Solar dynamics and turbulence occur at all heights of the solar atmosphere and could be described as stochastic processes. We propose that finite lifetime transients recurring at a certain place could trigger quasi-periodic processes in the associated structures. In this study, we developed a mathematical model for finite lifetime and randomly occurring transients, and found that quasi-periodic processes, with period longer than the time scale of the transients, are detectable intrinsically in form of trains. We simulate their propagation in an empirical solar atmospheric model with chromosphere, transition region and corona. We found that, due to the filtering effect of the chromospheric cavity, only the resonance period of the acoustic resonator is able to propagate to the upper atmosphere, such a scenario is applicable to slow magnetoacoustic waves in sunspots and active regions. If the thermal structure of the atmosphere is less wild and acoustic resonance does not take effect, the long period oscillations could propagate to the upper atmosphere. Such case would be more likely to occur at polar plumes.