# Characterizations of the Relative Behavior of Two Systems via Properties of Their Signature Vectors

- 1 Citations
- 2.3k Downloads

## Abstract

The signature of a system of components with independent and identically distributed (iid) lifetimes is a probability vector whose *i*th element represents the probability that the *i*th component failure causes the system to fail. (1985) introduced the concept and used it to characterize the class of systems that have increasing failure rates (IFR) when the components are iid IFR. (1999) showed that when signatures are viewed as discrete probability distributions, the stochastic, hazard rate or likelihood ratio ordering of two signature vectors implies the same ordering of the lifetimes of the corresponding systems in iid components. In this paper, these latter results are extended in a variety of ways. For example, conditions on system signatures are identified that are not only sufficient for such orderings of lifetimes to hold, but are also necessary. More generally, given any two coherent systems whose iid components have survival functions *S* _{ i } *(t)* and failure rates *r* _{ i } *(t)*, respectively, for *i*=1,2, the number and locations of crossings of the systems’ survival functions or failure rates in (0, ∞) can be fully specified in terms of the two system signatures. One is thus able to deduce how these systems compare to each other in real time, in contrast to the asymptotic comparisons one finds in the literature.

## Keywords and pharases

Coherent system mixed system hazard rate ordering stochastic ordering likelihood ratio ordering survival*k*-out-of-

*n*systems reliability crossing properties

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Barlow, R. E., and Proshan, F. (1981).
*Statistical Theory of Reliability*, To Begin With Press, Silver Springs, MD.Google Scholar - 2.Block, H., Dugas, M., and Samaniego F. J. (2004) Signature-related results on system failure rates and lifetimes,
*Technical Report No. 405*, Department of Statistics, University of California, Davis.Google Scholar - 3.Block, H., and Joe, H. (1997). Tail behavior of the failure rate functions of mixtures,
*Lifetime Data Analysis*,**3**, 269–288.zbMATHCrossRefGoogle Scholar - 4.Block, H., Li, Y., and Savits, T. (2003). Initial and final behavior of the failure rate functions for mixtures and systems,
*Journal of Applied Probability***40**, 721–740.zbMATHCrossRefMathSciNetGoogle Scholar - 5.Boland, P., and Samaniego, F. J. (2004). The signature of a coherent system and its applications in reliability, In
*Mathematical Reliability: An Expository Perspective*(Eds., R. Soyer, T. A. Mazzuchi, and N. D. Singpurwalla), Kluwer Academic Publishers, Boston.Google Scholar - 6.Kochar, S., Mukerjee, H., and Samaniego, F. J. (1999). The signature of a coherent system and its application to comparisons among systems,
*Naval Research Logistics*,**46**, 507–523.zbMATHCrossRefMathSciNetGoogle Scholar - 7.Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems,
*IEEE Transactions on Reliability*,**34**, 69–72.zbMATHCrossRefGoogle Scholar