Optimization of Threshold Functions over Streams

A common stream processing application is alerting, where the data stream management system (DSMS) continuously evaluates a threshold function over incoming streams. If the threshold is crossed, the DSMS raises an alarm. The threshold function is often calculated over two or more streams, such as combining temperature and humidity readings to determine if moisture will form on a machine and therefore cause it to malfunction. This requires taking a temporal join across the input streams. We show that for the broad class of functions called quasiconvex functions, the DSMS needs to retain very few tuples per-data-stream for any given time interval and still never miss an alarm. This surprising result yields a large memory savings during normal operation. That savings is also important if one stream fails, since the DSMS would otherwise have to cache all tuples in other streams until the failed stream recovers. We prove our algorithm is optimal and provide experimental evidence that validates its substantial memory savings.