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# Negative exponential distribution is the mathematical

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Unformatted text preview: . Normal distribution is the mathematical distribution that reflects the constant headway state (constant inter-arrival times). The comparison of two distribution, actual field data distribution and generated normal distribution, showed that the two distributions were quite different. Normal distribution fitted the data best for high traffic flow rates. For the analysis of intermediate headway state, May used the Pearson type III distribution as an example of the generalized mathematical model approach. The 38 comparison of the Pearson type III distribution and actual data distribution showed that the two distributions were about the same both for low and high flow rates. An example calculation procedure fo r the comparison graphs is given below for 698 vehicles per hour per diving lane data. The data used for the comparison graphs was set aside before the IAT distribution calculations. The data from 08/22/04 Sunday between 12:45 and 13:00 was used for the comparison. The 15 minute vehicle count for this set was 174 vehicles. This number first multiplied by 4 to get number of vehicles per hour and then it was multiplied by the corresponding correction factor for adjusting phantoms and misses. The adjusted number of vehicles per hour for the data found as 698 vehicles/hour/driving lane. The average IAT was 5.20 seconds with the standard deviation of 3.88 seconds. The minimum observed IAT for this 15 minute interval was 0.48 seconds and the maximum was 22.98 seconds. Histogram data with cumulative percentage values were calculated for the actual data set using MS Excel spreadsheet. For the OU fitting distribution, the adjusted number of vehicles per hour per lane was used. The IATs for cumulative percentage values 0%, 1%, 2%, 5%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 95%, 98%, 99%, and 100% (maximum) were calculated by linear interpolation using Table 8 corrected IATs table for driving lane. The values given in Table 10 were calculated for the IATs using the IAT distribution generated. 39 Table 10: Cumulative IATs Calculated using the OU Fitting Distribution Number of Vehicles per Hour Per Driving Lane = 698 Cumulative Percentage IAT (sec) 0.01% 0.10 1% 0.67 2% 0.82 5% 1.09 10% 1.39 20% 1.90 30% 2.45 40% 3.11 50% 3.90 60% 4.85 70% 6.01 80% 7.74 90% 10.70 95% 13.26 98% 16.87 99% 19.25 100% 25.68 The cumulative probabilities for the given IATs using negative exponential probability density function were calculated using MS Excel spreadsheet. The formula used for the calculation is shown in (1). f (t ) = λ × e − λt (1) where, t = IAT for which the probability is investigated (x ≥ 0.1 second, the minimum x value (IAT) was taken as 0.1) λ = 0.1922 reciprocal of the mean of the IATs for 15 minute time interval where no. of vphpl was 698) 40 The values used in Figure 9 a, b are given below in Table 11. Table 11: Cumulative Percentage Values used for Negative Exponential Distribution in OU Fitting Distribution Comparison Graph (Figure 9 a, b) Cumulative Percentage 0.01% 1% 2% 5% 10% 20% 30% 40% 50% 60% 70% 80% 90% 95% 98% 100% IAT (sec) 0.05 0.11 0.27 0.55 1.16 1.86 2.66 3.61 4.77 6.26 8.37 11.98 15.59 20.35 23.96 83.85 For the normal distribution, the MS Excel spreadsheet function was used for the calculation. MS Excel Normal Distribution function calculates the cumulative probability function, which is the integral from negative infinity to x in the formula (2). f ( x, µ , σ ) = ( x − µ )2 2σ 2 − 1 e 2π σ (2) x = IAT for which cumulative probability is investigated (x ≥ 0.1 second, the minimum x value (IAT) was taken as 0.1 seconds) 41 µ = 5.20 seconds (∞ < µ < ∞ ) (average of the IATs for 15 minute time interval where no. of vphpl was 698) σ =3.88 seconds (σ > 0 ) (standard deviation of the IATs for 15 minute time interval where no. of vphpl was 698) The values used in Figure 9 c, d are given below in Table 12. Table 12: Cumulative Percentage Values used for Normal Distribution in OU Fitting Distribution Comparison Graph (Figure 9 c, d) Cumulative Percentage 8.98% 13.91% 20.44% 28.50% 37.82% 47.92% 58.15% 67.86% 76.48% 83.64% 89.21% 93.26% 96.03% 97.79% 98.84% 99.43% 99.73% 99.88% 99.95% 99.98% 99.99% 100.00% IAT (sec) 0.1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 42 The cumulative probabilities for the given IATs using Pearson Type III distribution were calculated using Matlab. The probability density function for the Pearson Type III Distribution is given in (3). ? [?(t − a) ]K −1 e − ?(t− a) G(K) f(t) = (3) where, t = IAT for which the probability is investigated (t ≥ 0.1) λ = parameter that is a function of the mean time headway and the two user specified parameters, K and α . (λ = K =0.258, where t (average of the sample) t −α =5.20 seconds for 15 minute time interval where no. of vphpl was 698) K = user selected parameter between 0 and ∞ that affects the shape of the _ ^ distribution ( K = t− α =1.3777, where, t (average of the sample) =5.20 seconds, s s (standard deviation of the sample) = 3.88 sec...
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