Monitoring Event Frequencies
Abstract
The monitoring of event frequencies can be used to recognize behavioral anomalies, to identify trends, and to deduce or discard hypotheses about the underlying system. For example, the performance of a web server may be monitored based on the ratio of the total count of requests from the least and most active clients. Exact frequency monitoring, however, can be prohibitively expensive; in the above example it would require as many counters as there are clients. In this paper, we propose the efficient probabilistic monitoring of common frequency properties, including the mode (i.e., the most common event) and the median of an event sequence. We define a logic to express composite frequency properties as a combination of atomic frequency properties. Our main contribution is an algorithm that, under suitable probabilistic assumptions, can be used to monitor these important frequency properties with four counters, independent of the number of different events. Our algorithm samples longer and longer subwords of an infinite event sequence. We prove the almostsure convergence of our algorithm by generalizing ergodic theory from increasinglength prefixes to increasinglength subwords of an infinite sequence. A similar algorithm could be used to learn a connected Markov chain of a given structure from observing its outputs, to arbitrary precision, for a given confidence.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 arXiv:
 arXiv:1910.06097
 Bibcode:
 2019arXiv191006097F
 Keywords:

 Computer Science  Logic in Computer Science;
 Computer Science  Formal Languages and Automata Theory
 EPrint:
 This is an extended version of a paper presented at the 28th EACSL Annual Conference on Computer Science Logic (CSL 2020), which provides missing proofs in the appendix