By means of standardized statistical tests applied to empirical traffic data (recorded at the Expressway R1 in Prague, Czech Republic) we verify a hypothesis that vehicular clearances are distributed via GeneralizedWe formalize mathematical theory explaining recent results obtained by means of advanced statistical analysis applied to vehicular/pedestrian microstructure. For these purposes we use (and generalize) approaches applied in the theory of counting processes. Quantities standardly analyzed in vehicular headway modeling (headway, multi-headway, interval frequency, and statistical rigidity) are here reformulated into formal mathematical definitions and then analytical predictions for statistical rigidity of particle systems with GIG-distributed headways are compared with empirical behavior. We show that a connection between clearance distribution and rigidity in real-road data is not tight as in theoretical structures. This discrepancy is explained as a consequence of the fact that interaction rules acting in vehicular systems (unlike level processes studied) are not short-ranged, which supports a hypothesis that mutual interactions exist among several succeeding cars (as investigated in Krbálek et al 2018 Physica A 491, 112).