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67 Two-Lane Roundabout Delay Model 100 50% Utilization 90 100% Utilization Pedestrian Delay and Min. Delay (s) 80 Raw Data - Pre 70 Raw Data - Post PHB Raw Data - Post RCW 60 Raw Data - Min Delay 50 40 30 20 10 0 0% 20% 40% 60% 80% 100% PA_Dual This figure shows a chart of the developed mixed-priority delay model for the two-lane roundabout. The chart plots the relationship between the probability of encountering a dual crossing opportunity (gap or yield in both lanes) on the x-axis and the pedestrian delay in seconds on the y-axis. The graph shows two curves, representing pedestrians with 100% utilization of all crossing opportunities and pedestrians who only utilize 50% of opportunities. The graph further shows the field-observed data points for pretest and posttest (RCW and PHB), as well as the field-observed (theoretical) minimum delay for the pedestrians. Figure 27. Graphical comparison of two-lane roundabout delay model against field data. overall model and the PCROSS parameter are significant (p < form. In a comparison to the field-observed delay (filled cir- 0.0001). The adjusted R-square value suggests that 78.7% of cles), the 50% utilization curve appears to overpredict delay. the variability in the data is explained by the model, which is This is because the field-observed utilization rates (of dual very high given that inter-participant variability of crossing events) for blind pedestrians were very high at this site, on the performance was very high. order of 90% for both pretest and posttest. Consequently, the Figure 27 shows the fit of the model against field-observed field-observed delay matches more closely with the 100% data. Since the LN(PDUAL_ROSS) term represents a combination utilization curve, as it should. The 100% utilization curve of encounter and utilization parameters, it can be used to test expectedly is shifted downward compared to the 50% curve, the sensitivity of the different probability components. The corresponding to lower delay at the same PA_Dual encounter two curves contained in Figure 27 therefore show the predicted rate. The 100% curve matches well with the field-observed delay for 50% opportunity utilization and 100% utilization. minimum delay values. The latter approximates the delay a sighted pedestrian might The field-observed data for the posttest with PHB and have experienced if encountering the same crossing opportu- RCW installed show no observations at PA_Dual less than nities. The curves were created by varying PA_Dual from 0 to 40%, whereas the pretest data is distributed over almost the 1.0 while keeping the values of PU_DUAL constant at 0.5 and entire range. The corresponding delay figures for the posttest 1.0. The two curves are plotted against the raw average delay data points are low and match the predicted delay model. data for pretest and posttest conditions. Finally, the figure Consequently, the delay improvements of the PHB and RCW contains the theoretical average minimum delay for each data treatments are consistent with the increased PA_Dual prob- point. In the interpretation of the figure, the raw data should abilities. be compared against the 50% utilization curve, while the min- imum delay data should be compared against the 100% uti- Model Comparison lization curve. The figure again shows a predicted decrease in delay with This section compares the selected mixed-priority delay increase in the availability of dual crossing opportunities. The models for the three facility types. Since the models for CTLs shape of the curves is characteristic of the logarithmic model and single-lane roundabouts use the same variable definitions

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68 Table 11. Comparison of delay models for about (14.99) indicates that CTL delay is always higher than three facility types. the single-lane roundabout. However, due to the nature of the model, the relative difference is greater at low probabilities Single-Lane Two-Lane than at higher ones since the natural log decreases in absolute CTL Roundabout Roundabout value [e.g., ln(0.9) = 0.1]. As with a conventional linear Intercept 0.89 0.78 1.7* regression model, the intercept shifts the curve upward (pos- Ln(PCross) 17.75** 14.99 ** n/a itive coefficient) or downward (negative coefficient). How- Ln(PDual_Cross) n/a n/a -21.0** ever, due to the relatively low absolute value of the intercept Pr>F <.0001 <.0001 <.0001 compared to the coefficient for Ln(PCross), the effect of the DF 1 1 1 intercept on model form is negligible. In other words, in the R-Square 0.796 0.641 0.785 general model for y = a + b ln(x), the parameter "b" in this Adj. R-square 0.793 0.636 0.783 case weighs much more heavily than parameter "a." * Significant at p < 0.05 Following this discussion, the models predict greater ** Significant at p < 0.01 delay for channelized turn lanes than for single-lane round- abouts, assuming the same traffic patterns and pedestrian behavior. While the dependent variable for the two-lane and model form, they are directly comparable. The two-lane roundabout model is slightly different, its coefficient (21.0) roundabout model uses revised definitions for explanatory suggests that delays are greater at two-lane roundabouts variables, but a general comparison is still possible. Table 11 than at the other sites, assuming that traffic patterns and shows the final delay models for the three sites for comparison. user behavior are fixed. These comparisons are illustrated in All three models use natural-log transformed explanatory Figure 28, which shows the three delay curves as a function of variables [LN(PCross) and LN(PDual_Cross)] with negative the explanatory variables LN(PCross) and LN(PDual_Cross) coefficients. Since the natural log of a low number is a large on the x-axis. negative number [e.g., ln(0.1) = 2.3], this results in a high As discussed above, the curve for two-lane roundabout positive prediction (of delay) at low probabilities. With delay is always higher than the CTL curve, which in turn pre- increasing probabilities the absolute value of the natural log dicts higher delays than the single-lane roundabout curve. decreases, thus dropping the predicted delay. A greater coeffi- Due to the natural-log transformed model form, the relative cient therefore results in higher delays at low probabilities. For difference between the curves decreases with higher probabil- example, the fact that the CTL model has a higher absolute ities of crossing. All three curves expectedly approach a delay value coefficient (17.75) compared to the single-lane round- of zero as those probabilities approach 1.0 or 100%. Delay Model Comparison 100 90 CTL 80 Single-Lane RBT Pedestrian Delay (s) 70 Two-Lane RBT 60 50 40 30 20 10 0 0 0.2 0.4 0.6 0.8 1 P(Cross) or P(Dual_ Cross) This figure shows a comparison chart of the three developed mixed-priority delay models for the channelized turn lane, single-lane roundabout, and two-lane roundabout. The chart plots the relationship between the probability of encountering a (dual) crossing opportunity on the x-axis and the pedestrian delay in seconds on the y-axis. RBT = roundabout. Figure 28. Delay model comparison for three facility types.