Tunneling spectra of near optimally doped, submicron Bi$_2$Sr$_2$CaCu$_2$O$_{8+delta}$ intrinsic Josephson junctions are presented, and examined in the region where the superconducting gap evolves into pseudogap. The spectra are analyzed using a self-energy model, proposed by Norman {it et al.}, in which both quasiparticle scattering rate $Gamma$ and pair decay rate $Gamma_{Delta}$ are considered. The density of states derived from the model has the familiar Dynes form with a simple replacement of $Gamma$ by $gamma_+$ = ($Gamma$ + $Gamma_{Delta}$)/2. The $gamma_+$ parameter obtained from fitting the experimental spectra shows a roughly linear temperature dependence, which puts a strong constraint on the relation between $Gamma$ and $Gamma_{Delta}$. We discuss and compare the Fermi arc behavior in the pseudogap phase from the tunneling and angle-resolved photoemission spectroscopy experiments. Our results indicate an excellent agreement between the two experiments, which is in favor of the precursor pairing view of the pseudogap.
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