Comparisons of Multi-State Systems with Binary Components of Different Sizes
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Comparisons of Multi-State Systems with Binary Components of Different Sizes He Yi 1 & Narayanaswamy Balakrishnan 2 & Lirong Cui 3 Received: 16 January 2019 / Revised: 8 May 2020 Accepted: 17 June 2020 # Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract
The system signatures of multi-state coherent or mixed systems with i.i.d. (independent and identically distributed) binary-state components are considered in this paper. For any multistate coherent or mixed system with m i.i.d. components, the signature of its equivalent system of size m + l can be written as a mixture of signatures of several multi-state vector k -outof- m + l: F systems. This facilitates the comparison of two multi-state systems of different sizes upon transforming the signature of the system of smaller size to the signature of its equivalent system of larger size along the lines of Navarro et al. (2008) for the usual Samaniego signature. Some numerical examples are finally presented to illustrate all the results established here. Keywords System signature . Multi-state systems . Binary-state components . Equivalentsystems Mathematics Subject Classification (2010) Primary 60E15 . Secondary 62H10
1 Introduction Coherent systems are those whose components are all relevant to the state of the system, and improvement in the performance of any component cannot result in a reduction in the overall
* Lirong Cui [email protected] He Yi [email protected] Narayanaswamy Balakrishnan [email protected]
1
Beijing University of Chemical Technology, Beijing, China
2
McMaster University, Hamilton, Ontario, Canada
3
Beijing Institute of Technology, Beijing, China
Methodology and Computing in Applied Probability
reliability of the system. The structure function is an important concept that is used to describe coherent systems, but it lacks probabilistic interpretation; consequently, it is not so useful while establishing statistical and probabilistic results concerning comparisons of coherent systems. In this regard, Samaniego (1985) proposed the concept of signature for coherent systems, which has since become a useful tool in studying the performance and characteristics of coherent systems with i.i.d. components. Under the i.i.d. assumption, the signature of a coherent system of size n is defined as a vector s = (s1, s2, …, sn), in which the ith component si is the probability that the system failure is caused by the ith ordered component failure. Signatures have many extensions and applications in engineering reliability (see Samaniego 2007) and also see Yi and Cui (2018) for an alternative method for its computation. For coherent systems of the same size, many different stochastic comparison results exist in the literature; for example, Kochar et al. (1999) presented preservation theorems for stochastic ordering, hazard rate ordering and likelihood ratio ordering using signatures. Navarro et al. (2016) obtained ordering preservation results under the assumption of generalized distorted distributions for coherent syst
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