## Dynamic signatures and their use in comparing the reliability of new and used systems.(English)Zbl 1182.90036

Summary: The signature of a system with independent and identically distributed (i.i.d.) component lifetimes is a vector whose $$i$$th element is the probability that the $$i$$th component failure is fatal to the system. System signatures have been found to be quite useful tools in the study and comparison of engineered systems. In this article, the theory of system signatures is extended to versions of signatures applicable in dynamic reliability settings. It is shown that, when a working used system is inspected at time $$t$$ and it is noted that precisely $$k$$ failures have occurred, the vector $$\mathbf s \in[0,1]^{n-k}$$ whose $$j$$th element is the probability that the $$(k + j)$$th component failure is fatal to the system, for $$j = 1,2,2026;,n - k$$, is a distribution-free measure of the design of the residual system. Next, known representation and preservation theorems for system signatures are generalized to dynamic versions. Two additional applications of dynamic signatures are studied in detail. The well-known “new better than used” (NBU) property of aging systems is extended to a uniform (UNBU) version, which compares systems when new and when used, conditional on the known number of failures. Sufficient conditions are given for a system to have the UNBU property. The application of dynamic signatures to the engineering practice of “burn in” is also treated. Specifically, we consider the comparison of new systems with working used systems burned-in to a given ordered component failure time. In a reliability economics framework, we illustrate how one might compare a new system to one successfully burned-in to the $$k$$th component failure, and we identify circumstances in which burn-in is inferior (or is superior) to the fielding of a new system.

### MSC:

 90B25 Reliability, availability, maintenance, inspection in operations research
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