The Thirtieth Highest Hour When Hours Are Missing: An Exact Negative-Binomial Law for the Rank Shift, a Simple Rank Correction for the Design-Hour Volume, and a Test on Six Years of Interstate Counts
- 1Former Officer, CGGB
Abstract
Highway lanes are sized for the design-hour volume, usually the 30th highest hourly volume of the year, and the K-factor is that volume divided by the annual average daily traffic (AADT). Permanent counters lose hours to faults, and the 30th highest of the hours that were recorded is not the 30th highest hour of the year. I prove that if each hour is retained independently with probability p, the true rank of the k-th highest recorded hour is k plus a negative-binomial variable, with mean k/p and variance k(1 − p)/p squared. The 30th highest recorded hour is therefore, on average, the 30/p-th highest true hour, which biases the design volume downward. Using rank round(30p) instead removes the first-order bias. I also prove that the AASHTO month-by-day-of-week average for AADT is unbiased when missingness depends only on month and weekday, while the simple mean is not. On the UCI Metro Interstate Traffic Volume data (I-94 westbound, Minneapolis–Saint Paul, 2012–2018), the near-complete 2017 record gives a 30th-hour volume of 6,873 vehicles and a directional K30 of 0.0849. Simulated random deletion confirms the rank law to within 0.2 percent; at p = 0.5 the naive design volume is 104 vehicles per hour too low and the corrected rank cuts the bias to 6. Deleting whole weeks leaves the mean shift intact but more than doubles its variance. Under weekend-heavy day loss the simple AADT is 2.7 percent too high and the AASHTO average 0.3 percent.
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BibTeX
@misc{aguru2026thirtieth,
title = {{The Thirtieth Highest Hour When Hours Are Missing: An Exact Negative-Binomial Law for the Rank Shift, a Simple Rank Correction for the Design-Hour Volume, and a Test on Six Years of Interstate Counts}},
author = {Manoj Kumar Aguru},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23653382},
url = {https://zenodo.org/records/23653382},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Thirtieth Highest Hour When Hours Are Missing: An Exact Negative-Binomial Law for the Rank Shift, a Simple Rank Correction for the Design-Hour Volume, and a Test on Six Years of Interstate Counts
AU - Manoj Kumar Aguru
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23653382
UR - https://zenodo.org/records/23653382
AB - Highway lanes are sized for the design-hour volume, usually the 30th highest hourly volume of the year, and the K-factor is that volume divided by the annual average daily traffic (AADT). Permanent counters lose hours to faults, and the 30th highest of the hours that were recorded is not the 30th highest hour of the year. I prove that if each hour is retained independently with probability p, the true rank of the k-th highest recorded hour is k plus a negative-binomial variable, with mean k/p and variance k(1 − p)/p squared. The 30th highest recorded hour is therefore, on average, the 30/p-th highest true hour, which biases the design volume downward. Using rank round(30p) instead removes the first-order bias. I also prove that the AASHTO month-by-day-of-week average for AADT is unbiased when missingness depends only on month and weekday, while the simple mean is not. On the UCI Metro Interstate Traffic Volume data (I-94 westbound, Minneapolis–Saint Paul, 2012–2018), the near-complete 2017 record gives a 30th-hour volume of 6,873 vehicles and a directional K30 of 0.0849. Simulated random deletion confirms the rank law to within 0.2 percent; at p = 0.5 the naive design volume is 104 vehicles per hour too low and the corrected rank cuts the bias to 6. Deleting whole weeks leaves the mean shift intact but more than doubles its variance. Under weekend-heavy day loss the simple AADT is 2.7 percent too high and the AASHTO average 0.3 percent.
ER -