The randomized rumor spreading problem generates a big interest in the area of distributed algorithms due to its simplicity, robustness and wide range of applications. The two most popular communication paradigms used for spreading the rumor are Push and Pull algorithms. The former protocol allows nodes to send the rumor to a randomly selected neighbor at each step, while the latter is based on sending a request and downloading the rumor from a randomly selected neighbor, provided the neighbor has it. Previous analysis of these protocols assumed that every node could process all such push/pull operations within a single step, which could be unrealistic in practical situations. Therefore we propose a new framework for analysis rumor spreading accommodating buffers, in which a node can process only one push/pull message or push request at a time. We develop upper and lower bounds for randomized rumor spreading time in the new framework, and compare the results with analogous in the old framework without buffers.