Namespaces
Variants
Actions

Difference between revisions of "Consecutive k out of n-system"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
''consecutive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130200/c1302006.png" />-out-of-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130200/c1302007.png" /> structure, consecutive system''
+
{{TEX|done}}
 +
''consecutive $k$-out-of-$n$ structure, consecutive system''
  
An ordered sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130200/c1302008.png" /> components such that the system fails if and only if at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130200/c1302009.png" /> consecutive components fail. It is a consecutive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130200/c13020013.png" />-out-of-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130200/c13020014.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130200/c13020015.png" />-system if it works if at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130200/c13020016.png" /> consecutive components work. These systems are called circular, respectively linear, if the components are arranged in a circle, respectively on a line.
+
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]]).
 
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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Ch.A. Charalambides,  "Lucas numbers and polynomials of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130200/c13020021.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130200/c13020022.png" />"  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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130200/c13020023.png" />-out-of-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130200/c13020024.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130200/c13020025.png" />-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>
+
<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>

Latest revision as of 12:04, 12 August 2014

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
How to Cite This Entry:
Consecutive k out of n-system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Consecutive_k_out_of_n-system&oldid=32865
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article