The H I deficiency pattern of the spiral population in the Virgo cluster region reveals a significant number of galaxies at very large clustercentric distances with gaseous deficiencies comparable to those measured in the cluster centers. We have used the output of cosmological $N$-body simulations to investigate whether the gas-deficient galaxies on the outskirts of the Virgo cluster may have previously passed through its core. We find that the maximum radius reached by infalling galaxies as they bounce out of a Virgo-like cluster must be less than 2.5 virial radii, which results in a predicted velocity-distance diagram noticeably different from the one drawn by the data. The latter is fairly well reproduced, however, after including in the simulations distance errors at the 20% relative rms level. Yet, for several objects apparently over 5 Mpc in front or behind the Virgo center the assumption that they are instead close enough to the cluster to have passed already through its core strains the bounds of plausibility. Hence, unless these outlying H I-deficient Virgo's spirals have grossly incorrect distances, they cannot have lost their gas by interactions with the intracluster medium. This suggests that the evolution of spirals might begin already on the suburbs of clusters.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html