We revisit the problem of designing a full-state feedback controller for a deterministic finite state machine, so as to maximize a performance parameter R while simultaneously ensuring that the closed loop system satisfies a given performance objective involving a positive scaling parameter τ. Under some additional assumptions, we show that the problem of choosing τ to optimize R in closed loop admits an analytical solution. We demonstrate the use of this approach via a numerical example, showing substantial computational savings over the existing sampling based method. We also provide an intuitive, graph-theoretic interpretation of our result.