TRANSPORTATION RES EARCH RECORD 1225 99 Three-Dimensional Relationships Among Traffic Flow Theory Variables Ronnnr S. GrrcnRrsr AND FnEo L. Hen This paper is an investigation of the relationships among speed, flow, and occupancy, representing the three variables of tradi- tional theory for uninterrupted traflic flow. The variables were examined in three-dimensional space, rather than two at a time as has previously been the case. Scatter plots showing connected data points were positioned in space using a three-dimensional rectanguiar coordinaúe sysúem. Obìique views oi úhe daúa were projected as two-dimensional plots for presentation purposes. The resulting pictures were evaluated for points of agreement with traditional traflîc flow theory and with a possible new approach based on the cusp catastrophe theory. The results suggest that conventional theory is insufficient to explain the data and that the plotted data are visually consistent with the catastrophe theory m model of uninterrupted traflic flow. Traditional traffic flow theory presents speed-flow-concen- tration relationships as shown in Figure 1. This diagram (1) portrays the theoretical relationship between two variables. This representation implies that the underlying data can be adequately represented as three line functions using two- dimensional relationships and that the underlying data also represent a line function in three dimensions. This function Critical has been represented as a horseshoe shape, set at an angle Densi[y to each of the three projections (2,p. 50). This underlying three-dimensional representation has not recently been tested, although the data collection capabilities Critical Speed of current freeway traffic management systems (FTMSs) make this feasible now. In particular, several systems now collect FIGURE 1 Relationships among speed, density, and rate of speed as well as volume and occupancy data, making the use flow on uninterrupted flow facilities (1). of three independent variables possible. Earlier systems could obtain speed data only through a calculation that relied on lowing that, projections of the data are presented and described. the relationship investigated in this research. Next, there is a discussion of the relationship of the data to The purposes of this investigation were as follows: both traditional theory and catastrophe theory. Finally, a number of conclusions are presented. o To investigate empirically the three-dimensional rela- tionship among speed, flow, and occupancy; o To determine whether speed, flow, and occupancy rela- BACKGROUND tionships are best described using two-dimensional functions, a surface, or some other approach; and Improved data collection methods have often led to serious o To identify the points of agreement between observed questioning of traditional theory. This has happened for unin- data and traditional theory, and between the data and a recently terrupted traffic flow condition theories as well as other fields, proposed catastrophe theory representation of traffic flow. such as particle physics. The problem is the inability of tra- ditional theory to explain the data obtained with new equip- The first section of this paper briefly describes the difficulty ment or techniques. researchers have encountered in matching traffic flow theory Three examples of this in the developrnent of traffic flow with data. The second provides a description of the data and theory represent this type of concern. In 1965, use of newly its location source, collection method, and preparation. Fol- acquired data from Chicago freeways was made in an effort to calibrate the standard two-variable models (3). The results showed that the current theory did not matchihe data well. R. S. Gilchrist, RGP Transtech, Mississauga, Ontario, Canada L5N 1P'7. F. L. Hall, Department of Civil Engineering, McMaster Uni- However, an adaptation of the conventional theory was found versity, Hamilton, Ontario, Canada L8S 4L7. that allowed data and theory to coexist peacefully. 100 TRANSPORTATION RES EARCH RECORD 1225 In another study (4-ó), the data did not match the con- DATA ACQUISITION ventional theory well, but modifications were found to salvage the conventional theory. One such modification is the concept The data for this analysis were obtained from the Mississauga of two-regime models. Unfortunately, even with this modi- FTMS on the Queen Elizabeth Way (QEW), jusr west of fication, it was found that two sets of data collected at the Toronto. The area of the QEW studied is a suburban freeway same location required different sets of parameters. This linking a sizable commuting population in Mississauga, Oak- remains a concern in the modified theory. ville, and points west with major employment centers in met- The third example comes from the 1985 Highway Capacity ropolitan Toronto. The six-lane freeway is situated in gen- Manual (HCM) (1). Although not explicitly recognized as a erally level terrain. The data presented in this paper are obtained problem caused by new data acquisition techniques, the HCM from Station L6, on the east side of the Credit River, between does contain a discussion of the mismatch between data and the Mississauga Road entrance ramp and the exit ramp at theory in its coverage of the speed-flow relationship. There Highway 10 (see Figure 3). is explicit mention of previous attempts to fit a curve to data Eastbound travel into Toronto generates recurrent conges- and to test a variety of theoretical curves (Z). There is also tion between about 7:00 a.m. and 9:00 a.m. Interchange loca- implicit recognition of this ongoing concern since one section tions at Highway 10 and Cawthra Roåd contribute to conges- discusses work with discontinuous, or two-regime, models tion as traffic attempts to enter the main line from the ramps while other sections ignore this and rely on representations when the freeway is operating at or close to capacity. Ramp (see Figure 1). metering is in effect, but the ability of the metering process These approaches to matching data with theory have all to serve mainline traffic is limited by the availability of queue attempted to model the data using some form of a curve in space for the ramp traffic. Mainline queuing extends well west two-dimensional space, so three sets of curves must be devel- of Station 16. oped to explain traffic flow theory fully. It has been suggested Data used in this examination were taken from the FTMS that speed-flow-concentration relationships are better explained computer tapes for Friday, September 30, \987. All data are using a three-dimensional surface defined by the cusp cata- from the median, or left-most, side of the eastbound lanes. strophe theory (8-10). This model of traffic flow indicates A single lane was chosen to reduce the amount of data to be that data points representing speed, flow, and concentration examined, given that alarge number of different views might lie on a surface similar to a sheet of paper with a tear in one be desired for analysis. The median lane was selected because section (see Figure 2). The portion of the sheet to one side it has been indicated (10) fhat median-lane traffic operations of the tear is raised relative to the portion on the other side. provide a sensitive and reliable indicator of freeway operations. The raised portion represents higher speeds (uncongested data), This station has paired detectors in each lane, from which while the lower portion depicts lower speeds (occurring during speed, flow, and occupancy are transmitted to the central congestion). Transitions between uncongested and congested conditions can occur either by crossing the tear or going around the end of it. The former movement produces a sudden jump in one of the three variables at the same time that the other ¡ lo loronto two undergo continuous change. If the transition occurs by traversing the portion of the surface that has no tear, all three thro Rd. variables undergo continuous change. However, the work undertaken so far to test this theory (,. has relied on two-dimensional projections of the information, Trofic N Control i >@ I even though the underlying theory is explicitly three-dimen- Cdtr6 I sional. Inspection of the three variables concurrently is as E Hurontorio necessary to validate this new model as it is to provide a more St. :(Hwv. 101 l"'",.'" complete test of conventional theory. n \+ Stotion 16 > t\t -= .-"0"94--- o I = rc Ø ! o -ò' t o) o þo o \ U) c I o J or ./ QTW c o - o .8/ )a I Erin Mitts Pkwv. A so,thrrown Rrr To Ookville l[ NOT TO SCALE FIGUR¡j 2 Hall's perception of catastrophe theory surface (8-10). FIGURE 3 QEW Freeway in Mississauga, Ontario. Gilchrist and Hall 101 computer every 30 sec. The data used for plotting purposes coordinate system is placed in the plane of the page. (Figure included 410 30-sec intervals. The first point plotted was 6:00:30 5 shows this with the flow-occupancy plane in the plane of a.m.; the last was 9:25:30. Between Interval 1 and Interval the page.) The third dimension is viewed obliquely (speed in 75 (6:37:00), speeds were consistently over 80 km/hr. At about Figure 5). Projection lines in the third dimension are plotted Interval 75, there was a sudden drop in speed as traffic oper- parallel to each other (i.e., there is no apparent perspective ations became congested. At Interval 397 (about 9:16:30), to the view as there would be if the third dimension lines speeds again climbed above 80 km/hr. From about that inter- tended toward a "vanishing point"). val on, the freeway again operated under uncongested The three-dimensional plotting is left-handed; hence, with conditions. this type of reference system, the initial origin is located at A review of the data showed that speeds were missing dur- the bottom left corner.