| Summary: | The zeros of the reliability polynomials of circular consecutive-<i>k</i>-out-of-<i>n</i>:F systems are studied. We prove that, for any fixed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>, the set of the roots of all the reliability polynomials (for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mi>k</mi></mrow></semantics></math></inline-formula>) is unbounded in the complex plane. In the particular case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>, we show that all the nonzero roots are real, distinct numbers and find the closure of the set of roots. For every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mi>k</mi></mrow></semantics></math></inline-formula>, the expressions of the minimum root and the maximum root are given, both for circular as well as for linear systems.
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