Although neutrinos have mass, their absolute mass scale and the true nature (Majorana or Dirac) are still unknown. Neutrino and particle physics communities strongly prioritize the detection of neutrinoless double$-\beta$ decay. Neutrinoless double$-\beta$ decay is a second-order electroweak process (a hypothetical ultra-rare nuclear decay forbidden in the Standard Model), in which a nucleus decays through the emission of two electrons: $^N_Z A_{\beta\beta}\rightarrow ~_{Z+2}^{N-2}A + 2 e^-$, consequently violating the lepton number ($\Delta L = 2$). It is the most sensitive experimental probe to address the quest whether neutrinos are Majorana or Dirac particles. The observation of neutrinoless double$-\beta$ decay would not only establish the Majorana nature of neutrinos but also provide direct information on neutrino masses and probe the neutrino mass hierarchy. The present work would explore the required sensitivity for the upcoming projected neutrinoless double$-\beta$ decay experiments to probe the inverted mass hierarchy as well as non-degenerate normal mass hierarchy. We studied the required exposures of neutrinoless double$-\beta$ decay projects as a function of the expected background (following ``Discovery Potential at 3$\sigma$ with 50\% probability'' statistical scheme) before the experiments are performed. This work would address the crucial role of background suppression in the future neutrinoless double$-\beta$ decay experiments with sensitivity goals of approaching and covering non-degenerate normal mass hierarchy.