AISM 53, 189-202

© 2001 ISM

## Optimal threshold for the *k*-out-of-*n* monitor with dual failure modes

### Masaaki Sibuya^{1} and Kazuyuki Suzuki^{2}

^{1}Takachiho University, Suginami-ku, Tokyo 169-8508, Japan

^{2}The University of Electro-Communications, Chofu
City, Tokyo 182-8585, Japan

(Received January 4, 1999; revised December 6, 1999)

Abstract.
A monitor consists of $n$ identical sensors
working independently. Each sensor measures a variate of output or
environment of a system, and is activated if a variate is over a
threshold specified in advance for each sensor. The monitor
alarms if at least $k$ out of $n$ sensors are activated. The
performance of the monitor, the probabilities of failure to alarm
and false alarming, depends on the number $k$, the threshold
values and the probability distributions of the variate at normal
and abnormal states of the system. In this paper, a sufficient
condition on the pair of the distributions is given under which
the same threshold values for all the sensors are optimal. The
condition motivates new orders between probability distributions.
Solving an optimization problem an explicit condition is obtained
for maximizing or minimizing a symmetric function with the
constraint of another symmetric function.

Key words and phrases:
Dose response, increasing hazard function ratio, indicator variable, Lagrangian multiplier
method, monotone likelihood ratio, Neyman-Pearson lemma, stochastic order.

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