The Cherenkov Telescope Array (CTA) is the future large observatory in the very high energy (VHE) domain. Operating from 20 GeV to 300 TeV, it will be composed of tens of Imaging Air Cherenkov Telescopes (IACTs) displaced in a large area of a few square kilometers in both the southern and northern hemispheres. Thanks to the wide energy coverage and the tremendous boost in effective area (10 times better than the current IACTs), for the first time a VHE observatory will be able to detect transient phenomena in short exposures. The CTA/DATA On-Site Analysis (OSA) is the system devoted to the development of dedicated pipelines and algorithms to be used at the CTA site for the reconstruction, data quality monitoring, science monitoring and realtime science alerting during observations. The minimum exposure required to issue a science alert is not a general requirement of the observatory but is a function of the astrophysical object under study, because the ability to detect a given source is determined by the integral sensitivity which, in addition to the CTA Monte Carlo simulations, providing the energy-dependent instrument response (e.g. the effective area and the background rate), requires the spectral distribution of the science target. The OSA integral sensitivity is computed here for the most studied source at Gamma-rays, the Crab Nebula, for a set of exposures ranging from 1000 seconds to 50 hours, using the full CTA Southern array. The reason for the Crab Nebula selection as the first example of OSA integral sensitivity is twofold: (i) this source is characterized by a broad spectrum covering the entire CTA energy range; (ii) it represents, at the time of writing, the standard candle in VHE and it is often used as unit for the IACTs sensitivity. The effect of different Crab Nebula emission models on the CTA integral sensitivity is evaluated, to emphasize the need for representative spectra of the CTA science targets in the evaluation of the OSA use cases. Using the most complete model as input to the OSA integral sensitivity, we obtain a significant detection of the Crab nebula (about 10% of flux) even for a 1000 second exposure, for an energy threshold less than 10 TeV.