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Consider the problem of detecting anomalies among multiple stochastic processes. Each anomaly incurs a cost per unit time until it is identified. Due to the resource constraints, the decision-maker can select one process to probe and obtain a noisy observation. Each observation and switching across processes accompany a certain time delay. Our objective is to find a sequential inference strategy that minimizes the expected cumulative cost incurred by all the anomalies during the entire detection procedure under the error constraints. We develop a deterministic policy to solve the problem within the framework of the active hypothesis testing model. We prove that the proposed algorithm is asymptotic optimal in terms of minimizing the expected cumulative costs when the ratio of the single-switching delay to the single-observation delay is much smaller than the declaration threshold and is order-optimal when the ratio is comparable to the threshold. Not only is the proposed policy optimal in the asymptotic regime, but numerical simulations also demonstrate its excellent performance in the finite regime.

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