As a representative sequential pattern mining problem, counting the frequency of serial episodes from a streaming sequence has drawn continuous attention in academia due to its wide application in practice, e.g., telecommunication alarms, stock market, transaction logs, bioinformatics, etc. Although a number of serial episodes mining algorithms have been developed recently, most of them are neither stream-oriented, as they require multi-pass of dataset, nor time-aware, as they fail to take into account the time constraint of serial episodes. In this paper, we propose two novel one-pass algorithms, ONCE and ONCE+, each of which can respectively compute two popular frequencies of given episodes satisfying predefined time-constraint as signals in a stream arrives one-after-another. ONCE is only used for non-overlapped frequency where the occurrences of a serial episode in sequence are not intersected. ONCE+ is designed for the distinct frequency where the occurrences of a serial episode do not share any event. Theoretical study proves that our algorithm can correctly mine the frequency of target time constraint serial episodes in a given stream. Experimental study over both real-world and synthetic datasets demonstrates that the proposed algorithm can work, with little time and space, in signal-intensive streams where millions of signals arrive within a single second. Moreover, the algorithm has been applied in a real stream processing system, where the efficacy and efficiency of this work is tested in practical applications.