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112 I N N O VAT I O N S I N T R AV E L D E M A N D M O D E L I N G , V O L U M E 2 appear to account for some of the observed variation in MODEL SPEEDFLOW EQUATION speeds. CALIBRATION--V/C > 1.00 The Akcelik equation is of interest because it is not a smooth curve in v/c like the others. The Akcelik equa- The field data could not be used to evaluate the tionpredicted speed is sensitive to the link length in speedflow curve candidates for demands greater than addition to the v/c ratio. The Akcelik equation adds the capacity because the standard traffic counting procedure same delay to a link for a given v/c ratio, regardless of used could only count the served demand, not the the link length. (The assumption is that all the delay unserved demand. Thus a theoretical evaluation was occurs at the downstream signal at the end of the link. conducted of the speedflow curves comparing their pre- No delay accrues over the length of the link.) The result dicted delays for volumes greater than capacity against is that the Akcelik curve shows a bit more scatter (simi- the delays predicted by queuing theory. lar to the observed data) than the other curves, for which According to classical queuing theory, when demand the predicted speeds are not sensitive to link length. is greater than capacity, vehicles must wait in line until The reader will note that a simplified version of the the vehicles in front of them have had a chance to pass original Akcelik equation has been calibrated. The con- through the intersection. This theoretical average delay stant multiplier of 8 for the J calibration parameter has can be graphed and compared with the predictions pro- been subsumed within the J calibration parameter duced by the candidate speedflow curves. itself. The variable "capacity" from the Akcelik equa- Figure 3 illustrates this (the chart plots travel time per tion was dropped because the simplified equation fit segment, the inverse of speed, so that the theoretical the data better. Note also that, because the length of delay due to queuing can be included in the chart). Points analysis period is 1 h, it is no longer necessary to carry that fall on the horizontal portion of the queuing theory the time period duration variable, T. The final equation line represent traffic moving at free-flow speeds with no is shown below. delay. Points above this horizontal line represent speeds below free-flow speeds, with delay. The theoretical average delay due to queuing is the L thick solid line at the bottom of the chart. The line is flat S= until the real-world capacity of the link is reached, then L / S0 + 0.25 ( x - 1) + ( x - 1)2 + Jx (2) the predicted travel time increases rapidly, but linearly with increasing demand. The ideal speedflow curve would not cross the theo- where all variables are the same as defined before. retical solid line for queue delay. As can be seen, however, A statistical comparison of the equations is presented both the standard and fitted BPR curves cross the theo- in Table 3. This table shows the root-mean-square error retical queuing delay line. Both of these curves underesti- and the bias for each curve when compared against the mate the delay due to queuing when demand exceeds the observed data. The fitted equations (BPR, exponential, real-world capacity of an intersection at the end of a link. and Akcelik) naturally do better against the field data The fitted Akcelik curve is consistent with the queue than the standard BPR equation because they have been delay line, because the Akcelik curve is derived from clas- fitted to the data. While the standard BPR equation over- sical queuing theory. 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 CONCLUSIONS for the standard BPR curve is 16 mph (25.6 km/h), while the other curves have significantly lower RMS errors. There is a great deal of variation in the observed arterial The best-fitting curve, the Akcelik equation, has about a street segment speeds that cannot be explained solely on 40% better RMS error than the standard BPR equation. the basis of the v/c ratio for the signalized intersection at the terminus of the segment. The v/c ratio appears to TABLE 3 Quality of Fit to Observed 1-Hour Data explain about 30% of the variation. Other factors, such for v/c < 1.00 as signal timing offsets, affect the observed mean hourly Fitted speed on a segment. Standard Fitted Exponen- Fitted In evaluating data for demands less than the approach Fitted parameters BPR BPR tial Akcelik capacity, many equations, such as the fitted BPR, fitted S0 (free-flow speed) 40 mph 40 mph 40 mph 40 mph exponential, and the fitted Akcelik, performed equally A 0.15 2.248 1.0512 0.0019 B 4.00 1.584 1.185 well. The fitted Akcelik equation performed slightly bet- Bias (mph) 11.53 0.30 0.04 0.13 ter because it adds signal delay to the segment free-flow RMSE (mph) 16.00 9.83 9.84 9.40 travel time rather than treating delay as a multiplicative