Biochemical oscillations are ubiquitous in biology and allow organisms to properly time their biological functions. In this talk, I will discuss minimal Markov state models of non-equilibrium biochemical networks that support oscillations. Two biologically relevant observables in these networks are the coherence and time period of oscillations, which are quantities are expected to depend on the detailed makeup and arrangement of transition rates in the Markov state model. First, using a perturbation theory that is valid for single-cycle networks, we obtain analytical expressions for the coherence and period which reveal that many of these details become irrelevant in the limit where a high chemical affinity drives the system out of equilibrium. This allows the coherence and time period of oscillations to be robustly maintained and tuned in the limit of high affinity. Second, I will discuss how our perturbation theory can be applied to networks with multiple cycles. We can design these more complex networks so that they maintain a constant period of oscillations when the affinity changes, a phenomenon known as input compensation.