Processing large data streams is now a major topic in data management. The data involved can be truly massive, and the required analyses complex. In a stream of sequential events such as stock feeds, sensor readings, or IP traffic measurements, data tuples pertaining to recent events are typically more important than older ones. This can be formalized via time-decay functions, which assign weights to data based on the age of data. Decay functions such as sliding windows and exponential decay have been studied under the assumption of well-ordered arrivals, i.e., data arrives in non-decreasing order of time stamps. However, data quality issues are prevalent in massive streams (due to network asynchrony and delays etc.), and correct arrival order is not guaranteed.We focus on the computation of decayed aggregates such as range queries, quantiles, and heavy hitters on out-of-order streams, where elements do not necessarily arrive in increasing order of timestamps. Existing techniques such as Exponential Histograms and Waves are unable to handle out-of-order streams. We give the first deterministic algorithms for approximating these aggregates under popular decay functions such as sliding window and polynomial decay. We study the overhead of allowing out-of-order arrivals when compared to well-ordered arrivals, both analytically and experimentally. Our experiments confirm that these algorithms can be applied in practice, and compare the relative performance of different approaches for handling out-of-order arrivals.
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