We propose estimators of the buffer overflow probability in queues fed by a Markov-modulated input process and serviced by an autocorrelated service process. These estimators are based on large-deviations asymptotics for the overflow probability. We demonstrate that the proposed estimators are less likely to underestimate the overflow probability than the estimator obtained by certainty equivalence. As such, they are appropriate in situations where the overflow probability is associated with quality of service (QoS) and we need to provide firm QoS guarantees. We also show that as the number of observations increases to infinity the proposed estimators converge with probability one to the appropriate target, and thus, do not lead to underutilization of the system in this limit.