Stationary Traffic Models and Freeway Geometry
By
Michael J. Cassidy
University
of California
Institute
of Transportation Studies
416C
McLaughlin Hall
Berkeley,
CA 94720
Ph:
(510) 642-7702
FAX:
(510) 642-1246
Cassidy@ce.berkeley.edu
Funded by a UCTC Year 13 Research Grant
Occupancies and flows were jointly sampled from numerous freeway segments in nearly stationary, unqueued traffic. The data from one segment were averaged across all lanes there and plotted. Each plot was compared with one sampled at a neighboring freeway segment, with the two segments differing only in their number of travel lanes. Such comparisons were repeated for a total of five pairs of segments on five freeways in and near Toronto, Canada and in California. All occupancy-flow relations were piece-wise linear in form for average flows up to about 2,000 vehicles per hour per lane. Only when traffic became moderately dense did average vehicle speeds diminish with increasing occupancies. The occupancy above which these speed diminutions occurred was the same for both segments in a pair. Notably, however, the average vehicle speed corresponding to a given occupancy was always higher on the segment with the larger number of lanes.
The occupancies and flows used in this work were jointly extracted (from nearly stationary, unqueued traffic on neighboring freeway segments) using conventional loop detectors. A freeway segment’s number of travel lanes, henceforth designated n, differed from that of its neighbor, but design standards were identical for both segments in a pair. The detectors used for measuring these data were located some distances from ramps. All measurements occurred during fair weather.
By comparing the scatter-plot of occupancies and flows (averaged across all lanes) on one segment with that of its neighboring segment, the effects of n were isolated. Observed differences in the shapes of two neighboring relations are thus attributed solely to n.
Such differences were made plainly visible by a careful method of sampling. Only average values of the data from sustained, near-stationary traffic conditions were used in the scatter-plots. This eliminated much of the noise that might have otherwise obscured subtle distinctions in neighboring relations.
To distinguish near-stationary periods from non-stationary ones, the detector data were processed in ways to obtain curves of cumulative vehicle counts made by detectors at location x by time t, N(x, t) and analogous curves of cumulative occupancy, T(x, t).
Each set of N- and T-curves was measured from the same detector and over the same time interval. Further, each N- and T-curve in a set was re-scaled by the constants α and β so that they had the same numerical value at time te, the ending time used in constructing the curves; αN(x, te) = βT(x, te).
The slopes of the piece-wise linear curves of N and T are the flows and occupancy rates, respectively. To identify visually any changes in these, a function of the form bo∙t was subtracted from both re-scaled curves.
The resulting curves revealed whether or not traffic conditions were nearly stationary. When the N-curve displayed a linear trend, the period was marked by nearly constant or “quasi-linear” vehicle arrivals. When both the N- and T-curves in a set were (nearly) superimposed, fluctuations in flows (i.e., wiggles on the N-curve) were highly correlated with fluctuations in occupancy (i.e., fluctuation on the T-curve). It follows that the vehicles represented by these curves had nearly the same physical lengths (as seen by the detectors) and that they all traveled at nearly the same speed. This is a definition of near-stationarity (albeit a rather restrictive one for reasons explained momentarily).
The occupancy-flow plots constructed in this work consisted of data selected in the above manner. The sampling intervals used here thus varied from one observation to the next; these intervals were dictated by the periods that traffic remained nearly stationary. Periods when the re-scaled N-curve was not quasi-linear, and/or when the N- and T-curves were not nearly superimposed, were considered to be non-stationary. Data from these periods were discarded.
By using the above diagnostic to identify periods from which to extract data, any nearly stationary periods that included fast and slow vehicles and/or long and short vehicles may have been excluded unjustifiably. Since the detector data were collected over 30-sec sampling intervals, more discriminating tests could not be used.
Also of note, the diagnostic
rendered simple the task of distinguishing queued from unqueued traffic states;
(only the latter were used in this study).
This is because the arrival of a queue’s tail to a measurement location
is marked by changes in the slopes of the N- and T-curves that occur abruptly
and in opposite directions.
The scatterplots obtained in the ways just described indicated that relations between freeway occupancies and flows are piece-wise linear in form over (nearly) the entire range of unqueued conditions. These observations mean that waves can arise in unqueued traffic and travel forward at speeds lower than those of the vehicles. It follows that shocks can form in uncongested traffic. But these waves and shocks would only appear when traffic is moderately dense. In light traffic, vehicle speeds are insensitive to occupancies and flows, such that changing states are carried forward with vehicles.
Pair-wise comparisons of neighboring relations show the average vehicle speed corresponding to a given occupancy (or flow) was higher on the segment with larger n. It is not clear this can be attributed entirely to greater opportunities afforded fast vehicles to over-take slower ones. Notably, the higher free flow speeds observed on segments of greater n are not obviously linked to over-taking. After all, a free flow speed prevails even in very light freeway traffic when, for any n, over-taking occurs without delay. And in denser traffic (when speeds fell below the free flow one), the vehicle speed for a given occupancy was higher on the segment of greater n even where this added n did not improve over-taking.
So the observed effects of n on vehicle speed may have as much (or more) to do with driver psychology than with over-taking. All else being equal, drivers seem inclined to adopt a higher speed when traveling on freeway segments that are wider; i.e., on segments that have larger n.
Of further note, the occupancy beyond which average vehicle speeds diminished was always the same for both relations in a pair. That speeds became sensitive to occupancies above some “threshold” might be linked to vehicle over-taking. If such is the case, occupancies above the threshold coincide with vehicle spacings that are small enough to hinder over-taking maneuvers. But this observed sensitivity of speeds could be more a car-following effect whereby, in sufficiently dense traffic, drivers adjust their speeds in response to the vehicle spacings.
Finally, findings reported here qualitatively match information currently provided in certain traffic handbooks. The speed-flow models in newer editions of the Highway Capacity Manual (1994 and 2000) display vehicle speeds that are insensitive to flows in light traffic. Furthermore, different models are now provided for freeway segments with different free flow vehicle speeds. Free flow speeds, in turn, are said to be influenced by design standards (i.e., roadway alignments), speed limits and, notably, the number of travel lanes, n. The influence of n reported in these versions of the Capacity Manual is consistent with our present findings.
The P.I. has produced a manuscript (co-authored with S. Anani) titled Stationary models of unqueued freeway traffic and some effects of freeway geometry. This manuscript is currently under review for journal publication.