Gene-expression bimodality, as a potential mechanism generating phenotypic cell diversity, can enhance the survival of cells in a fluctuating environment. Previous studies have shown that intrinsic or extrinsic regulations could induce bimodal gene expressions, but it is unclear whether this bimodality can occur without regulation. Here we develop an interpretable and tractable model, namely a generalized telegraph model (GTM), which considers silent transcription intervals and translational bursting, each being characterized by a general distribution. Using the queuing theory, we derive the analytical expressions of protein distributions, and show that non-exponential inactive times and translational bursting can lead to two peaks of the protein distribution away from the origin, which are different from those occurring in classical telegraph models. We also find that both silent-interval noise and translational burst-size noise can amplify gene-expression noise and induce diverse dynamic expression patterns. Our results not only provide an alternative mechanism of phenotypic switching but also could be used in explaining the bimodal phenomenon in experimental observations.