We study the reliabilities of circular $({n, f, k}):F({G})$ and $\langle {n, f, k} \rangle:F({G})$ systems. These systems involve two common failure criteria which are more suitable for real world applications. Explicit formulas for the reliabilities of these circular systems are obtained for Markov-dependent components using the distribution theory of runs. The obtained results are also important for the distribution theory of runs. Moreover, the presented formulas consist of combinatorial terms, and therefore enable easier calculations. Some numerical results and the results of a simulation study are presented.