We revisit the security analysis of the permutation-based deterministic random bit generator (DRBG) discussed by Coretti et al. at CRYPTO 2019. Specifically, we prove that their construction, based on the sponge construction, and hence called Sponge-DRBG in this paper, is secure up to \(O\left( \min \left\{ 2^{\frac{c}{2}}, 2^{\frac{\lambda }{2}}\right\} \right) \) queries in the seedless robustness model, where \(\lambda \) is the required min-entropy and c is the sponge capacity. This significantly improves the provable security bound from the existing \(O\left( \min \left\{ 2^{\frac{c}{3}}, 2^{\frac{\lambda }{2}}\right\} \right) \) to the birthday bound. We also show that our bound is tight by giving matching attacks. As the Multi-Extraction game-based reduction proposed by Chung et al. at Asiacrypt 2024 is not applicable to Sponge-DRBG in a straightforward manner, we further refine and generalize the proof technique so that it can be applied to a broader class of DRBGs to improve their provable security. We also propose a new permutation-based DRBG, dubbed POSDRBG, with almost the optimal output rate 1, outperforming the output rate \(\frac{r}{n}\) of Sponge-DRBG, where n is the output size of the underlying permutation and \(r=n-c\) . We prove that POSDRBG is tightly secure up to \(O\left( \min \left\{ 2^{\frac{c}{2}}, 2^{\frac{\lambda }{2}}\right\} \right) \) queries. Thus, to the best of our knowledge, POSDRBG is the first permutation-based DRBG that achieves the optimal output rate of 1, while maintaining the same level of provable security as Sponge-DRBG in the seedless robustness model.