Among the most employed tools to improve the performance of any wireless communication system are the multiple-input and multiple-output antennas. To achieve that task, they overcome the fading phenomenon which can affect any signal during its propagation to make it very strong for reliable communication. The signal-to-noise ratio represents the cornerstone without it, no performance criterion can be evaluated. Taking into account the complexity of evaluating a matrix of SNR by the use of a MIMO system, we have opted for a MIMO system with space-time block coding which gives an output SNR in the form of a scalar instead of a matrix product. In this regard, we have derived the effective capacity expressions for the Weibull fading channel by employing an original and tight approximate probability density function, which has been already derived in our previous works, of the output SNR of a MIMO system with space-time block coding. The channels are assumed independent and not necessarily identically distributed. Additionally, we have proposed a high-SNR approximation that is streamlined and makes it possible to analyze the system's performance in terms of its fading characteristics simply. In the end, we have checked the accuracy of our proposed expressions via Monte Carlo simulation methods.