Unformatted text preview: .
Normal distribution is the mathematical distribution that reflects the constant
headway state (constant interarrival 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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 Spring '13
 Mr.Kau
 Normal Distribution, ........., Cumulative distribution function, Driving Lane

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