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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, 8:158:30 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For each lane, three time periods were randomly selected and withdrawn for later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, interarrival 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 I76 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 I76 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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 Spring '13
 Mr.Kau

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