View source for Consecutive k out of n-system

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''consecutive $k$-out-of-$n$ structure, consecutive system''

An ordered sequence of $n$ components such that the system fails if and only if at least $k$ consecutive components fail. It is a consecutive $k$-out-of-$n$: $G$-system if it works if at least $k$ consecutive components work. These systems are called circular, respectively linear, if the components are arranged in a circle, respectively on a line.

The reliability of such systems, which in simple cases amounts to probabilities of runs of consecutive successes or failures of Bernoulli trials, has connections with [[Fibonacci polynomials|Fibonacci polynomials]] and Lucas-type polynomials (see [[Lucas polynomials|Lucas polynomials]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Ch.A. Charalambides,   "Lucas numbers and polynomials of order $k$ and the length of the longest circular success run"  ''Fibonacci Quart.'' , '''29'''  (1991)  pp. 290–297</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.N. Philippou,   F.S. Makri,   "Longest circular runs with an application in reliability via the Fibonacci-type polynomials of order $k$"  G.E. Bergum (ed.)  et al. (ed.) , ''Applications of Fibonacci Numbers'' , '''3''' , Kluwer Acad. Publ.  (1990)  pp. 281–286</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Preuss,   "On the reliability of generalized consecutive systems"  ''Nonlin. Anal. Th. Meth. Appl.'' , '''30''' :  8  (1997)  pp. 5425–5429</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.A. Pekoez,   S.M. Ross,   "A simple derivation of extended reliability formulas for linear and circular consecutive $k$-out-of-$n$: $F$-systems"  ''J. Appl. Probab.'' , '''32'''  (1995)  pp. 554–557</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  Ch.A. Charalambides,   "Success runs in a circular sequence of independent Bernoulli trials"  A.P. Godbole (ed.)  St.G. Papastavrides (ed.) , ''Runs and Patterns in Probability'' , Kluwer Acad. Publ.  (1994)  pp. 15–30</TD></TR></table>