- Email: [email protected]
Comparing the MTBF of four systems with standby components
~
Microelectron. Reliab., Voh 35, No. 7, pp. 1031-1035, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0026-2714/95 $9.50+.00
1Pergamon
COMPARING THE MTBF OF FOUR SYSTEMS WITH STANDBY COMPONENTS FAN C. MENG Institute of Statistical Science, Academia Sinica Talpei 11529, Taiwan
(Received for publication 5 November 1994)
Abstract.
The mean time between failures (MTBF) of four systems arising in
standby redundancy enhancement are compared. A general ordering relationship between their M T B F is obtained. In deriving this result the usual assumption of exponential life distribution of components is removed, and the components can assume arbitrary life distributions.
1. I n t r o d u c t i o n A commonly used technique in improving system reliability (system life) is through redundancy operations.
When a component, is placed as an active redundancy with
one component in a system the two components are working in parallel; while if the spare component is in standby redundancy the spare begins to work only when the original component has failed.
Both of the two types of redundancy can enhance
system performance. On the other hand, series redundancy can he used in some safety monitoring systems, in which the redundant is connected in series with one component and this position functions if and only if the component and the redundant are both functioning. Consider a series system composed of n independent components xi, i = l . . . . . ~: the structure function of the system is denoted by (~,(x) = ¢ ~ ( x l , . . - , :rn) -- minl
1031
Fan C. Meng
1032
(a) System I
(b) System II
(c) System III
(d) System IV
Fig. 1 System l and system II are obtained by implementing the n spares as active redundancies in the series system ~s(x) at the system level and at the component level respectively; while system III and system IV are assembled by implementing the n spares as series redundancies in the parallel system 4v(x) at the component level and at the system level respectively. The structure functions of the four systems are represented as follows. SystemI: q ~ 1 ( x , z ) = 0 ~ ( Z l , . . . , x = ) V ~ ( z l , . . . , z ~ ) ; System II: Oil(x,z) = q~s(xl V z l , . . . , x ~ V z~,); SystemIII: O n 1 ( X , Z ) = q ~ p ( x l A z t , ' ' ' , x ~ A z n ) ; System IV: q~1v(x,z) = ~v(xl,.-. ,x~) A ~p(zl . . . . . z~), where 'V' and 'A' stand for maximum and minimum respectively. Assuming that the 2n independent components xl, • •., x=. zl ,. -., z~ have exponential life distributions, some ordering relations between the MTBF of the four systems have been obtained by Yamashiro et al. [4] [5] and Meng [2]. {e.g., see the recent paper [2] for the details.) Compared with parallel redundancy, standby redundancy is easier to implement and often used in mechanical designs. In this note we study the case that the components z~ . . . . . z~ are used as standby redundancy components in the systems ~b~(x) and ¢Sp(x), and obtain a general ordering relationship between the MTBF of the four redundant systems arising in these reliability operations.
2. Main R e s u l t s
Suppose that the n spare components zl . . . . . z,~ are to be implemented as standby redundancies in the series and the parallel systems to improve (expected) system lifetimes. The MTBF of the following four systems arising in this consideration are of
Standby components
1033
our interest. In the sequel, the assumption that the lifetimes of the components are exponentially distributed made in Yamashiro et al. [4] [5] and Meng [2] are removed; the symbol xl N zl means that component zl is in standby with component xl; and similarly, ¢(.) ~ ~b(.) means that system ¢(.) is in standby with system ¢(.). System I' (IV') is assembled by first connecting the components z l , . . . , zn in series (parallel), and then implementing it as a standby system with the original series (parallel) system comprising the components x l , . . . , x~ (namely, redundancy at the system level): while system II' (III') is obtained by connecting component zi as a standby spare with component xi in the original series (parallel) system, i = 1 , . . . , n. (namely, redundancy at the component level.) The structure functions of the four systems are represented as follows. System I': ( ~ p ( X , Z ) = ¢ s ( X l . . . . , X n ) ~J ~)s(Zl . . . . . Zn) ; System [I': ¢ l l , ( X , Z ) = ¢ s ( x l O z l , " ' , x , ~ z ~ ) ; System III': CgF(X,Z) = ¢v(xl ~ z l , ' ' - , x ~ ~ z~); System IV': ¢iy,(x,z) = Cv(Xl,...,z~) ~ Cv(zl . . . . . z~). Let Fi(t) and Gi(t) denote the (absolutely continuous) life distribution functions of components xi and zi; and let fi(t) and gi(t) denote their density functions respectively. Let R1,(t), Ru,(t), RHI,(t) and R1v,(t) denote the reliability functions of systems I'-IV' a.t time t; and let Ol,,Og,,OHl, and 01v, denote their mean lifelengths (MTBF), i.e.,
0(.) = f o R()(t)dt. We obtain the following ordering relationship between the MTBF of the four systems.
T h e o r e m 1. 01,
<
01i, < Oill, < Oiv,.
P r o o f . Let T = ( T I , . . . , T~) and T' = (T~ . . . . . T,'~) denote the random lifetimes of the components x = ( x l , . . . , xn)andz = (zl . . . . . z,~) respectively. From the two inequalities min { t + t ' } > min t +
l<_i<_n
-- l
mint'
l<_i
for allt. t';
and max{t+t'}
1
< maxt+ -- 1<_i
maxt'
l
for a l l t , t',
it follows that the expected lifetimes, E min {T + T'} > E rain T + E rain T ' for all T, T': t
_
+ T'} _< E max T + E max T' for aU T, T'. z
l __.i< To
1034
Fan C. Meng We thus conclude that for a series (parallel) system standby redundancy at the component level is more (less) effective than at the system level. (see Shen and Xie [3].) That is, 01, < 0ii,; and 0tit, < Orv,. We now show that Oil, < Otli,. For given time t >_ 0, let R i ( t ) (i = 1 , . . . , n) denote the resulting reliability of p o s i t i o n i at time t from the implementation of component zi as a standby to component x,. Clearly,
R~(t)
=
1 -
a~(t
-
*)L(*)d*.
The reliability functions of system II' and system III' are R.,(t/=
fo
a~(t - x)f~(./d~l,
iI/o
a,(t - x)f~(x)dx.
II[1 i=l
and R,ll,(t) = 1 -
i=1
Let a, = J~ G , ( t - x ) f ~ ( x ) d x
(i = 1. . . . , n). Then, since 0 _< al,
I-I(1 - ai) < rain (1 - al) < 1 i=1
--
l < i < n -
--
.,aN _< 1,
ai. ~=1
Hence RIp(t) < Rlll'(t), and 011, < 0II1, holds.
Conclusions.
A general ordering relationship between the MTBF of four redundant
systems arising in standby redundancy enhancement is derived. Consider the situation that system is repaired after its failure. Some components are repaired as new; while the other components assume various aging processes. The assumption of exponential life distribution of components, tacitly made in Yamashiro [4] [5], Meng [2] and many others (e.g., see [1]), makes it easy to analyze the MTBF of a system. (due to the memory-less
property.)
In this note this assumption is removed, and our results hold
for arbitrary life distributions of the components.
REFERENCES 1. A. Kumar and M. Agarwal. A review of standby redundant systems. IEEE Trans. Reliab. R-29, No. 4,290-294 (1980). 2. F.C. Meng. On comparison of MTBF between four redundant systems. Microelectron. Reliab. 33, No. 13, 1987-1990 (1993). 3. K. Shen and M. Xie. The effectiveness of adding standby redundancy at system & component levels. IEEE Trans. Reliab. 40, No. l, 53-55 (1991).
1035
Standby components 4. M. Yamashiro, M. Satoh and Y. Yuasa. Comparison of MTBF in the parallel string configuration and the quad configuration systems. Microelectron. Reliab. 32, No. 4, 557-559 (1992). 5. M. Yamashiro, M. Satoh and Y. Yuasa. Parallel-series and series-parallel redundant systems consisting of units having two kinds of failure rates. Reliab. 32, No. 5, 611-613 (1992).
Microelectron.
Microelectron. Reliab., Voh 35, No. 7, pp. 1031-1035, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0026-2714/95 $9.50+.00
1Pergamon
COMPARING THE MTBF OF FOUR SYSTEMS WITH STANDBY COMPONENTS FAN C. MENG Institute of Statistical Science, Academia Sinica Talpei 11529, Taiwan
(Received for publication 5 November 1994)
Abstract.
The mean time between failures (MTBF) of four systems arising in
standby redundancy enhancement are compared. A general ordering relationship between their M T B F is obtained. In deriving this result the usual assumption of exponential life distribution of components is removed, and the components can assume arbitrary life distributions.
1. I n t r o d u c t i o n A commonly used technique in improving system reliability (system life) is through redundancy operations.
When a component, is placed as an active redundancy with
one component in a system the two components are working in parallel; while if the spare component is in standby redundancy the spare begins to work only when the original component has failed.
Both of the two types of redundancy can enhance
system performance. On the other hand, series redundancy can he used in some safety monitoring systems, in which the redundant is connected in series with one component and this position functions if and only if the component and the redundant are both functioning. Consider a series system composed of n independent components xi, i = l . . . . . ~: the structure function of the system is denoted by (~,(x) = ¢ ~ ( x l , . . - , :rn) -- minl
1031
Fan C. Meng
1032
(a) System I
(b) System II
(c) System III
(d) System IV
Fig. 1 System l and system II are obtained by implementing the n spares as active redundancies in the series system ~s(x) at the system level and at the component level respectively; while system III and system IV are assembled by implementing the n spares as series redundancies in the parallel system 4v(x) at the component level and at the system level respectively. The structure functions of the four systems are represented as follows. SystemI: q ~ 1 ( x , z ) = 0 ~ ( Z l , . . . , x = ) V ~ ( z l , . . . , z ~ ) ; System II: Oil(x,z) = q~s(xl V z l , . . . , x ~ V z~,); SystemIII: O n 1 ( X , Z ) = q ~ p ( x l A z t , ' ' ' , x ~ A z n ) ; System IV: q~1v(x,z) = ~v(xl,.-. ,x~) A ~p(zl . . . . . z~), where 'V' and 'A' stand for maximum and minimum respectively. Assuming that the 2n independent components xl, • •., x=. zl ,. -., z~ have exponential life distributions, some ordering relations between the MTBF of the four systems have been obtained by Yamashiro et al. [4] [5] and Meng [2]. {e.g., see the recent paper [2] for the details.) Compared with parallel redundancy, standby redundancy is easier to implement and often used in mechanical designs. In this note we study the case that the components z~ . . . . . z~ are used as standby redundancy components in the systems ~b~(x) and ¢Sp(x), and obtain a general ordering relationship between the MTBF of the four redundant systems arising in these reliability operations.
2. Main R e s u l t s
Suppose that the n spare components zl . . . . . z,~ are to be implemented as standby redundancies in the series and the parallel systems to improve (expected) system lifetimes. The MTBF of the following four systems arising in this consideration are of
Standby components
1033
our interest. In the sequel, the assumption that the lifetimes of the components are exponentially distributed made in Yamashiro et al. [4] [5] and Meng [2] are removed; the symbol xl N zl means that component zl is in standby with component xl; and similarly, ¢(.) ~ ~b(.) means that system ¢(.) is in standby with system ¢(.). System I' (IV') is assembled by first connecting the components z l , . . . , zn in series (parallel), and then implementing it as a standby system with the original series (parallel) system comprising the components x l , . . . , x~ (namely, redundancy at the system level): while system II' (III') is obtained by connecting component zi as a standby spare with component xi in the original series (parallel) system, i = 1 , . . . , n. (namely, redundancy at the component level.) The structure functions of the four systems are represented as follows. System I': ( ~ p ( X , Z ) = ¢ s ( X l . . . . , X n ) ~J ~)s(Zl . . . . . Zn) ; System [I': ¢ l l , ( X , Z ) = ¢ s ( x l O z l , " ' , x , ~ z ~ ) ; System III': CgF(X,Z) = ¢v(xl ~ z l , ' ' - , x ~ ~ z~); System IV': ¢iy,(x,z) = Cv(Xl,...,z~) ~ Cv(zl . . . . . z~). Let Fi(t) and Gi(t) denote the (absolutely continuous) life distribution functions of components xi and zi; and let fi(t) and gi(t) denote their density functions respectively. Let R1,(t), Ru,(t), RHI,(t) and R1v,(t) denote the reliability functions of systems I'-IV' a.t time t; and let Ol,,Og,,OHl, and 01v, denote their mean lifelengths (MTBF), i.e.,
0(.) = f o R()(t)dt. We obtain the following ordering relationship between the MTBF of the four systems.
T h e o r e m 1. 01,
<
01i, < Oill, < Oiv,.
P r o o f . Let T = ( T I , . . . , T~) and T' = (T~ . . . . . T,'~) denote the random lifetimes of the components x = ( x l , . . . , xn)andz = (zl . . . . . z,~) respectively. From the two inequalities min { t + t ' } > min t +
l<_i<_n
-- l
mint'
l<_i
for allt. t';
and max{t+t'}
1
< maxt+ -- 1<_i
maxt'
l
for a l l t , t',
it follows that the expected lifetimes, E min {T + T'} > E rain T + E rain T ' for all T, T': t
_
+ T'} _< E max T + E max T' for aU T, T'. z
l __.i< To
1034
Fan C. Meng We thus conclude that for a series (parallel) system standby redundancy at the component level is more (less) effective than at the system level. (see Shen and Xie [3].) That is, 01, < 0ii,; and 0tit, < Orv,. We now show that Oil, < Otli,. For given time t >_ 0, let R i ( t ) (i = 1 , . . . , n) denote the resulting reliability of p o s i t i o n i at time t from the implementation of component zi as a standby to component x,. Clearly,
R~(t)
=
1 -
a~(t
-
*)L(*)d*.
The reliability functions of system II' and system III' are R.,(t/=
fo
a~(t - x)f~(./d~l,
iI/o
a,(t - x)f~(x)dx.
II[1 i=l
and R,ll,(t) = 1 -
i=1
Let a, = J~ G , ( t - x ) f ~ ( x ) d x
(i = 1. . . . , n). Then, since 0 _< al,
I-I(1 - ai) < rain (1 - al) < 1 i=1
--
l < i < n -
--
.,aN _< 1,
ai. ~=1
Hence RIp(t) < Rlll'(t), and 011, < 0II1, holds.
Conclusions.
A general ordering relationship between the MTBF of four redundant
systems arising in standby redundancy enhancement is derived. Consider the situation that system is repaired after its failure. Some components are repaired as new; while the other components assume various aging processes. The assumption of exponential life distribution of components, tacitly made in Yamashiro [4] [5], Meng [2] and many others (e.g., see [1]), makes it easy to analyze the MTBF of a system. (due to the memory-less
property.)
In this note this assumption is removed, and our results hold
for arbitrary life distributions of the components.
REFERENCES 1. A. Kumar and M. Agarwal. A review of standby redundant systems. IEEE Trans. Reliab. R-29, No. 4,290-294 (1980). 2. F.C. Meng. On comparison of MTBF between four redundant systems. Microelectron. Reliab. 33, No. 13, 1987-1990 (1993). 3. K. Shen and M. Xie. The effectiveness of adding standby redundancy at system & component levels. IEEE Trans. Reliab. 40, No. l, 53-55 (1991).
1035
Standby components 4. M. Yamashiro, M. Satoh and Y. Yuasa. Comparison of MTBF in the parallel string configuration and the quad configuration systems. Microelectron. Reliab. 32, No. 4, 557-559 (1992). 5. M. Yamashiro, M. Satoh and Y. Yuasa. Parallel-series and series-parallel redundant systems consisting of units having two kinds of failure rates. Reliab. 32, No. 5, 611-613 (1992).
Microelectron.