This paper presents new results on sequential hypothesis tests for Markov models of time series data. In particular, a technique for sequential hypothesis testing for Markov models inferred using concepts of symbolic dynamics is developed. These models are created by discretizing the phase space of a dynamical system and the system dynamics is approximated as a finite memory Markov chain on the discrete state-space. We present sequential update rules for log-likelihood ratio statistic of Markov models under the setting of binary hypothesis testing and analyze the stochastic evolution of this statistic. The proposed technique allows us to choose a lower bound on the performance of the detector and guarantees that the test will terminate in finite time. The study is motivated by time-critical detection problems with physical systems, where a temporal model is trained and a fast reliable decision with large volumes of streaming data is desired during operation. The proposed technique is first illustrated through a simulation example. Furthermore, the ideas are tested on pressure time-series data obtained from a laboratory-scale swirl stabilized combustor, where some controlled protocols are used to induce instability. The proposed framework is used to detect and estimate onset of instability during combustion. We compare the performance with maximum-likelihood classifier and show that the proposed technique gives reliable detection of instability using fewer observations.