To control mosquito-borne diseases such as dengue, malaria, and Zika, {\it Wolbachia}-infected male mosquitoes have been released in open areas to suppress wild mosquito population driven by cytoplasmic incompatibility (CI). In this work, we initiate a preliminary assessment on how the CI intensity $\xi$, and the mating competitiveness $\mu$ of released males relative to wild males, impact the suppression efficacy by a delay differential equation model. Our analysis identifies a threshold CI intensity $\xi_0\in (0, 1)$ as an increasing function of the natural reproduction rate of the wild mosquitoes, and a threshold value $r^*$ for the ratio $r(t)$ between the numbers of released males and wild males. The population suppression fails when $\xi\le \xi_0$, and succeeds when $\xi>\xi_0$ and $r(t)\ge r^*$. Our analyses indicate that $\xi$ plays a more important role than $\mu$ in the population suppression. For instance, a slight decrease of $\xi$ from 1 to 0.92 is more devastating than halving $\mu$ from 1 to 0.5. In our estimation of the optimal starting date for infected male release to target a more than $95\%$ wild population reduction during the peak season of dengue in Guangzhou, we find that the optimal date is almost independent of $\mu$ but is sensitive to $\xi$. If CI is complete, then starting about two months ahead can be an optimal option for less financial and labor costs. A slight reduction in the CI intensity requires a considerably earlier starting date.
Keywords: Wolbachia; cytoplasmic incompatibility; delay differential equation; mosquito population suppression; mosquito-born diseases.