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Analytical evaluation of reliability models for multiplex systems
Microelectron. Reliab., Vol. 35, No. 6, pp. 981-983, 1995 Copyright © 1995 ElsevierScience Ltd Printed in Great Britain. All rights reserved 0026-2714/95 $9.50+ .00 0026-2714(94)00063-8
Pergamon
ANALYTICAL
EVALUATION OF RELIABILITY MODELS MULTIPLEX SYSTEMS
FOR
M. A. EL-DAMCESE Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt (Received for publication 6 May 1994) Abstract--System reliability can be modelled by a continuous time homogeneous Markov process whenever the components of such a system have constant failure and repair rates. This work presents a generalized Markovian model for a system of n-identical/nonrepairable components. The transition probability matrix and the associated set of 2" differential equations are fully described. As an illustration, a system of four-identical/repairable components is completely analysed.
1. INTRODUCTION Markov reliability models are frequently used for systems having a small number of components [1-3]. For complex systems however, the M a r k o v approach is associated with considerable difficulties as to memory requirements and solution time. Several ideas on how to solve the corresponding huge system of equations have been addressed [4-7]. This paper suggests the use of Laplace transforms to solve such systems of equations analytically. Reliability formulae for some selected configurations of a definite number of components are provided.
2. BASIC ASSUMPTIONS AND MARKOV STATE DEFINITIONS
(iii) If x: component success state ~: component failure state, then So = { x l , x2 . . . . .
x n - 1, xn},
S1 = {-~1, x2 . . . . .
x n - 1, Xn},
$2 = {X1, 3¢2. . . . .
Xn- 1, Xn} . . . . .
S2°-2 = { ~ , , ~ . . . . . & _ , , x . } ,
S2. 1={&,0~2..... & 1,&}. (iv) The set of all system states (f/) is partitioned into two subsets, one for operational system-states (U) and the other for failed system-states (D). When the system is operational, one can write S e U. System reliability at time t can be written as [8]:
2.1. Basic assumption
R(t) = Pr{S ~ V during (0, t] I S e U at t = 0}.
The transition probability P~,~ from state i at time t, Si(t) to state j at time t + dt, Sj(t + dt), takes the following special values:
3. ANALYTICAL EVALUATION 3.1. Transition probability matrix
(i) P,4 = 2 fit, when the transition includes only the failure of the one component with a constant failure rate 2. (ii) P~4 =/~ 6t, when the transition includes only the repair of the one component with a constant repair rate/~. (iii) P~,j = 0, when the transition includes more than one failure or repair.
Under the assumptions described before, the transition probability matrix for a system of n-identical components is given by:
o 1 P=
2
o I 1 -nA 2 o 1 (no 0
2
..
,~. 1)2
0 1 -(n-2)2
2"-2
2"-1
0
0
..
0
0
..
0
0 0
2.2. Markov state definitions
2" - 2
(i) The number of states for n-components configuration is given by 2" = ~",,=o (~"), where m is the number of components failed in any state and (,7,) is the combinational formula n!/(n - m)!m!, the following notations and definitions are usually utilized: (ii) System states: Sj, j = 0, 1, 2 . . . . . 2" - 1. 981
2"-I
0 o
0
1-2
0
0
0
I
(1) 3.2. State probability differential equations The state probability Pj, j = 0, 1, 2 . . . . . 2" - 1 can be viewed as a result of solving a set of 2" first order
982
M.A. EL-DAMCESE
linear differential equations given by the following identity:
d/dt P(t) = P(t)V(t)
(2)
where
P ( t ) - system-state probability vector at time t, whose entries are the system state probabilities at t. V(t) = transition-rate matrix, whose entries are the component failure rate. F r o m equation (2) we can find that
4. C A S E
4.1. System with four-identical~repairable components 4.1.1, Transition probability matrix. Under the above assumptions, the transition probability matrix for reliability with repair of a four-identical component system is given by
P=
L=_o: d/dt Pj(t) = - n2Po(t)
(3)
STUDIES
0
0
1
2
• ..
14
15
1
1 - 42
2
2
...
0
0
2
p
0 ...
0
0
1-32-p
.
14
0
0
0 ...
15
0
0
0
1-3p-2
•..
0
#
1 - 4#
(12) O
d/dt P)(t) = 2Po(t) - (n - 1)2Pj(t)
(4)
4.1.2. State probability differential equations. The following system of 24 differential equations can be easily obtained
(5)
j --- o:
n < j_~<_ 2" - - 1 :
d/dt Pj(t) = m2Pj_ l(t) - (n -
m)2Pi(t),
d/dt Po(t) = -42Po(t) + 4gP,(t) form=2,3 ..... n,j=[~ (~)]+1~=~
(13)
0
where
d/dt P)(t) = 2Po(t ) - [32 +//]P)(t) + 3#Ps(t)
(14)
4 < j ~ < 15:
Pj(t) = P~+,(t) . . . . .
Pk,
d/dt Pj(t) = m2Pj_ l(t) - [(n - m)2 + m#]P)(t) + (n -- m)pPk(t )
Using the Laplace transform technique with the following boundary conditions:
for m = 2,3,4, j = Po(0) = 1
and
Pj(O) = 0 for j > 0
(6)
+ 1, k = i=1
1
~, P j ( t ) = 1.
(7)
j=0
Po(t) = 4 e -4~t - e -3~t - e 2~, _ e - , t + (122 + 20p) x [e -2"' - e-4"']/2a - (62 + lOp)
Depending on the value of j, the solution can be derived as
x [e ~' -- e-3~']/2a + (24//4) x [1/24 -- e-~r/6 + e
j = O:
2~t/4 -
e-3~'/6
+ e-4~'/24]/a4 + [ 3823 + 90//3 + 130#2)"
(8)
Po(t) = e x p [ - nat]
+ 158//22][e-'t/3 - e-2~t + e -3~r
O
Pj(t) = e x p [ - ( n - 1)2t] - e x p [ - n 2 t ] n
+ 1 i=1
solution using Laplace transform technique with the boundary conditions (6) and (7) gives:
and 2n
(15)
- e 4at/3]/2a3 - [1422 + 30p 2 + 322//]
(9)
x [ e - " / 6 - e-2,t
1:
for m = 2 , 3 . . . . . n , j =
(10)
3e-3~t/2
2 e - 4 " / 3 ] / a 2.
-
Pj(t) = Pn-"(1 - p),n
+
(16)
Pj(t) = ).[e -2~' -- e - ' " ] / a -- 2[e -~' -- e-3"t]/2a -- [22 + 7 2 / / ] [ e - " / 6 a 2 -- e-2~'/a2
+1
+ 3e- 3"f/2a 2 -- 2e-4"t/3a 2]
ki=l
where
+ [282//2 + 423 + 1422//] P = exp[-2t].
(11)
Following the evaluation of system state probabilities one can easily derive the expression for system reliability. The expression is mainly dependent on the configuration under consideration.
x [ e - " / 6 a 3 -- e-2~t/a3 --
+
3e-3~'/2a3
2e-4~,/3a 3] + 242//3[--e-~'/6a 4
+ 1/24a 4 + e 2"t/4a 4 -- e 3~/6a 4 + e-4~t/24a4],
j = 1. . . . . 4.
(17)
Reliability models
(a)
I
0
R (t)
Co)
983
---Exp [-20kt]
It
L
R (t) = 3 e l ~ t
0
- e 13~t - 2 e lT~t + e 2°~t
, @o
(c)
R (t) = I - I 1 - exp[-htl } ~ Fig. 1. Some selected configurations for 20-identical/nonrepairable components. 14
l (a)
exs(t) = 1 -
O
~ Pj(t),
(20)
j=O
-
where a = A. + / t .
(21)
R'(t) = Po(t)
4.2. Some selected configurations (b)
Figures 1 and 2 show different configurations for 20-identicai/nonrepairable components and fouridentical/repairable components, respectively. Formulae for reliability without repair R(t), and reliability with repair R'(t) are indicated under each configuration.
•
R'(t) = Po(t)+3Pl(t)+3P5 (t)
Fig. 2. Some selected configurations for four-identical/ repairable components. REFERENCES
Pj(t) = - 222[e- ~'/6a 2 - e - 2at~a2 + 3e- 3~'/2a2
_ 2e-4~,/3a z] + [223 + 14).2#] × [e ~t/6a3 -- e- 2~t/a3 + 3e- 3~t/2a3 _ 2e-4~r/3a 3] + 2422p2[1/24 -- e-~t/6 + e - 2 " / 4 _ e-3~,/6 + e - 4 " / 2 4 ] / a 4, j = 5 . . . . . 10
(18)
P~(t) = 623[e-'t/6a3 -- e - 2"/a3 + 3e- 3"/2a3 + 2e-4"/3a3] + (24231t) × [I/24 - e - " / 6 + e - 2"'/4 - e-3~'/6 + e - 4 " / 2 4 ] / a 4, j = 11 . . . . . 14 (19)
1. A. Birolini, Reliability and availability of 2-item redundant systems, IEEE Trans. Reliab. R-24 (1985). 2. N. R. Mann et al., Methods for Statistical Analysis of Reliability and Life Data. John Wiley, New York (1974). 3. B. V. Gnedenko, Y. K. Belajev and A. D. Solovev, Mathematical Method in Reliability Theory. Nauka, Moskva (1965) [in Russian]. 4. K. C. Kapur and L. R. Lamberson, Reliability in Enoineerino Design. John Wiley, New York (1977). 5. G. Cafaro, F. Corsi and F. Vacce, Multistate Markov models and structural properties of the transition-rate matrix, IEEE Trans. Reliab. R-35 (1986). 6. R. D. Guild and E. G. Tourigny, Reliability, reliability with repair and availability of four identical element multiplex systems, Nucl. Technol. 41 (1978). 7. A. Lesanovsky, Multistate Markov models for systems with dependent units, IEEE Trans. Reliab. R-37(5) (1988). 8. D. Dyer, Unification of reliability/availability/repairability models for Markov systems, IEEE Trans. Reliab. R-38(2) (1989).
Pergamon
ANALYTICAL
EVALUATION OF RELIABILITY MODELS MULTIPLEX SYSTEMS
FOR
M. A. EL-DAMCESE Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt (Received for publication 6 May 1994) Abstract--System reliability can be modelled by a continuous time homogeneous Markov process whenever the components of such a system have constant failure and repair rates. This work presents a generalized Markovian model for a system of n-identical/nonrepairable components. The transition probability matrix and the associated set of 2" differential equations are fully described. As an illustration, a system of four-identical/repairable components is completely analysed.
1. INTRODUCTION Markov reliability models are frequently used for systems having a small number of components [1-3]. For complex systems however, the M a r k o v approach is associated with considerable difficulties as to memory requirements and solution time. Several ideas on how to solve the corresponding huge system of equations have been addressed [4-7]. This paper suggests the use of Laplace transforms to solve such systems of equations analytically. Reliability formulae for some selected configurations of a definite number of components are provided.
2. BASIC ASSUMPTIONS AND MARKOV STATE DEFINITIONS
(iii) If x: component success state ~: component failure state, then So = { x l , x2 . . . . .
x n - 1, xn},
S1 = {-~1, x2 . . . . .
x n - 1, Xn},
$2 = {X1, 3¢2. . . . .
Xn- 1, Xn} . . . . .
S2°-2 = { ~ , , ~ . . . . . & _ , , x . } ,
S2. 1={&,0~2..... & 1,&}. (iv) The set of all system states (f/) is partitioned into two subsets, one for operational system-states (U) and the other for failed system-states (D). When the system is operational, one can write S e U. System reliability at time t can be written as [8]:
2.1. Basic assumption
R(t) = Pr{S ~ V during (0, t] I S e U at t = 0}.
The transition probability P~,~ from state i at time t, Si(t) to state j at time t + dt, Sj(t + dt), takes the following special values:
3. ANALYTICAL EVALUATION 3.1. Transition probability matrix
(i) P,4 = 2 fit, when the transition includes only the failure of the one component with a constant failure rate 2. (ii) P~4 =/~ 6t, when the transition includes only the repair of the one component with a constant repair rate/~. (iii) P~,j = 0, when the transition includes more than one failure or repair.
Under the assumptions described before, the transition probability matrix for a system of n-identical components is given by:
o 1 P=
2
o I 1 -nA 2 o 1 (no 0
2
..
,~. 1)2
0 1 -(n-2)2
2"-2
2"-1
0
0
..
0
0
..
0
0 0
2.2. Markov state definitions
2" - 2
(i) The number of states for n-components configuration is given by 2" = ~",,=o (~"), where m is the number of components failed in any state and (,7,) is the combinational formula n!/(n - m)!m!, the following notations and definitions are usually utilized: (ii) System states: Sj, j = 0, 1, 2 . . . . . 2" - 1. 981
2"-I
0 o
0
1-2
0
0
0
I
(1) 3.2. State probability differential equations The state probability Pj, j = 0, 1, 2 . . . . . 2" - 1 can be viewed as a result of solving a set of 2" first order
982
M.A. EL-DAMCESE
linear differential equations given by the following identity:
d/dt P(t) = P(t)V(t)
(2)
where
P ( t ) - system-state probability vector at time t, whose entries are the system state probabilities at t. V(t) = transition-rate matrix, whose entries are the component failure rate. F r o m equation (2) we can find that
4. C A S E
4.1. System with four-identical~repairable components 4.1.1, Transition probability matrix. Under the above assumptions, the transition probability matrix for reliability with repair of a four-identical component system is given by
P=
L=_o: d/dt Pj(t) = - n2Po(t)
(3)
STUDIES
0
0
1
2
• ..
14
15
1
1 - 42
2
2
...
0
0
2
p
0 ...
0
0
1-32-p
.
14
0
0
0 ...
15
0
0
0
1-3p-2
•..
0
#
1 - 4#
(12) O
d/dt P)(t) = 2Po(t) - (n - 1)2Pj(t)
(4)
4.1.2. State probability differential equations. The following system of 24 differential equations can be easily obtained
(5)
j --- o:
n < j_~<_ 2" - - 1 :
d/dt Pj(t) = m2Pj_ l(t) - (n -
m)2Pi(t),
d/dt Po(t) = -42Po(t) + 4gP,(t) form=2,3 ..... n,j=[~ (~)]+1~=~
(13)
0
where
d/dt P)(t) = 2Po(t ) - [32 +//]P)(t) + 3#Ps(t)
(14)
4 < j ~ < 15:
Pj(t) = P~+,(t) . . . . .
Pk,
d/dt Pj(t) = m2Pj_ l(t) - [(n - m)2 + m#]P)(t) + (n -- m)pPk(t )
Using the Laplace transform technique with the following boundary conditions:
for m = 2,3,4, j = Po(0) = 1
and
Pj(O) = 0 for j > 0
(6)
+ 1, k = i=1
1
~, P j ( t ) = 1.
(7)
j=0
Po(t) = 4 e -4~t - e -3~t - e 2~, _ e - , t + (122 + 20p) x [e -2"' - e-4"']/2a - (62 + lOp)
Depending on the value of j, the solution can be derived as
x [e ~' -- e-3~']/2a + (24//4) x [1/24 -- e-~r/6 + e
j = O:
2~t/4 -
e-3~'/6
+ e-4~'/24]/a4 + [ 3823 + 90//3 + 130#2)"
(8)
Po(t) = e x p [ - nat]
+ 158//22][e-'t/3 - e-2~t + e -3~r
O
Pj(t) = e x p [ - ( n - 1)2t] - e x p [ - n 2 t ] n
+ 1 i=1
solution using Laplace transform technique with the boundary conditions (6) and (7) gives:
and 2n
(15)
- e 4at/3]/2a3 - [1422 + 30p 2 + 322//]
(9)
x [ e - " / 6 - e-2,t
1:
for m = 2 , 3 . . . . . n , j =
(10)
3e-3~t/2
2 e - 4 " / 3 ] / a 2.
-
Pj(t) = Pn-"(1 - p),n
+
(16)
Pj(t) = ).[e -2~' -- e - ' " ] / a -- 2[e -~' -- e-3"t]/2a -- [22 + 7 2 / / ] [ e - " / 6 a 2 -- e-2~'/a2
+1
+ 3e- 3"f/2a 2 -- 2e-4"t/3a 2]
ki=l
where
+ [282//2 + 423 + 1422//] P = exp[-2t].
(11)
Following the evaluation of system state probabilities one can easily derive the expression for system reliability. The expression is mainly dependent on the configuration under consideration.
x [ e - " / 6 a 3 -- e-2~t/a3 --
+
3e-3~'/2a3
2e-4~,/3a 3] + 242//3[--e-~'/6a 4
+ 1/24a 4 + e 2"t/4a 4 -- e 3~/6a 4 + e-4~t/24a4],
j = 1. . . . . 4.
(17)
Reliability models
(a)
I
0
R (t)
Co)
983
---Exp [-20kt]
It
L
R (t) = 3 e l ~ t
0
- e 13~t - 2 e lT~t + e 2°~t
, @o
(c)
R (t) = I - I 1 - exp[-htl } ~ Fig. 1. Some selected configurations for 20-identical/nonrepairable components. 14
l (a)
exs(t) = 1 -
O
~ Pj(t),
(20)
j=O
-
where a = A. + / t .
(21)
R'(t) = Po(t)
4.2. Some selected configurations (b)
Figures 1 and 2 show different configurations for 20-identicai/nonrepairable components and fouridentical/repairable components, respectively. Formulae for reliability without repair R(t), and reliability with repair R'(t) are indicated under each configuration.
•
R'(t) = Po(t)+3Pl(t)+3P5 (t)
Fig. 2. Some selected configurations for four-identical/ repairable components. REFERENCES
Pj(t) = - 222[e- ~'/6a 2 - e - 2at~a2 + 3e- 3~'/2a2
_ 2e-4~,/3a z] + [223 + 14).2#] × [e ~t/6a3 -- e- 2~t/a3 + 3e- 3~t/2a3 _ 2e-4~r/3a 3] + 2422p2[1/24 -- e-~t/6 + e - 2 " / 4 _ e-3~,/6 + e - 4 " / 2 4 ] / a 4, j = 5 . . . . . 10
(18)
P~(t) = 623[e-'t/6a3 -- e - 2"/a3 + 3e- 3"/2a3 + 2e-4"/3a3] + (24231t) × [I/24 - e - " / 6 + e - 2"'/4 - e-3~'/6 + e - 4 " / 2 4 ] / a 4, j = 11 . . . . . 14 (19)
1. A. Birolini, Reliability and availability of 2-item redundant systems, IEEE Trans. Reliab. R-24 (1985). 2. N. R. Mann et al., Methods for Statistical Analysis of Reliability and Life Data. John Wiley, New York (1974). 3. B. V. Gnedenko, Y. K. Belajev and A. D. Solovev, Mathematical Method in Reliability Theory. Nauka, Moskva (1965) [in Russian]. 4. K. C. Kapur and L. R. Lamberson, Reliability in Enoineerino Design. John Wiley, New York (1977). 5. G. Cafaro, F. Corsi and F. Vacce, Multistate Markov models and structural properties of the transition-rate matrix, IEEE Trans. Reliab. R-35 (1986). 6. R. D. Guild and E. G. Tourigny, Reliability, reliability with repair and availability of four identical element multiplex systems, Nucl. Technol. 41 (1978). 7. A. Lesanovsky, Multistate Markov models for systems with dependent units, IEEE Trans. Reliab. R-37(5) (1988). 8. D. Dyer, Unification of reliability/availability/repairability models for Markov systems, IEEE Trans. Reliab. R-38(2) (1989).