Journal of Combinatorial Optimization

, Volume 32, Issue 4, pp 985–1001

# $$(\alpha , \tau )$$-Monitoring for event detection in wireless sensor networks

Article

## Abstract

Detecting abnormal events is one of the fundamental issues in wireless sensor networks (WSNs). In this paper, we investigate $$(\alpha ,\tau )$$-monitoring in WSNs. For a given monitored threshold $$\alpha$$, we prove that (i) the tight upper bound of $$\Pr [{S(t)} \ge \alpha ]$$ is $$O\left( {\exp \left\{ { - n\ell \left( {\frac{\alpha }{{nsup}},\frac{{\mu (t)}}{{nsup}}} \right) } \right\} } \right)$$, if $$\mu (t) < \alpha$$; and (ii) the tight upper bound of $$\Pr [{S(t)} \le \alpha ]$$ is $$O\left( {\exp \left\{ { - n\ell \left( {\frac{\alpha }{{nsup}},\frac{{\mu (t)}}{{nsup}}} \right) } \right\} } \right)$$, if $$\mu (t) > \alpha$$, where $$\Pr [X]$$ is the probability of random event $$X,\, S(t)$$ is the sum of the monitored area at time $$t,\, n$$ is the number of the sensor nodes, $$sup$$ is the upper bound of sensed data, $$\mu (t)$$ is the expectation of $$S(t)$$, and $$\ell ({x_1},{x_2}) = {x_1}\ln \left( {\frac{{{x_1}}}{{{x_2}}}} \right) + (1 - {x_1})\ln \left( {\frac{{1 - {x_1}}}{{1 - {x_2}}}} \right)$$. An instant $$(\alpha ,\tau )$$-monitoring scheme is then developed based on the upper bound. Moreover, approximate continuous $$(\alpha , \tau )$$-monitoring is investigated. We prove that the probability of false negative alarm is $$\delta$$, if the sample size is for a given precision requirement, where is the fractile of a standard normal distribution. Finally, the performance of the proposed algorithms is validated through experiments.

## Keywords

Monitoring Distributed algorithm Sensor networks

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