Recent software is mostly constructed by reusing and composing existing components. Software components are usually stateful and therefore described using behavioral models such as finite state machines. Asynchronous communication is a classic interaction mechanism used for such software systems. However, analysing communicating systems interacting asynchronously via reliable FIFO buffers is an undecidable problem. A typical approach is to check whether the system is bounded, and if not, the corresponding state space can be made finite by limiting the presence of communication cycles in behavioral models or by fixing buffer sizes. In this paper, we focus on infinite systems and we do not restrict the system by imposing any arbitrary bounds. We introduce a notion of stability and prove that once the system is stable for a specific buffer bound, it remains stable whatever larger bounds are chosen for buffers. This enables us to check certain properties on the system for that bound and to ensure that the system will preserve them whatever larger bounds are used for buffers. We also prove that computing this bound is undecidable but show how we succeed in computing these bounds for many typical examples using heuristics and equivalence checking.