Intriguing collective spontaneous cascade emissions have recently been realized. In despite of much success, a depth understanding of the complexity is still lacking. With this motivation, a new simple cascade superradiance model is developed in this work. The existing model of identical two-level atoms is reexamined with a new insight. Temporal evolutions of average time delays and the fluctuations are introduced and the superradiance time delays are obtained in four different ways. These formulations allow to extend the two-level model to a cascade three-level model. The correlated two-mode emissions and the characteristics are discussed in detail. In the future, the correlated emissions from the collective atoms may be used, for example, in quantum noise quenching.