The spatial structure of transmitters in wireless networks plays a key role in evaluating the mutual interference and hence the performance. Although the Poisson point process (PPP) has been widely used to model the spatial configuration of wireless networks, it is not suitable for networks with repulsion. The Ginibre point process (GPP) is one of the main examples of determinantal point processes that can be used to model random phenomena where repulsion is observed. Considering the accuracy, tractability and practicability tradeoffs, we introduce and promote the $β$-GPP, an intermediate class between the PPP and the GPP, as a model for wireless networks when the nodes exhibit repulsion. To show that the model leads to analytically tractable results in several cases of interest, we derive the mean and variance of the interference using two different approaches: the Palm measure approach and the reduced second moment approach, and then provide approximations of the interference distribution by three known probability density functions. Besides, to show that the model is relevant for cellular systems, we derive the coverage probability of the typical user and also find that the fitted $β$-GPP can closely model the deployment of actual base stations in terms of the coverage probability and other statistics.