Burst suppression is an electroencephalogram (EEG) pattern associated with profoundly inactivated brain states characterized by cerebral metabolic depression. This pattern is distinguished by short-duration band-limited electrical activity (bursts) interspersed between relatively near-isoelectric periods (suppressions). Prior work in neurophysiology suggests that burst and suppression segments are respectively associated with consumption and regeneration of adenosine triphosphate resource in cortical networks. This indicates that once a suppression (or, burst) segment begins, the propensity to switch out of the state gradually increases with duration spent in the state. Prior EEG monitoring frameworks that track the brain state during burst suppression by tracking the estimated fraction of time spent in suppression, relative to bursts, do not incorporate this information. In this work, we incorporate this information within a hidden semi-Markov model (HSMM) wherein two states (burst & suppression) stochastically switch between each other using sojourn-time dependent transition probabilities. We demonstrate the HSMM's utility in analyzing clinical data by estimating the state probabilities, the optimal state sequence, and the brain's metabolic activation level characterized by parameters governing sojourn-time dependence in transition probabilities. The HSMM-based approach proposed here provides a novel statistical framework that advances the state-of-the-art in analyzing burst suppression EEG.