The 45-Rally Parabola: A Mathematical Model of Set Intensity in Modern Volleyball - A Semiotic Model of Physical and Mental Load Based on Rally Point System and FIVB Performance Benchmarks
Year: 2026
Type: preprint
Abstract: Modern volleyball is governed by the rally point system, where every rally produces one point. A standard set ending 25-20 therefore consists of 45 rallies. This study proposes a simple parabolic model to describe set intensity: f(x) = -x^2 + 45x + 1/4 where x is the rally number (1 to 45) and f(x) is the relative intensity index of the set. The constant c = 1/4 represents the baseline intensity after warm-up. Players do not start from zero physical condition, because the FIVB official warm-up of 10 minutes at the net has already been completed before rally 1. Therefore the model satisfies f(0) = 1/4 and f(45) = 1/4, representing the baseline intensity at the start and end of the set. Using the first derivative f'(x) = -2x + 45 = 0, the vertex is found at x = 22.5, between rally 22 and 23, with a peak index value of f(22.5) = 506.5. This study also shows that 45% is the central benchmark of modern volleyball. Receive 35-45%, Setter Excellent 38-42%, and Attack 40-45% all converge at 45%. Teams below 45% are below tournament average, teams above 45% are above average or championship level. From the 45 rallies, only about 50% become setter attempts. Approximately 40% end due to unforced errors and 10% end due to ace or block points, leaving 22.5 combined setter attempts, or about 11 attempts per team. This explains why the normal range is 10-15 attempts per team per set, while 22-23 attempts only occur in long deuce sets. The peak index value 506.5 is numerically consistent with the estimated total duration of one set plus baseline: 45 rallies x 11.25 seconds = 506.25 seconds, plus 1/4 baseline = 506.5. This numerical correspondence is used as a mathematical semiotic, not as a unit equivalence.
Author: Sembiring, Heryanto
Language: English
Subfield: Orthopedics and Sports Medicine
Field: Medicine
Domain: Health Sciences
Open Access status: green
OpenAlex ID: W7225406130