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The Galactic center Nuclear Star Cluster is one of the densest stellar clusters in the Galaxy. The stars in its inner portions orbit the supermassive black hole associated with the compact radio source Sgr A* at the orbital speeds of several thousand kilometers per second. The B-type star S2 is currently the best case to test the general relativity as well as other theories of gravity, based on its stellar orbit. Yet, its orbital period of approximate to 16years and the eccentricity of approximate to 0.88 yields the relativistic pericenter shift of approximate to 11', which is observationally still difficult to reliably measure due to possible Newtonian perturbations as well as reference-frame uncertainties. A naive way to solve this problem is to find stars with smaller pericenter distances, r_p <~ 1529 Schwarzschild radii (120 AU), and thus with more prominent relativistic effects. In this paper, we show that to detect stars on relativistic orbits is progressively less likely, given the volume shrinkage and the expected stellar density distributions. Finally, one arrives at a sparse region where the total number of bright stars is expected to fall below 1. One can, however, still potentially detect stars crossing this region. In this paper, we provide a simple formula for the detection probability of a star crossing a sparse region. We also examine an approximate timescale in which the star reappears in the sparse region, i.e., a 'waiting' timescale for observers.