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In this paper, we consider how we can detect patterns in data streams that are serial Markovian, where target behaviors are Markovian, but targets may switch from one Markovian behavior to another. We want to reliably and promptly detect behavior changes. Traditional Markov-model-based pattern detection approaches, such as hidden Markov models, use maximum likelihood techniques over the entire data stream to detect behaviors. To detect changes between behaviors, we use statistical pattern matching calculations performed on a sliding window of data samples. If the window size is very small, the system will suffer from excessive false-positive rates. If the window is very large, change-point detection is delayed. This paper finds both necessary and sufficient bounds on the window size. We present two methods of calculating window sizes based on the state and transition structures of the Markov models. Two application examples are presented to verify our results. Our first example problem uses simulations to illustrate the utility of the proposed approaches. The second example uses models extracted from a database of consumer purchases to illustrate their use in a real application.