The problem of quickly detecting signals among multiple sequentially observed data streams, subject to constraints on the familywise false-alarm and missed-detection probabilities, is considered under an active sampling setup where only one stream can be sampled at each time instant. Under this setup, it is in general impossible to minimize, even asymptotically, the expected decision time simultaneously for every stream. We therefore consider minimizing the expected time until either the kth signal declaration or termination simultaneously for every k. We first establish non-asymptotic lower bounds for these criteria. For k > |B|, where B denotes the true subset of signals, we show that these lower bounds are attained asymptotically as the error levels vanish by any policy that stops sampling and makes a decision for each stream according to a local Sequential Probability Ratio Test. For 1 <= k <= |B|, however, the performance depends critically on the sampling rule, and we propose one with a simple exploration mechanism that prevents switching away from signals prematurely because of early unfavorable observations. We show that, under a mild relative-rate condition on the two error levels, the resulting policy attains all the lower bounds simultaneously. Numerical studies are presented to compare the proposed policy with an oracle policy and a follow-the-leader policy.

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