Consecutive k out of n-system

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 and Lucas-type polynomials (see Lucas polynomials).

References

[a1]	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
[a2]	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
[a3]	W. Preuss, "On the reliability of generalized consecutive systems" Nonlin. Anal. Th. Meth. Appl. , 30 : 8 (1997) pp. 5425–5429
[a4]	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
[a5]	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