A lossy channel system (LCS) is a computational model where nondeterministic finite automata communicate over a faulty network. They are notable in that certain LCS-related verification problems achieve a computational complexity of \(\mathbf{F}_{\omega^\omega}\)-complete, using a complexity class analog of the fast-growing hierarchy. Priority channel systems allow the automata to mark certain messages as more important than others, yielding a more powerful system with \(\mathbf{F}_{\varepsilon_0}\)-complete verification problems.