly unreliability, is typically measured in some fashion by the standard deviation of the travel time.

Hyder first assumed that variability over a short link segment gets worse as delay becomes worse. Links with little delay have little variation and links with a lot of delay can have much variation. They further assumed that this could be a nonlinear relationship; they adopted a power function of average travel time divided by free travel time. Presumably, we already know how to compute these two numbers. Hyder’s own floating car studies suggested that the standard deviation of travel time is pretty much linear with this ratio (or linear in the percent delay). Hyder’s first assumption is plausible and elegant, but admittedly empirical in its actual choice of inputs.

Secondly, Hyder assumed that very long links would have comparatively less variation than short links. For example, a minor incident along a road that caused say 1 minute of delay, would have a greater relative impact on a 1-block segment than a 1-mile segment. If we were to assume that a long segment was made up of many short segments and that unusual delays in those short segments are probabilistically independent of each other, then error theory would tell us that the relative variability in a long segment is the reciprocal of the square root of the number of short segments contained within. This is called the “square root of N law”.

HOWEVER, short link segments are not necessarily independent of each other. An obvious example is a blockage that can back traffic up across multiple segments. Without knowing the exact cause and duration of every queue, we would need to surmise that long links do better, relatively, than short links, but not to the extent suggested by the “square root of N law”. A good proxy for the number of short segments in a link is distance. So Hyder multiplied their equation with distance raised to a negative power, with that power being as low as -0.5 (according to error theory) but often somewhat closer to zero.

So with that explanation behind us, Hyder’s equation is:

where CV is the coefficient of variation (the ratio of the standard deviation to the mean), t is average travel time, t0 is free travel time, L is link length or distance, and everything else are empirical coefficients. Presumably, these coefficients could vary by functional class or by the type of traffic control.

If you are concerned about such things, Hyder’s CV equation will not cause any issues by itself with equilibrium traffic assignments.

Link impedance can be combined to obtain path impedances. (I am ignoring node impedances for now, but the principles are the same.) But this is not a simple process of addition. To get path impedances you would need to consider how link delays are correlated.

The distance between pairs of links on a given path heavily influences the amount of correlation. Links that are widely separated in distance are sure to be uncorrelated, but links in immediate proximity to one another are sure to have some correlation, especially when there are long delays and the possibility of backups.

To find the amount of travel time variation in a path it is necessary to add together variances of individual links and covariances between links. (A covariance is an unnormalized correlation coefficient.) We cannot simply add impedances because they contain the standard deviation of travel time, which is the square root of the variance. Doing all this adding could be a big job since there are as many variances and covariances in a path as the square of the number of links. If a path had 100 links (not an unusually large number for sure), then there would be a need to calculate and add together 10,000 numbers.

Hyder made a very useful assumption that is reasonable and hugely simplifying. They assumed no correlation between links that are nonadjacent. For the most part, any such correlation is likely small and ignorable. So it is necessary to consider only the covariances between successive links in a path, of which there are one less than the number of links. So the daunting problem of the previous paragraph collapses to needing to add together just 199 numbers, not 10,000 numbers.

This simplification of covariances also aids path building, since there is no need to store and reference all these covariances during the process of choosing which link needs to be next included in a path. Actually, only one covariance is needed for any link being newly considered for inclusion in a path.

Of course, once the variances and covariances are added, finding the path standard deviation and path impedance is a simple calculation.

Keep in mind that path building algorithms need to be quite sophisticated to build shortest paths because impedances are not additive.

Alan Horowitz, Whitefish Bay, March 4, 2016