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On the reliability of generalized consecutive systems
N&ineorAmly~i~.
Theory.
Methods
& Applicnriom, Proc. 2nd
PII: SO362-546X(%)00114-9
Vol. 30, No. 8, pp. 5425-5429, 1997 World Congressof Nonlinear Analysts 0 1997 Elsevier Science Ltd Printed in Great Britain. All rightsreserved 0362-546X/97 $17.00 + 0.00
ON THE RELIABILITY OF GENERALIZED CONSECUTIVE SYSTEMS WOLFGANG PREXJSS Pachbereich
lnformatik
Key wora3 and phrases: consecutively comected system
und Mathematik,
Hochschule
fiir Technik & Wichaft,
01069 Dresden,
Germany
reliability, consecutive-k-out-of-n:P system, connected-X-out-of-@&F lattice system, system, consecutive k-witbin-m-out-of-n:P system, k-within-(r,s)-out-of-(m,n):F lattice
1. INTRODUCTION
The consecutive-k-out-of-n:F system and its generalizations have attracted considerable attention in a great number of papers (see References in [2], [4], [ 121). A CONSECUTIYE-k-OUT-OF-n:F SYSTEM consists of a sequence of n (linearly or cyclically) ordered components which either fail or operate. The system fails whenever k consecutive components are failed, 1< k
Pr(z, =J] =
pi4
,
Epij=
1
(i=O,l,...,rl)
j=O
where Z, =j
0 = 1, . . . , k(o) means that the component (0 is directly connected to {i+ 1, i+2,.. ., m)
5425
Second World Congress
5426
of Nonlinear Analysts
(m = min(i+j, n+ 1)) by absolutely reliable arcs. Zi = 0 represents the failure state of component (0. The Zi are assumed to be independent and the system is considered to be failed if there is no path from the component (0) (the source) to the sink (n+l). In case of the linear CCSMC the source is not absolutely reliable. A generalization of the circular consecutive-k-out-of-n:F system is the circular CCSMC including only n elements {l, . . . , n}, i.e. there is no source and no sink. In the circular case the system is functioning if for every component (j) there is a component (i) such that 0) is within the present range r of (0, i.e. zi = r (
Another one dimensional generalization of the consecutive system is the CONSECUTIVE kWITHIN-m-OUT-OF-n:F SYSTEM consisting of n linearly or cyclically ordered independent p, or failed with components { 1,. . .,n}. The component (9 is either functioning with probability probability qi = l-p, . The whole system fails iff among any m subsequent components there are k or more failed ones. In case of k = m the system is a consecutive-m-out-of-n:F system. This type of system is a mathematical model for situations which occur in quality control or “sliding window” detection. It has been studied in several papers of Papastavridis, Koutras, Sfakianakis, Kounias, Hillaris, Psillakis, Cai (see Ref. in [2],[4]) where reliability bounds and, in special cases, formulas for its reliability have been established. In section 2 of the present paper we give recursive procedures evaluating the exact reliability of the linear and circular consecutive k-within-m-out-of-n:F systems whose component reliabilities may be unequal. A generalization of the previous system and also of the connected-(r,s)-out-of-(m,n):F lattice system is the k-WIT-(r,s)-ovTF-(m,n):F LATTICE SYSTEM (see [4]) whose (two state-) components are ordered like the elements of a (m&-matrix. The components are assumed to be independent and to have equal reliabilities. The system fails if at least one (r,s)-submatrix of its components contains k or more failed components. In section 3 we give lower and upper bounds for the system’s reliability. Further systems connected with the consecutive system are investigated in the papers [6]-[ 111.
2. CONSECUTIVE k-WITHIN-n-OUT-OF-n:F
SYSTEM
Linear consecutive k-within-m-out-of-n:Fsystem (see [3]). We need some denotations: xi
-
4
-
EIj
-
Btm
-
a binary number, iE { 1,. . . , m}, xi = l-x,; a binary random variable characterizing the component (j) , &(X=1) = pj, Pr&=O) = qj , pi + qj = 1 , jE {l,..., n}; the event that the linear consecutive k-within-m-out-of-&i-I+ 1):F system consisting of the components {I , ._. , j} is in working condition, j-l+ 1 km; the set of all m-element binary vectors (xl,. .., x,,J such that x, + . . . + x,,, > m-k.
Then the Reliability Rli, of the system is given by
where the probabilities
Pr(E,,. IX,_,,,+1 = x,, . . . , X,, = x,,J are computed by recursion from the following
Second World Congress of Nonlinear Analysts equations:
,
x1
+ .. .
5427
+x,,,
, x, + 1..+x,>m-k
and, forj = m, . . . , n-l
, if
0
=
I
xl+...+x,
Z+i’lj I Xi-l+, =O*Xj_m+a =X1,‘.‘,Xj=X,_1)Qj_a+l +
+ MErj I Xj_n+r=‘,‘j_~+~=~~,..‘,Xj=X~_~)~j-~ol+~, if
x1 +... +x,>m-k
On the basis of these formulas the system’s reliability can be found by means of a corresponding algorithm which is particularly quick for k = 2 . Circular consecutive k-within-m-out-ofn:F
system (see [5]).
Let us introduce the following denotations: 4
-
the binary random variable characterizing the component 0) , 1) = pi, Pr&=O) = qj , jE { l,...) n); the two-argument operator defined for s,tE ( 1,. . . ,n) , s Cl3t = (s+t-1)mod n + 1 ; the cardinallity of the set {s,s@l,..., t) , s,tE {l,..., n},
Pr&=
63
-
v(&G
-
v(s,t)
4,
-
Rcir
-
%,r
-
a;,”
-
=
t-s+1
, s.5t
, s>t 1 n-(s-t+l) event; the linear consecutive k-within-m-out-of-v(s,t):F system consisting of the components (s,scB 1,. ..,t> is in working condition ; reliability of the circular consecutive k-within-m-out-of-n:F system consisting of the components {1,. . . ,n} ; the set of all l-element binary vectors (x,,. .., XJ such that 1. x, + . . . + x, 2 l-k+1 , lm ; the set of all n-element binary vectors (xl,. .., x,,) such that sz m-k+1 for each jE{l,...,n) xjetO + ... + xj@(ml)
The reliability R,, of the system is given as follows: If n 2 2m-1 then Rti=
J’W&-, c Cv,,...y~,....;r,-d~Y~_~
I XI=~l~...J,,,_l =+Jn_m+l=Y~,...& Ill-1 ’
=Y,_J
x
Second World Congress of Nonlinear Analysts
5428
where the conditional probabilities P&J?,,,, 1 X,=x,,. by recursion from the following equations:
.,X,,., =x,,,.~,X~_~+~ =y,,. . .,X,., =ym,)
are computed
and, forjE(2m2,...,n-2)
pr(E,j+l I Xl =xI,**.,Xm-l=Xm_l,Xj-m+3 =Yl *...‘Xj+l'Ym_1) =
’0 M’ij
if
(x,+...+x,_i~m-k-l)V@,+...+y,_,Sm-k-l)
I Xi =xi ,-**yXm_i ‘xm-1 ,Xj_m+z= I ,Xj_n+s =Yt 9*.*,X,=Ym_&P,-m+a if
(xi+...+x,_i >m-k)h@,+...+y,_,
=m-k)
zz
If n < 2m-1 then R,,
has to be computed by enumeration from the formula
Rb =
c (+.xJ~
l!j(p,xj
+qjij)
i-1
The algorithm based on these formulas is particularly quick for k = 2 too. In this case some of the equations acquire much simpler form, which significantly accelerates the computations. Also larger values m are then admissible.
3. k-WITHIN-(r,s)-oUT-oF~m,~):F
LATTICE SYSTEM
The upper and lower bounds for the k-within-(r,s)-out-of-(m,n):F identical components are given by (see [4])
where
lattice system’s reliability
with
Second World Congress
of Nonlinear Analysts
5429
and p is the reliability of the components. The approximation of the system’s reliability by its lower and upper bounds is a rough one. Yet its accuracy increases if the size of the @,$)-matrix increases vs. the size of the (m&-matrix; the difference rs-k decreases; the component reliability tends to 1. REFERENCES 1. 2. 3. 4. 5. 6. I. 8. 9. 10. 11 12.
13. 14.
T.K. Boebme, A. Kossow and W. Preuss, A generalization of consecutive-k-out-of-n:F systems, IEEE Transacrionr on Reliability, vol. R-41, 3, pp. 451-457 (1992). M. Chao, J. Fu and M. Koutras, Survey of reliability studies of consecutive-k-out-of-n:F & related systems, IEEE Transactions on Reliabiliry, vol. 44, No.1 , pp. 120-127 (1995). J. Malinowski and W. Preuss, A recursive algorithm evaluating the exact reliability of a consecutive-k-within-m-out-of-n:F system, Microelecfron. Rehb., vol. 35, No. 12, pp. 1461-1465 (1995). J. Malinowski and W. Preuss, On the reliability of generalized consecutive systems-A survey, Inrernur. Journal of Reliability, Quality and Safety Engineering, vol. 2, No. 2, pp. 187-201 (1995). J. Malinowski and W. Preuss, A recursive algorithm evaluating the exact reliability of a circular consecutive-k-witm-outof-n:F system, Microelectron. Reliab. (accepted). J. Malinowski and W. Preuss, Reliability evaluation for tree-structured systems with multistate components. Microelecrr. Reliab., vol. 36, No. 1, pp. 9-17 (1996). J. Malinowski and W. Preuss, Reliability of reverse-tree-structured systems with multistate components, Microelectr. Reliab., vol. 36, No. 1, pp. l-7 (1996). J. Mahowski and W. Preuss. Reliability of a 2-way linear consecutively connected system with multistate components, Microelectr. Reliab. (accepted). J. Malinowski and W. Preuss, Reliability of 2-way circular consecutively connected system with multistate components, Microelectr. Reliab. (submitted). J. Malinowski and W. Preuss, Reliability increase of consecutive k-out-of-n:F and related systems through components’ rearrangement, Microeledr. Reliab. (accepted). J. Malinowski and W. Preuss, A parallel algorithm evaluating the reliability of a system with known minimal cuts (paths), Microelectr. Reliab. (accepted). W. Preuss and T.K. Boehme, On reliability analysis of consecutive-k-out-of-n+ systems and their generalizations - A survey, Proc. of lth International Conference on “Approximations, Probability and Related Fields”, Plenum 94, Pub]. Company. M. Sfakianakis and S.G. Papastavridis, Reliability of a general consecutive R-out-of-n:P system, IEEE Transactions on Reliability, vol. 42, No. 3, pp. 491-4% (1993). H. Yamamoto and M. Miyakawa, Reliability of a linear connected-(r,s)-out-of-(m,n):P lattice system, IEEE Tram. Reliab. vol 44, pp. 333-336 (1995).
Theory.
Methods
& Applicnriom, Proc. 2nd
PII: SO362-546X(%)00114-9
Vol. 30, No. 8, pp. 5425-5429, 1997 World Congressof Nonlinear Analysts 0 1997 Elsevier Science Ltd Printed in Great Britain. All rightsreserved 0362-546X/97 $17.00 + 0.00
ON THE RELIABILITY OF GENERALIZED CONSECUTIVE SYSTEMS WOLFGANG PREXJSS Pachbereich
lnformatik
Key wora3 and phrases: consecutively comected system
und Mathematik,
Hochschule
fiir Technik & Wichaft,
01069 Dresden,
Germany
reliability, consecutive-k-out-of-n:P system, connected-X-out-of-@&F lattice system, system, consecutive k-witbin-m-out-of-n:P system, k-within-(r,s)-out-of-(m,n):F lattice
1. INTRODUCTION
The consecutive-k-out-of-n:F system and its generalizations have attracted considerable attention in a great number of papers (see References in [2], [4], [ 121). A CONSECUTIYE-k-OUT-OF-n:F SYSTEM consists of a sequence of n (linearly or cyclically) ordered components which either fail or operate. The system fails whenever k consecutive components are failed, 1< k
Pr(z, =J] =
pi4
,
Epij=
1
(i=O,l,...,rl)
j=O
where Z, =j
0 = 1, . . . , k(o) means that the component (0 is directly connected to {i+ 1, i+2,.. ., m)
5425
Second World Congress
5426
of Nonlinear Analysts
(m = min(i+j, n+ 1)) by absolutely reliable arcs. Zi = 0 represents the failure state of component (0. The Zi are assumed to be independent and the system is considered to be failed if there is no path from the component (0) (the source) to the sink (n+l). In case of the linear CCSMC the source is not absolutely reliable. A generalization of the circular consecutive-k-out-of-n:F system is the circular CCSMC including only n elements {l, . . . , n}, i.e. there is no source and no sink. In the circular case the system is functioning if for every component (j) there is a component (i) such that 0) is within the present range r of (0, i.e. zi = r (
Another one dimensional generalization of the consecutive system is the CONSECUTIVE kWITHIN-m-OUT-OF-n:F SYSTEM consisting of n linearly or cyclically ordered independent p, or failed with components { 1,. . .,n}. The component (9 is either functioning with probability probability qi = l-p, . The whole system fails iff among any m subsequent components there are k or more failed ones. In case of k = m the system is a consecutive-m-out-of-n:F system. This type of system is a mathematical model for situations which occur in quality control or “sliding window” detection. It has been studied in several papers of Papastavridis, Koutras, Sfakianakis, Kounias, Hillaris, Psillakis, Cai (see Ref. in [2],[4]) where reliability bounds and, in special cases, formulas for its reliability have been established. In section 2 of the present paper we give recursive procedures evaluating the exact reliability of the linear and circular consecutive k-within-m-out-of-n:F systems whose component reliabilities may be unequal. A generalization of the previous system and also of the connected-(r,s)-out-of-(m,n):F lattice system is the k-WIT-(r,s)-ovTF-(m,n):F LATTICE SYSTEM (see [4]) whose (two state-) components are ordered like the elements of a (m&-matrix. The components are assumed to be independent and to have equal reliabilities. The system fails if at least one (r,s)-submatrix of its components contains k or more failed components. In section 3 we give lower and upper bounds for the system’s reliability. Further systems connected with the consecutive system are investigated in the papers [6]-[ 111.
2. CONSECUTIVE k-WITHIN-n-OUT-OF-n:F
SYSTEM
Linear consecutive k-within-m-out-of-n:Fsystem (see [3]). We need some denotations: xi
-
4
-
EIj
-
Btm
-
a binary number, iE { 1,. . . , m}, xi = l-x,; a binary random variable characterizing the component (j) , &(X=1) = pj, Pr&=O) = qj , pi + qj = 1 , jE {l,..., n}; the event that the linear consecutive k-within-m-out-of-&i-I+ 1):F system consisting of the components {I , ._. , j} is in working condition, j-l+ 1 km; the set of all m-element binary vectors (xl,. .., x,,J such that x, + . . . + x,,, > m-k.
Then the Reliability Rli, of the system is given by
where the probabilities
Pr(E,,. IX,_,,,+1 = x,, . . . , X,, = x,,J are computed by recursion from the following
Second World Congress of Nonlinear Analysts equations:
,
x1
+ .. .
5427
+x,,,
, x, + 1..+x,>m-k
and, forj = m, . . . , n-l
, if
0
=
I
xl+...+x,
Z+i’lj I Xi-l+, =O*Xj_m+a =X1,‘.‘,Xj=X,_1)Qj_a+l +
+ MErj I Xj_n+r=‘,‘j_~+~=~~,..‘,Xj=X~_~)~j-~ol+~, if
x1 +... +x,>m-k
On the basis of these formulas the system’s reliability can be found by means of a corresponding algorithm which is particularly quick for k = 2 . Circular consecutive k-within-m-out-ofn:F
system (see [5]).
Let us introduce the following denotations: 4
-
the binary random variable characterizing the component 0) , 1) = pi, Pr&=O) = qj , jE { l,...) n); the two-argument operator defined for s,tE ( 1,. . . ,n) , s Cl3t = (s+t-1)mod n + 1 ; the cardinallity of the set {s,s@l,..., t) , s,tE {l,..., n},
Pr&=
63
-
v(&G
-
v(s,t)
4,
-
Rcir
-
%,r
-
a;,”
-
=
t-s+1
, s.5t
, s>t 1 n-(s-t+l) event; the linear consecutive k-within-m-out-of-v(s,t):F system consisting of the components (s,scB 1,. ..,t> is in working condition ; reliability of the circular consecutive k-within-m-out-of-n:F system consisting of the components {1,. . . ,n} ; the set of all l-element binary vectors (x,,. .., XJ such that 1. x, + . . . + x, 2 l-k+1 , l
The reliability R,, of the system is given as follows: If n 2 2m-1 then Rti=
J’W&-, c Cv,,...y~,....;r,-d~Y~_~
I XI=~l~...J,,,_l =+Jn_m+l=Y~,...& Ill-1 ’
=Y,_J
x
Second World Congress of Nonlinear Analysts
5428
where the conditional probabilities P&J?,,,, 1 X,=x,,. by recursion from the following equations:
.,X,,., =x,,,.~,X~_~+~ =y,,. . .,X,., =ym,)
are computed
and, forjE(2m2,...,n-2)
pr(E,j+l I Xl =xI,**.,Xm-l=Xm_l,Xj-m+3 =Yl *...‘Xj+l'Ym_1) =
’0 M’ij
if
(x,+...+x,_i~m-k-l)V@,+...+y,_,Sm-k-l)
I Xi =xi ,-**yXm_i ‘xm-1 ,Xj_m+z= I ,Xj_n+s =Yt 9*.*,X,=Ym_&P,-m+a if
(xi+...+x,_i >m-k)h@,+...+y,_,
=m-k)
zz
If n < 2m-1 then R,,
has to be computed by enumeration from the formula
Rb =
c (+.xJ~
l!j(p,xj
+qjij)
i-1
The algorithm based on these formulas is particularly quick for k = 2 too. In this case some of the equations acquire much simpler form, which significantly accelerates the computations. Also larger values m are then admissible.
3. k-WITHIN-(r,s)-oUT-oF~m,~):F
LATTICE SYSTEM
The upper and lower bounds for the k-within-(r,s)-out-of-(m,n):F identical components are given by (see [4])
where
lattice system’s reliability
with
Second World Congress
of Nonlinear Analysts
5429
and p is the reliability of the components. The approximation of the system’s reliability by its lower and upper bounds is a rough one. Yet its accuracy increases if the size of the @,$)-matrix increases vs. the size of the (m&-matrix; the difference rs-k decreases; the component reliability tends to 1. REFERENCES 1. 2. 3. 4. 5. 6. I. 8. 9. 10. 11 12.
13. 14.
T.K. Boebme, A. Kossow and W. Preuss, A generalization of consecutive-k-out-of-n:F systems, IEEE Transacrionr on Reliability, vol. R-41, 3, pp. 451-457 (1992). M. Chao, J. Fu and M. Koutras, Survey of reliability studies of consecutive-k-out-of-n:F & related systems, IEEE Transactions on Reliabiliry, vol. 44, No.1 , pp. 120-127 (1995). J. Malinowski and W. Preuss, A recursive algorithm evaluating the exact reliability of a consecutive-k-within-m-out-of-n:F system, Microelecfron. Rehb., vol. 35, No. 12, pp. 1461-1465 (1995). J. Malinowski and W. Preuss, On the reliability of generalized consecutive systems-A survey, Inrernur. Journal of Reliability, Quality and Safety Engineering, vol. 2, No. 2, pp. 187-201 (1995). J. Malinowski and W. Preuss, A recursive algorithm evaluating the exact reliability of a circular consecutive-k-witm-outof-n:F system, Microelectron. Reliab. (accepted). J. Malinowski and W. Preuss, Reliability evaluation for tree-structured systems with multistate components. Microelecrr. Reliab., vol. 36, No. 1, pp. 9-17 (1996). J. Malinowski and W. Preuss, Reliability of reverse-tree-structured systems with multistate components, Microelectr. Reliab., vol. 36, No. 1, pp. l-7 (1996). J. Mahowski and W. Preuss. Reliability of a 2-way linear consecutively connected system with multistate components, Microelectr. Reliab. (accepted). J. Malinowski and W. Preuss, Reliability of 2-way circular consecutively connected system with multistate components, Microelectr. Reliab. (submitted). J. Malinowski and W. Preuss, Reliability increase of consecutive k-out-of-n:F and related systems through components’ rearrangement, Microeledr. Reliab. (accepted). J. Malinowski and W. Preuss, A parallel algorithm evaluating the reliability of a system with known minimal cuts (paths), Microelectr. Reliab. (accepted). W. Preuss and T.K. Boehme, On reliability analysis of consecutive-k-out-of-n+ systems and their generalizations - A survey, Proc. of lth International Conference on “Approximations, Probability and Related Fields”, Plenum 94, Pub]. Company. M. Sfakianakis and S.G. Papastavridis, Reliability of a general consecutive R-out-of-n:P system, IEEE Transactions on Reliability, vol. 42, No. 3, pp. 491-4% (1993). H. Yamamoto and M. Miyakawa, Reliability of a linear connected-(r,s)-out-of-(m,n):P lattice system, IEEE Tram. Reliab. vol 44, pp. 333-336 (1995).