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April, 1979 Multivalued State Component Systems
Sheldon M. Ross
Ann. Probab. 7(2): 379-383 (April, 1979). DOI: 10.1214/aop/1176995096


Consider a system that is composed of $n$ components, each of which is operating at some performance level. We suppose that there exists a nondecreasing function $\phi$ such that $\phi(x_1, \cdots, x_n)$ denotes the performance level of the system when the $i$th component's performance level is $x_i, i = 1, \cdots, n$. We allow both $x_i$ and $\phi(x_1, \cdots, x_n)$ to be arbitrary nonnegative numbers and extend many of the important results of the usual binary model to this more general framework. In particular, we obtain a fundamental inequality for $E\lbrack\phi(X_1, \cdots, X_n)\rbrack$ when $\phi$ is binary, which can, among other things, be used to generate a host of inequalities concerning increasing failure rate average distributions including, as a special case, the convolution and system closure theorem. We also define the concept of an increasing failure rate average stochastic process and prove the analog of the closure theorem; and then also do the same for new better than used stochastic processes.


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Sheldon M. Ross. "Multivalued State Component Systems." Ann. Probab. 7 (2) 379 - 383, April, 1979.


Published: April, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0395.60078
MathSciNet: MR525062
Digital Object Identifier: 10.1214/aop/1176995096

Primary: 60K10
Secondary: 62N05

Keywords: closure theorem , increasing failure rate average , Multivalued , stochastic process

Rights: Copyright © 1979 Institute of Mathematical Statistics


Vol.7 • No. 2 • April, 1979
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