Innovations in Travel Demand Modeling, Volume 2: Papers (2008)

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appear to account for some of the observed variation in speeds. The Akcelik equation is of interest because it is not a smooth curve in v/c like the others. The Akcelik equa- tion–predicted speed is sensitive to the link length in addition to the v/c ratio. The Akcelik equation adds the same delay to a link for a given v/c ratio, regardless of the link length. (The assumption is that all the delay occurs at the downstream signal at the end of the link. No delay accrues over the length of the link.) The result is that the Akcelik curve shows a bit more scatter (simi- lar to the observed data) than the other curves, for which the predicted speeds are not sensitive to link length. The reader will note that a simplified version of the original Akcelik equation has been calibrated. The con- stant multiplier of 8 for the J calibration parameter has been subsumed within the J calibration parameter itself. The variable “capacity” from the Akcelik equa- tion was dropped because the simplified equation fit the data better. Note also that, because the length of analysis period is 1 h, it is no longer necessary to carry the time period duration variable, T. The final equation is shown below. (2) where all variables are the same as defined before. A statistical comparison of the equations is presented in Table 3. This table shows the root-mean-square error and the bias for each curve when compared against the observed data. The fitted equations (BPR, exponential, and Akcelik) naturally do better against the field data than the standard BPR equation because they have been fitted to the data. While the standard BPR equation over- estimates arterial speeds by an average of 11.5 mph (bias) (18.4 km/h), the other curves overestimate arterial speeds by less than 1⁄2 mph on average. The RMS error for the standard BPR curve is 16 mph (25.6 km/h), while the other curves have significantly lower RMS errors. The best-fitting curve, the Akcelik equation, has about a 40% better RMS error than the standard BPR equation. MODEL SPEED–FLOW EQUATION CALIBRATION—V/C > 1.00 The field data could not be used to evaluate the speed–flow curve candidates for demands greater than capacity because the standard traffic counting procedure used could only count the served demand, not the unserved demand. Thus a theoretical evaluation was conducted of the speed–flow curves comparing their pre- dicted delays for volumes greater than capacity against the delays predicted by queuing theory. According to classical queuing theory, when demand is greater than capacity, vehicles must wait in line until the vehicles in front of them have had a chance to pass through the intersection. This theoretical average delay can be graphed and compared with the predictions pro- duced by the candidate speed–flow curves. Figure 3 illustrates this (the chart plots travel time per segment, the inverse of speed, so that the theoretical delay due to queuing can be included in the chart). Points that fall on the horizontal portion of the queuing theory line represent traffic moving at free-flow speeds with no delay. Points above this horizontal line represent speeds below free-flow speeds, with delay. The theoretical average delay due to queuing is the thick solid line at the bottom of the chart. The line is flat until the real-world capacity of the link is reached, then the predicted travel time increases rapidly, but linearly with increasing demand. The ideal speed–flow curve would not cross the theo- retical solid line for queue delay. As can be seen, however, both the standard and fitted BPR curves cross the theo- retical queuing delay line. Both of these curves underesti- mate the delay due to queuing when demand exceeds the real-world capacity of an intersection at the end of a link. The fitted Akcelik curve is consistent with the queue delay line, because the Akcelik curve is derived from clas- sical queuing theory. CONCLUSIONS There is a great deal of variation in the observed arterial street segment speeds that cannot be explained solely on the basis of the v/c ratio for the signalized intersection at the terminus of the segment. The v/c ratio appears to explain about 30% of the variation. Other factors, such as signal timing offsets, affect the observed mean hourly speed on a segment. In evaluating data for demands less than the approach capacity, many equations, such as the fitted BPR, fitted exponential, and the fitted Akcelik, performed equally well. The fitted Akcelik equation performed slightly bet- ter because it adds signal delay to the segment free-flow travel time rather than treating delay as a multiplicative S L L S x x Jx = + −( ) + −( ) +⎡⎣⎢ ⎤ ⎦⎥/ .0 2 0 25 1 1 112 INNOVATIONS IN TRAVEL DEMAND MODELING, VOLUME 2 TABLE 3 Quality of Fit to Observed 1-Hour Data for v/c < 1.00 Fitted Standard Fitted Exponen- Fitted Fitted parameters BPR BPR tial Akcelik S0 (free-flow speed) 40 mph 40 mph 40 mph 40 mph A 0.15 2.248 1.0512 0.0019 B 4.00 1.584 1.185 Bias (mph) 11.53 0.30 0.04 0.13 RMSE (mph) 16.00 9.83 9.84 9.40