OPTIMAL THRESHOLD FOR THE k-OUT-OF-n MONITOR WITH DUAL FAILURE MODES

MASAAKI SIBUYA^{1} AND KAZUYUKI SUZUKI^{2}

1Takachiho University, Suginami-ku, Tokyo 169-8508, Japan 2The University of Electro-Communications, Chōfu 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.