# 025 second at every fifteen minutes a line listing

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Unformatted text preview: location. For each period, a table and a graph for the cumulative IAT distribution were established. An example of a graph is shown in Figure 6. 27 120.00% Cumulative Percentage 100.00% 80.00% 60.00% 40.00% Number of Vehicles per Hour= 677 Average=5.36 s Standard Deviation= 4.67 s Minimum= 0.60 s Maximum= 28.95 s 20.00% 0.00% 0 5 10 15 20 25 30 35 Interarrival Time (s) Figure 6: An Actual Cumulative IAT Distribution Observed during a 15 Minute Time Period (08/20/04 Friday, 8:15-8:30 AM) For each time period a mean and standard deviation of the IATs were determined. For each lane, three time periods were randomly selected and withdrawn for later validation of the model. Additionally, three time periods for the passing lane were removed because they constituted outliers. Hourly flow rates, average IATs, average speeds, and standard deviations for the time intervals used in this study are shown in Appendix A. For each of the marked periods the number of vehicles passing was reported; this number was multiplied by 4 and then by the multiplication factors given in Table 1 according to the lane of the vehicles to obtain a traffic flow rate in vehicles per hour per lane (vphpl). The cumulative value of 0% was assigned an IAT of 0.1 seconds. From all the time periods in a given set, for the 1% cumulative value, the IATs in seconds were 28 determined and plotted as a function of the hourly traffic count. A hyperbolic leastsquares fit was made to determine the mathematical relationship between the IAT and the hourly traffic count for the 1% cumulative IAT. The same procedure was used to determine the mathematical relationships for the cumulative percent iles of 2%, 5%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 95%, 98%, 99%, and 100% (taken as the maximum IAT recorded). Graphs of the IATs as a function of volume at each percentile level are shown in Appendix B, both for driving and passing lanes. The percentile values were then rearranged into a table of cumulative IATs with each percentile forming a column and each hourly traffic count forming a row. From these tables, graphs of cumulative IATs versus hourly traffic counts were plotted using Microsoft Excel. For each percentile, inter-arrival time versus hourly traffic count data were fitted using a hyperbolic fit of the form y= (a/x) + b. The hyperbolic fits for determining cumulative IAT distributions for given percentiles for driving and passing lane are given in Table 3 and Table 4 respectively. Each of these fits may be used to compute an IAT at that cumulative percentile for a given hourly traffic count. The R2 values for each fit equation are also shown in Table 3 and Table 4. 29 Table 3: Hyperbolic Fit Formulae used in Excel Spreadsheet for Determining Cumulative IATs for Selected Percentiles for I-76 Westbound Driving Lane (X is hourly traffic count in vphpl and Y is cumulative IAT in seconds.) Cumulative Percentage Model 0% 0.1 1% Y = 122.06/X + 0.4947 2% Y = 183.83/X + 0.5566 5% Y = 198.45/X + 0.8047 10% Y = 469.05/X + 0.7137 20% Y = 892.65/X + 0.6247 30% Y = 1254.82/X + 0.6556 40% Y = 1701.76/X - 0.6734 50% Y = 2322.24/X + 0.5729 60% Y = 3208.26/X + 0.2549 70% Y = 4295.84/X -0.1387 80% Y = 5390.20/X + 0.0199 90% Y = 7592.15/X - 0.1768 95% Y = 10848.2/X - 2.2824 98% Y = 12050.26/X - 0.3884 99% Y = 12842.42/X + 0.8547 100% (max) Y = 13495.82/X + 6.3496 * IAT value for 0% was arbitrarily set to 0.1s. R2 * 0.1824 0.3272 0.3352 0.6722 0.8410 0.9091 0.9248 0.9442 0.9542 0.9641 0.9642 0.9167 0.9180 0.8842 0.8293 0.6329 30 Table 4: Hyperbolic Fit Formulae used in Excel Spreadsheet for Determining Cumulative IATs for Selected Percentiles for I-76 Westbound Passing Lane (X is hourly traffic count in vphpl and Y is cumulative IAT in seconds.) Cumulative Percentage 0% 1% 2% 5% 10% 20% 30% 40% 50% 60% 70% 80% 90% 95% 98% 99% 100% (max) Model 0.1 Y = 12.22/X + 0.4753 Y = 29.33/X + 0.5211 Y = 57.77/X + 0.5776 Y = 281.99/X + 0.1617 Y = 443.98/X + 0.1633 Y = 762.81/X - 0.0794 Y = 1266.43/X - 0.3614 Y = 1976.05/X - 0.6918 Y = 3166.15/X -1.3412 Y = 4388.64/X -1.0848 Y = 6357.94/X -1.0673 Y = 9498.70/X - 0.2287 Y = 10960.47/X + 5.3074 Y = 11411.96/X + 14.6193 Y = 11656.48/X + 21.5875 Y = 12419.46/X + 32.9255 R2 * 0.0301 0.0931 0.2743 0.3636 0.6756 0.7104 0.8680 0.8916 0.9036 0.9378 0.9319 0.8455 0.8692 0.8042 0.7491 0.6723 * IAT value for 0% was arbitrarily set to 0.1s. Using the hyperbolic fit distributions, a spreadsheet was created that allows a user to type in a volume level in vphpl and extract cumulative distribution function values at 1%, 2%, 5%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 95%, 98%, 99%, and 100%. Using the values at these percentiles a cumulative density function for IATs is generated. A table of IAT distribution at cumulative percentile levels as a function of hourly traffic counts for driving lane is given in Table 5 and for passing lane is given in Table 6. 31 Table 5: IATs in Seconds based on Hyperbolic Formulae Developed for Driving Lane (Min 184 vehicles/hour, Max 782 vehicles/hour) Cumulative Percentage 1% 2% 5% 10% 20% 30% 40% 50% 60% 70% 80% 90% 95%...
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