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Optimizing replacement policy for a cold-standby system with waiting repair times
Applied Mathematics and Computation 214 (2009) 133–141
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Optimizing replacement policy for a cold-standby system with waiting repair times Jishen Jia a, Shaomin Wu b,* a b
Department of Scientific Research, Hennan Mechanical and Electrical Engineering College, Xinxiang 453002, PR China School of Applied Sciences, Cranfield University, Bedfordshire MK43 0AL, UK
a r t i c l e
i n f o
Keywords: Geometric process Cold-standby system Long-run cost per unit time Replacement policy Maintenance policy
a b s t r a c t This paper presents the formulas of the expected long-run cost per unit time for a coldstandby system composed of two identical components with perfect switching. When a component fails, a repairman will be called in to bring the component back to a certain working state. The time to repair is composed of two different time periods: waiting time and real repair time. The waiting time starts from the failure of a component to the start of repair, and the real repair time is the time between the start to repair and the completion of the repair. We also assume that the time to repair can either include only real repair time with a probability p, or include both waiting and real repair times with a probability 1 p. Special cases are discussed when both working times and real repair times are assumed to be geometric processes, and the waiting time is assumed to be a renewal process. The expected long-run cost per unit time is derived and a numerical example is given to demonstrate the usefulness of the derived expression. Ó 2009 Elsevier Inc. All rights reserved.
0. Introduction A two-component cold-standby system is composed of a primary component and a backup component, where the backup component is only called upon when the primary component fails. Cold-standby systems are commonly used for non-critical applications. However, cold-standby systems are one of most important structures in the reliability engineering and have been widely applied in reality. An example of such a system is the data backup system in computer networks. Reliability analysis and maintenance policy optimisation for cold-standby systems has attracted attentions from many researchers. Zhang and Wang [11,12] and Zhang et al. [10] derived the expected long-run cost per unit time for a repairable system consisting of two identical components and one repairman when a geometric process depicting working times is assumed. Utkin [5] proposed imprecise reliability models of cold-standby systems when he assumed that arbitrary probability distributions of the component time to failure are possible and they are restricted only by available information in the form of lower and upper probabilities of some events. Coit [1] described a solution methodology to optimal design configurations for non-repairable series–parallel systems with cold-standby redundancy when he assumed non-constant component hazard functions and imperfect switching. Yu et al. [9] considers a framework to optimally design a maintainable previous term cold-stand by next term system, and determine the maintenance policy and the reliability character of the components. Due to various reasons, repair might start immediately after a component fails. In some scenarios, from the failure of a component to the completion of repair, there might be two periods: waiting time and real repair time. The waiting time starts from the failure of the component to the start of repair; and the real repair time is the time between the start to repair
* Corresponding author. E-mail address: shaomin.wu@cranfield.ac.uk (S. Wu). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.03.064
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J. Jia, S. Wu / Applied Mathematics and Computation 214 (2009) 133–141
and the completion of the repair. This is especially true for cold-standby systems as they are not critical enough for a standby repairman be equipped for it. For example, when a component fails to work, its owner will call its contracted maintenance company or return the component to its suppler for repair. The waiting time can be the time spent by repairmen from their working place to the place where the component fails, or the time on delivering the failed component to its supplier. Usually, the waiting time can be seen as a random variable independent of the age of the component, whereas the real repair time can become longer and longer when the component becomes older. On the other hand, the working time of the component can become shorter and shorter due to various reasons such as ageing and deterioration. Such working and real repair time patterns can be depicted by geometric processes as many authors have studied [2–4,7,8,10–13]. The geometric processes introduced by Lam [4] define an alternative to the non-homogeneous Poisson processes: a sequence of random variables {Xk, k = 1, 2, . . .} is a geometric process if the distribution function of Xk is given by F(ak1t) for k = 1, 2, . . . and a is a positive constant. Wang and Pham [6] later refer the geometric process as a quasi-renewal process. Finkelstein [2] develops a very similar model: he defines a general deteriorating renewal process such that Fk+1(t) 6 Fk(t). Wu and Clements-Croome [7] extend the geometric process by replacing its parameter ak1 with a1ak1 + b1bk1, where a > 1 and 0 < b < 1. The geometric process has been applied in reliability analysis and maintenance policy optimisation for various systems by many authors; for example, see Wang and Pham [6], and Wu and Clements-Croome [8]. This paper presents the formulas of the expected long-run cost per unit time for a cold-standby system that consists of two identical components with perfect switching. When a component fails, a repairman will be called in to bring the component back to a certain working state. The time to repair is composed of two different time periods: waiting time and real repair time. The waiting time starts from the component failure to the start to repair, and the real repair time is the time between the start to repair and the completion of the repair. Both the working times and real repair times are assumed to be geometric processes, and the waiting time is assumed to be a renewal process. We also assume that the time to repair can either include only real repair time with a probability p, or include both waiting time and real repair time with a probability 1 p. The expected long-run cost per unit time is derived and a numerical example is given to demonstrate the usefulness of the derived expression. The paper is structured as follows. The coming section introduces geometric processes defined by Lam [4], denotation and assumptions. Section 3 discusses special cases. Section 4 offers numerical examples. Concluding remarks are offered in the last section. 1. Definitions and model assumptions This section first borrows the definition of the geometric process from Lam [4], and then makes assumptions for the paper. 1.1. Definition Definition 1. Assume n, g are the two random variables. For arbitrary real number a, there is
Pðn P aÞ > Pðg P aÞ; then n is called stochastically bigger than g. Similarly, if n stochastically smaller than g. Definition 2 [4]. Assume that {Xn, n = 1, 2, . . .} is a sequence of independent non-negative random variables. If the distribution function of Xn is F(an1t), for some a > 0 and all, n = 1, 2, . . ., then {Xn, n = 1, 2, . . .} is called a geometric process. Obviously, if a > 1, then {Xn, n = 1, 2, . . .} is stochastically decreasing, if a < 1, then {Xn, n = 1, 2, . . .} is stochastically increasing, and if a = 1, {Xn, n = 1, 2, . . .} is a renewal process.
1.2. Assumptions and denotation The following assumptions are assumed to hold in what follows. A. At the beginning, both components are new, component 1 is working and component 2 is under cold standby. B. When both of the two components are in good condition, one is working and the other is under cold standby. When the working component fails, a repairman repairs the failed component immediately with probability p, or repairs it with a waiting time with probability 1 p. As soon as the working component fails, the standby one will start to work. Assume the switching is perfect. After the failed one has been repaired, it is either put in use the other one fails or put in standby if the other is working. If one fails while the other is still under repair, the failed one must wait for repair until the repair for the first failed is completed.
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J. Jia, S. Wu / Applied Mathematics and Computation 214 (2009) 133–141
C. The time interval from the completion of the (n 1)th repair to that of the nth repair of component i is called the nth cycle of component i, where i = 1, 2; n = 1, 2, . . .. Denote the working time and the repair time of component i in the nth ðiÞ cycle (i = 1, 2; n = 1, 2, . . .) as X ðiÞ n and Y n , respectively, the waiting time of component i (i = 1, 2) in the nth cycle as ðiÞ ðiÞ ; n ¼ 1; 2; . . .g, the cumulative distribution functions of X ðiÞ fZ ðiÞ n n , Y n and Z n , as Fn(x) Gn(x), and S(x), respectively. ðiÞ ðiÞ ; Y ; and Z (i = 1, 2, and n = 1, 2, . . .) are statistically independent. D. X ðiÞ n n n E. When a replacement is required, a brand new but identical component will be used to replace, and the replacement time is negligible. F. Denote the repair cost per unit time of two components as Cm, the working reward per unit time as Cw, the replacement cost as Cr.
2. Expected cost under replacement policy N Fig. 1 shows a typical scenario, given that the above-mentioned assumptions hold. In what follows, we consider a replacement policy N, where a replacement is carried out if the number of failures reaches N for component 1. Denote the time between the (n 1)th replacement and the nth replacement of the system as Tn. Obviously, {T1, T2, . . .} forms a renewal process. Let C(N) be the expected long-run cost per unit time of the system under the policy N. Because {T1, T2, . . .} is a renewal process, the interval time between two consecutive replacements is a renewal cycle. Then, according to renewal reward theorem, we can know that the long-run average cost per unit time is given by
CðNÞ ¼
Expected cost incurred in a cycle : Expected length of a cycle
ð1Þ
Let W be the length of a renewal cycle of the system, then ð1Þ
W ¼ X1 þ
N1 X ð1Þ ð1Þ ð2Þ ð1Þ ð1Þ ð2Þ ð1Þ ½maxfZ i þ Y i ; X i gIfAi g þ maxfY i ; X i gIfBi g i¼1
N2 X ð2Þ ð2Þ ð1Þ ð2Þ ð2Þ ð1Þ ð2Þ ð1Þ ½maxfZ i þ Y i ; X iþ1 gIfAi g þ maxfY i ; X iþ1 gIfBi g þ X N : þ i¼1
The expected length of a renewal cycle is ð1Þ
ð1Þ
EðWÞ ¼ E½X 1 þ E½X N þ
N1 X
ð1Þ
ð1Þ
ð2Þ
ð1Þ
ð1Þ
ð2Þ
ð1Þ
E½maxfZ i þ Y i ; X i gIfAi g þ maxfY i ; X i gIfBi g
i¼1
þ
N2 X
ð2Þ E½maxfZ i
ð2Þ
ð1Þ
ð2Þ
ð2Þ
ð1Þ
ð2Þ
þ Y i ; X iþ1 gIfAi g þ maxfY i ; X iþ1 gIfBi g:
ð2Þ
i¼1
Let C be the cost of a renewal cycle of the system under the policy N, then
C ¼ Cr þ Cm Cx
( N1 X
" N X
N2 X
i¼1 ð1Þ Xi
i¼1 ð1Þ
ð1Þ
Yi þ
ð2Þ
)
i¼1
þ
N1 X
ð2Þ
ð2Þ
ð1Þ
ð2Þ
ð2Þ
ð2Þ
ð1Þ
ð2Þ
Y i þ ½Y N1 IfAg þ ðX N Z N1 ÞIfAgIðAN1 Þ þ ½Y N1 IfBg þ X N IfBgIðBN1 Þ
ð2Þ Xi
# ð3Þ
;
i¼1 ð2Þ
ð1Þ
ð2Þ
ð2Þ
ð1Þ
ð2Þ
ð1Þ
ð2Þ
where A ¼ fX N Z N1 > Y N1 g, A ¼ fX N Z N1 < Y N1 g, B ¼ fX N Y N1 > 0g, and B ¼ fX N Y N1 < 0g.
Fig. 1. A possible progressive figure of the system.
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J. Jia, S. Wu / Applied Mathematics and Computation 214 (2009) 133–141
If X and Y are two independent non-negative random variables and their cumulative distribution functions are F(x) and G(x), respectively, we have following three lemmas. Denote E(C) as the expected value of C. By substituting the numerator and denominator of Eq. (1) with E(C) and E(W), respectively, we have
CðNÞ ¼
EðCÞ : EðWÞ
ð4Þ
Then the optimal replacement number can be obtained by minimising the value of C(N) in Eq. (4). Lemma 1
Z
EðmaxfX; YgÞ ¼ EX þ
1
Z0 1
¼ EY þ
FðxÞ½1 GðxÞdx
ð5Þ
GðxÞ½1 FðxÞdx:
ð6Þ
0
The proof of Lemma 1 is given in Appendix. Lemma 2
E½IfY > XgX þ E½If0 < Y < XgY ¼
Z
1
½1 FðxÞ½1 GðxÞdx:
ð7Þ
0
The proof of Lemma 2 is given in Appendix. Similarly, we have Lemma 3
E½ðY XÞIfY X > 0g ¼
Z
1
½1 GðxÞFðxÞdx:
ð8Þ
0
3. Special cases and discussion ðiÞ n1 t) and G(bn1t), respectively, where a > 1, 0 < b < 1. fY ðiÞ Denote the distributions of X ðiÞ n and Y n as F(a n ; n ¼ 1; 2; . . .g constitutes an increasing geometric process, whereas fX ðiÞ n ; n ¼ 1; 2; . . .g constitutes a decreasing geometric process. Then we have the following theorem. n1 n1 n1 Theorem 1. Assume F n ðtÞ ¼ Fðan1 tÞ ¼ 1 exp a k t , Gn ðtÞ ¼ Gðb tÞ ¼ 1 exp b l t , and SðtÞ ¼ 1 exp ct ,
(t P 0), respectively. Then the expected length of a renewal cycle is given by
EðWÞ ¼ k þ þ2
k þ ð2N 3Þð1 pÞc aN1 N2 X i¼1
l b
i1
þ
l b
N2
þ ð1 pÞk
N2
þ
pk2 b aN2 ðb
l
N2
k þ aN2 lÞ
þ
N2 X
! 1 l2 aN2 ðcaN2 þ kÞ ðaN2 l þ bN2 kÞð2bN2 ck þ lk þ aN2 lcÞ
2
ð1 pÞk2
i¼1
1
2 i1
ðai l þ b
i1
kÞð2b
ck þ lk þ ai lcÞ i1
þp
1 1 þ ai ðcai þ kÞ ai1 ðcai1 þ kÞ
þ
!
1 ðai1 l þ b !!
i1
i1
kÞð2b
ck þ lk þ ai1 lcÞ
kð1 þ aÞ klðai l þ ai1 l þ 2b kÞ i1 i1 ai ðb k þ ai lÞðb k þ ai1 lÞ
ð9Þ
and the expected cost is a cycle is
EðCÞ ¼ C m 2
N 2 X
l i1
i¼1
b
þ Cr Cx 2
þ
l b
N2
þ ð1 pÞ
N1 X k k þ N1 i1 a a i¼1
!
2l N2
b
aN1 lc2 N2
ðaN1 c þ kÞðb
k2 l
!
c lÞ ðaN1 c þ kÞðbN2 k þ aN1 lÞ
þ
!
kl N2
b
k þ aN1 l
p
ð10Þ
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J. Jia, S. Wu / Applied Mathematics and Computation 214 (2009) 133–141
and the expected long-run cost per unit time is given by
CðNÞ ¼
EðCÞ : EðWÞ
ð11Þ
If one sets a = 1 and b = 1, the above results E(W) and E(C) will be the situations where the components can be repaired as good as new.
4. Numerical examples 4.1. Parameter set 1 If we set a = 1.8, b = 0.98, k = 100, l = 10, c = 5, Cx = 500, Cm = 20, Cr = 5000, and p = 0.8, then the optimum number for a replacement will be N = 6, and the corresponding expected long-run cost per unit time is 433.41. The expected long-run cost per unit time is shown in Table 1, which corresponds to Fig. 1. 4.2. Parameter set 2 If we set a = 1.1, b = 0.98, k = 100, l = 1, c = 0.2, Cx = 500, Cm = 20, Cr = 5000, and p = 0.8, then the optimum number for a replacement will be N = 35, and the corresponding expected long-run cost per unit time is 491.85. The expected long-run cost per unit time is shown in Table 2, which corresponds to Fig. 2. Compare Figs. 2 and 3, we can find that the optimum replacement time becomes longer in the second situation. In both situations, we can easily find an optimum replacement time point. However, due to the complexity of Eq. (11), we are not able to prove that there exists a unique optimal value N.
Table 1 The expected long-run cost per unit time versus replacement times for parameter set 1. Times
Cost rate
Times
Cost rate
Times
Cost rate
Times
Cost rate
Times
Cost rate
1 2 3 4 5 6 7 8 9 10 11 12 13 14
88.74 107.92 279.47 389.09 428.42 433.41 424.96 411.2 395.37 378.99 362.83 347.25 332.41 318.37
15 16 17 18 19 20 21 22 23 24 25 26 27 28
305.11 292.59 280.78 269.62 259.06 249.07 239.6 230.61 222.07 213.94 206.21 198.84 191.8 185.08
29 30 31 32 33 34 35 36 37 38 39 40 41 42
178.66 172.52 166.64 161 155.6 150.42 145.44 140.66 136.06 131.63 127.38 123.27 119.32 115.51
43 44 45 46 47 48 49 50 51 52 53 54 55 56
111.84 108.29 104.87 101.56 98.37 95.28 92.29 89.4 86.6 83.89 81.27 78.73 76.27 73.88
57 58 59 60 61 62 63 64 65 66 67 68 69 70
71.56 69.32 67.14 65.02 62.97 60.97 59.04 57.15 55.33 53.55 51.82 50.14 48.5 46.91
Table 2 The expected long-run cost per unit time versus replacement times for parameter set 2. Times
Cost rate
Times
Cost rate
Times
Cost rate
Times
Cost rate
Times
Cost rate
1 2 3 4 5 6 7 8 9 10 11 12 13 14
17.37 18.96 37.01 58.53 83.54 111.78 142.75 175.68 209.62 243.55 276.47 307.51 336.03 361.63
15 16 17 18 19 20 21 22 23 24 25 26 27 28
384.14 403.59 420.14 434.04 445.6 455.12 462.91 469.24 474.35 478.46 481.74 484.34 486.39 487.99
29 30 31 32 33 34 35 36 37 38 39 40 41 42
489.22 490.15 490.84 491.32 491.63 491.8 491.85 491.79 491.65 491.43 491.13 490.77 490.36 489.88
43 44 45 46 47 48 49 50 51 52 53 54 55 56
489.36 488.78 488.15 487.48 486.76 485.99 485.18 484.33 483.43 482.5 481.52 480.5 479.44 478.35
57 58 59 60 61 62 63 64 65 66 67 68 69 70
477.22 476.05 474.85 473.61 472.34 471.03 469.7 468.33 466.93 465.5 464.05 462.56 461.05 459.5
Note: times in Tables 1 and 2 stands for replacement times; cost rate stands for the expected long-run cost per unit time.
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J. Jia, S. Wu / Applied Mathematics and Computation 214 (2009) 133–141
Fig. 2. The change of C(N) over N for parameter set 1.
Fig. 3. The change of C(N) over N for parameter set 2.
5. Conclusions Cold-standby systems are a category of important reliability structure in engineering. Searching an optimal replacement time point for such systems is of interest and important. This paper derived the expected long-run cost per unit time for a cold-standby system when time to repair is composed of two time periods: waiting time and real repair time. We also considered a special scenario where the working times and real repair times are geometric processes. Numerical examples are given to demonstrate the usefulness of the derived expression. Appendix. Proof of Lemma 1 Proof. Because X, Y are two independent random variables, therefore
EðmaxfX; YgÞ ¼
Z Z Z
maxfx; yg f ðxÞgðyÞdxdy ¼ D
Z
Z
yf ðxÞgðyÞdxdy þ x6y
y
yf ðxÞgðyÞdxdy þ
0
1
x
Z
Z Z
xf ðxÞgðyÞdxdy x>y
Z 1 xf ðxÞgðyÞdydx ¼ ygðyÞFðyÞdy þ xGðxÞf ðxÞdx 0 0 0 0 0 0 Z 1 Z 1 Z 1 Z 1 ¼ xFðxÞd½1 GðxÞ þ xGðxÞf ðxÞdx ¼ ½xf ðxÞ½1 GðxÞ þ xGðxÞf ðxÞdx þ FðxÞ½1 GðxÞdx 0 0 0 0 Z 1 ¼ EX þ FðxÞ½1 GðxÞdx ¼
1
Z
Z Z
1
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J. Jia, S. Wu / Applied Mathematics and Computation 214 (2009) 133–141
where f ðxÞ ¼ dFðxÞ and gðyÞ ¼ dGðyÞ . dx dy
h
Proof of Lemma 2
Proof. As X and Y are two independent non-negative random variables,
E½IfY > XgX ¼
Z Z
xf ðxÞgðyÞdxdy ¼
Z
x
¼
Z
1
Z
0
x
x½1 GðxÞd½1 FðxÞ ¼
E½If0 < Y < XgY ¼
Z Z
yf ðxÞgðyÞdxdy ¼
xf ðxÞ½1 GðxÞdx
1
½1 GðxÞ xgðxÞ½1 FðxÞdx; 0
Z
0
1 0
Z
1 0
Z xf ðxÞgðyÞdy dx ¼
1
1
ygðyÞ½1 FðyÞdy ¼
Z
0
1
xgðxÞ½1 FðxÞdx
0
and
E½IfY > XgX þ E½If0 < Y < XgY ¼
Z
1
½1 FðxÞ½1 GðxÞdx:
0
Proof of Theorem Proof. According to the above theorems and formulas (2) and (3), we have ð1Þ
ð1Þ
EðWÞ ¼ E½X 1 þ E½X N þ
N 1 X
ð1Þ
ð1Þ
ð2Þ
ð1Þ
ð1Þ
ð2Þ
ð1Þ
E½maxfZ i þ Y i ; X i gIfAi g þ maxfY i ; X i gIfBi g
i¼1
þ
N2 X
ð2Þ
ð2Þ
ð1Þ
ð2Þ
ð2Þ
ð1Þ
ð2Þ
E½maxfZ i þ Y i ; X iþ1 gIfAi g þ maxfY i ; X iþ1 gIfBi g
i¼1 ð1Þ
ð1Þ
¼ E½X 1 þ E½X N þ
N1 X ð1Þ ð1Þ ð2Þ ð1Þ ð2Þ fE½maxfZ i þ Y i ; X i gð1 pÞ þ E½maxfY i ; X i gpg i¼1
þ
N2 X
ð2Þ
ð2Þ
ð1Þ
ð2Þ
ð1Þ
fE½maxfZ i þ Y i ; X iþ1 gð1 pÞg þ E½maxfY i ; X iþ1 gpg
i¼1
( ¼
ð1Þ E½X 1
þ
ð1Þ E½X N
þ
N 1 X
ð1Þ E½maxfZ i
þ
ð1Þ ð2Þ Y i ; X i g
N2 X
þ
i¼1
( þ
N1 X
ð1Þ ð2Þ E½maxfY i ; X i g
( ð1Þ
þ
N2 X
) ð2Þ ð1Þ E½maxfY i ; X iþ1 g
ð1Þ
N1 X
ð1Þ
ð1Þ
E½Z i þ Y i þ
i¼1 N1 X
ð1Þ
E½Y i þ
i¼1
¼kþ
þ
k aN1
k aN1
N2 Z X i¼1
p
N1 Z X i¼1
ð2Þ
E½Y i þ
N1 Z X i¼1
þ ð2N 3Þð1 pÞc þ 2
1
Hi ðtÞ½1 F i ðtÞdt þ 0
Gi ðtÞ½1 F i ðtÞdt þ
0
i¼1
N2 Z X i¼1
l b
i1
þ
ð2Þ
ð2Þ
E½Z i þ Y i þ
i¼1
1
N2 X
N2 X
l N2
b
þ
N1 Z X i¼1
N2 Z X i¼1
)
1
Hi ðtÞ½1 F iþ1 ðtÞdt ð1 pÞ
0
)
1
Gi ðtÞ½1 F iþ1 ðtÞdt p 0
1
½ð1 pÞHi ðtÞ þ pGi ðtÞ½1 F i ðtÞdt 0
1
½ð1 pÞHi ðtÞ þ pGi ðtÞ½1 F iþ1 ðtÞdt
0
i¼1
þ
N2 X i¼1
N2 Z X
¼kþ
ð1 pÞ
i¼1
¼ E½X 1 þ E½X N þ
þ
þ
ð2Þ ð1Þ Y i ; X iþ1 g
i¼1
i¼1
(
) ð2Þ E½maxfZ i
0
þ ð2N 3Þð1 pÞc þ 2
N2 X i¼1
1
½ð1 pÞHi ðtÞ þ pGðb
i1
l b
i1
þ
l N2
b
þ
Z
1
½ð1 pÞHN1 ðtÞ þ pGðb
N2
tÞ½1 FðaN2 tÞdt
0
tÞ½2 Fðai tÞ Fðai1 tÞdt
ð12Þ
140
J. Jia, S. Wu / Applied Mathematics and Computation 214 (2009) 133–141
and
# Z 1 Z 1 N2 N2 N1 þ ð1 pÞ ½1 R ðtÞ½1 Gðb tÞdt þ p ½1 Gðb tÞ½1 Fða tÞdt þ Cr N i1 0 0 i¼1 b i¼1 b " # N N1 X X k k Cx þ ai1 i¼1 ai1 i¼1 ( ) Z 1 Z 1 N2 X l l N2 N2 N1 ¼ Cm 2 þ N2 þ ð1 pÞ ½1 RN ðtÞ½1 Gðb tÞdt þ p ½1 Gðb tÞ½1 Fða tÞdt þ C r i1 0 0 b i¼1 b " # N 1 X k k ; Cx 2 þ ai1 aN1 i¼1
EL ¼ C m
" N1 X
l
þ i1
N2 X
l
ðiÞ
ðiÞ
ð1Þ
ð2Þ
where t P 0, Hi(t) and RN(x) represent the cumulative distribution functions of the random variables Z i þ Y i and X N Z N1 , respectively. Hence we have Hi (x) = S(t)*Gi(t), and RN(x) = FN(t)*[1 S(t)], where ‘‘*” indicates convolution, and
Hi ðtÞ ¼ SðtÞ Gðb
i1
" #) Z t( i1 b u tÞ ¼ 1 exp ðt uÞ d 1 exp
l
0
t ¼ 1 exp
l
c
(
cbi1 þ l
c
i1
exp
b
l
"
! t
exp
i1
2b
l
þ
1
! #)
c
t
;
N1 Z t a u 1 exp ðt uÞ d exp k c 0 N1 t k a t 1þ ; exp ¼ exp t exp k þ aN1 c c k c
RN ðtÞ ¼ FðaN1 tÞ ½1 SðtÞ ¼
where
Z
1
½ð1 pÞHN1 ðtÞ þ pGðb
0
N2
tÞ½1 FðaN2 tÞdt
"
# N2 1 l2 pk2 b þ ; ¼ ð1 pÞk N2 N2 N2 N2 N2 a ðca þ kÞ ðaN2 l þ b kÞð2b ck þ lk þ aN2 lcÞ aN2 ðb k þ aN2 lÞ 2
Z
1
½ð1 pÞHi ðtÞ þ pGðb
0
i1
tÞ½2 Fðai tÞ Fðai1 tÞdt
(
1 1 1 l2 ½ þ i1 i1 ai ðcai þ kÞ ai1 ðcai1 þ kÞ ðai l þ b kÞð2b ck þ lk þ ai lcÞ ) " # i1 1 kð1 þ aÞ klðai l þ ai1 l þ 2b kÞ i1 þp ; þ i1 i1 i1 ai ðai1 l þ b kÞð2b ck þ lk þ ai1 lcÞ ðb k þ ai lÞðb k þ ai1 lÞ
¼ ð1 pÞk2
Z
1
N2
½1 RN ðtÞ½1 Gðb
tÞdt
0
¼
Z
1
t 2 exp
c
0
¼
2l b
N2
( N2 ) N1 k a t b exp exp t dt t exp k þ aN1 c k c l
aN1 lc2 ðaN1
N2
c þ kÞðb
c lÞ
k2 l ðaN1
c þ kÞðbN2 k þ aN1 lÞ
and
Z
1
N2
½1 Gðb
0
N1
tÞ½1 Fða
tÞdt ¼
Z 0
1
( N2 ) N1 a t b kl exp t dt ¼ N2 : exp k l b k þ aN1 l
References [1] D.W. Coit, Cold-standby redundancy optimization for nonrepairable systems, IIE Trans. (Inst. Ind. Eng.) 33 (6) (2001) 471–478. [2] M.S. Finkelstein, A scale model of general repair, Microelectron. Reliab. 33 (1) (1993) 41–44. [3] J. Jia, S. Wu, A replacement policy for a repairable system with its repairman having multiple vacations, Comput. Ind. Eng. (2008), doi:10.1016/ j.cie.2008.11.003.
J. Jia, S. Wu / Applied Mathematics and Computation 214 (2009) 133–141 [4] [5] [6] [7] [8] [9] [10] [11]
141
Y. Lam, A note on the optimal replacement problem, Adv. Appl. Prob. 20 (2) (1988) 479–482. L.V. Utkin, Imprecise reliability of cold standby systems, Int. J. Qual. Reliab. Manage. 20 (6) (2003) 722–739. H. Wang, H. Pham, A quasi renewal process and its applications in imperfect maintenance, Int. J. Syst. Sci. 27 (10) (1996) 1055–1062. S. Wu, D. Clements-Croome, A novel repair model for imperfect maintenance, IMA J. Manage. Math. 17 (3) (2006) 235–243. S. Wu, D. Clements-Croome, Optimal maintenance policies under different operational schedules, IEEE Trans. Reliab. 54 (2) (2005) 338–346. H. Yu, F. Yalaoui, E˙. Châtelet, C. Chu, Optimal design of a maintainable cold-standby system, Reliab. Eng. Syst. Safety 92 (1) (2007) 85–91. Y.L. Zhang, G.J. Wang, Z.C. Ji, Replacement problems for a cold standby repairable system, Int. J. Syst. Sci. 37 (1) (2006) 17–25. Y.L. Zhang, G.J. Wang, A bivariate optimal repair-replacement model using geometric process for cold standby repairable system, Eng. Optim. 38 (6) (2006) 609–619. [12] Y.L. Zhang, G.J. Wang, A deteriorating cold standby repairable system with priority in use, Eur. J. Oper. Res. 183 (1) (2007) 278–295. [13] Y.L. Zhang, S. Wu, Reliability analysis for a k/n(F) system with repairable repair-equipment, Appl. Math. Model. 33 (7) (2009) 3052–3067.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Optimizing replacement policy for a cold-standby system with waiting repair times Jishen Jia a, Shaomin Wu b,* a b
Department of Scientific Research, Hennan Mechanical and Electrical Engineering College, Xinxiang 453002, PR China School of Applied Sciences, Cranfield University, Bedfordshire MK43 0AL, UK
a r t i c l e
i n f o
Keywords: Geometric process Cold-standby system Long-run cost per unit time Replacement policy Maintenance policy
a b s t r a c t This paper presents the formulas of the expected long-run cost per unit time for a coldstandby system composed of two identical components with perfect switching. When a component fails, a repairman will be called in to bring the component back to a certain working state. The time to repair is composed of two different time periods: waiting time and real repair time. The waiting time starts from the failure of a component to the start of repair, and the real repair time is the time between the start to repair and the completion of the repair. We also assume that the time to repair can either include only real repair time with a probability p, or include both waiting and real repair times with a probability 1 p. Special cases are discussed when both working times and real repair times are assumed to be geometric processes, and the waiting time is assumed to be a renewal process. The expected long-run cost per unit time is derived and a numerical example is given to demonstrate the usefulness of the derived expression. Ó 2009 Elsevier Inc. All rights reserved.
0. Introduction A two-component cold-standby system is composed of a primary component and a backup component, where the backup component is only called upon when the primary component fails. Cold-standby systems are commonly used for non-critical applications. However, cold-standby systems are one of most important structures in the reliability engineering and have been widely applied in reality. An example of such a system is the data backup system in computer networks. Reliability analysis and maintenance policy optimisation for cold-standby systems has attracted attentions from many researchers. Zhang and Wang [11,12] and Zhang et al. [10] derived the expected long-run cost per unit time for a repairable system consisting of two identical components and one repairman when a geometric process depicting working times is assumed. Utkin [5] proposed imprecise reliability models of cold-standby systems when he assumed that arbitrary probability distributions of the component time to failure are possible and they are restricted only by available information in the form of lower and upper probabilities of some events. Coit [1] described a solution methodology to optimal design configurations for non-repairable series–parallel systems with cold-standby redundancy when he assumed non-constant component hazard functions and imperfect switching. Yu et al. [9] considers a framework to optimally design a maintainable previous term cold-stand by next term system, and determine the maintenance policy and the reliability character of the components. Due to various reasons, repair might start immediately after a component fails. In some scenarios, from the failure of a component to the completion of repair, there might be two periods: waiting time and real repair time. The waiting time starts from the failure of the component to the start of repair; and the real repair time is the time between the start to repair
* Corresponding author. E-mail address: shaomin.wu@cranfield.ac.uk (S. Wu). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.03.064
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J. Jia, S. Wu / Applied Mathematics and Computation 214 (2009) 133–141
and the completion of the repair. This is especially true for cold-standby systems as they are not critical enough for a standby repairman be equipped for it. For example, when a component fails to work, its owner will call its contracted maintenance company or return the component to its suppler for repair. The waiting time can be the time spent by repairmen from their working place to the place where the component fails, or the time on delivering the failed component to its supplier. Usually, the waiting time can be seen as a random variable independent of the age of the component, whereas the real repair time can become longer and longer when the component becomes older. On the other hand, the working time of the component can become shorter and shorter due to various reasons such as ageing and deterioration. Such working and real repair time patterns can be depicted by geometric processes as many authors have studied [2–4,7,8,10–13]. The geometric processes introduced by Lam [4] define an alternative to the non-homogeneous Poisson processes: a sequence of random variables {Xk, k = 1, 2, . . .} is a geometric process if the distribution function of Xk is given by F(ak1t) for k = 1, 2, . . . and a is a positive constant. Wang and Pham [6] later refer the geometric process as a quasi-renewal process. Finkelstein [2] develops a very similar model: he defines a general deteriorating renewal process such that Fk+1(t) 6 Fk(t). Wu and Clements-Croome [7] extend the geometric process by replacing its parameter ak1 with a1ak1 + b1bk1, where a > 1 and 0 < b < 1. The geometric process has been applied in reliability analysis and maintenance policy optimisation for various systems by many authors; for example, see Wang and Pham [6], and Wu and Clements-Croome [8]. This paper presents the formulas of the expected long-run cost per unit time for a cold-standby system that consists of two identical components with perfect switching. When a component fails, a repairman will be called in to bring the component back to a certain working state. The time to repair is composed of two different time periods: waiting time and real repair time. The waiting time starts from the component failure to the start to repair, and the real repair time is the time between the start to repair and the completion of the repair. Both the working times and real repair times are assumed to be geometric processes, and the waiting time is assumed to be a renewal process. We also assume that the time to repair can either include only real repair time with a probability p, or include both waiting time and real repair time with a probability 1 p. The expected long-run cost per unit time is derived and a numerical example is given to demonstrate the usefulness of the derived expression. The paper is structured as follows. The coming section introduces geometric processes defined by Lam [4], denotation and assumptions. Section 3 discusses special cases. Section 4 offers numerical examples. Concluding remarks are offered in the last section. 1. Definitions and model assumptions This section first borrows the definition of the geometric process from Lam [4], and then makes assumptions for the paper. 1.1. Definition Definition 1. Assume n, g are the two random variables. For arbitrary real number a, there is
Pðn P aÞ > Pðg P aÞ; then n is called stochastically bigger than g. Similarly, if n stochastically smaller than g. Definition 2 [4]. Assume that {Xn, n = 1, 2, . . .} is a sequence of independent non-negative random variables. If the distribution function of Xn is F(an1t), for some a > 0 and all, n = 1, 2, . . ., then {Xn, n = 1, 2, . . .} is called a geometric process. Obviously, if a > 1, then {Xn, n = 1, 2, . . .} is stochastically decreasing, if a < 1, then {Xn, n = 1, 2, . . .} is stochastically increasing, and if a = 1, {Xn, n = 1, 2, . . .} is a renewal process.
1.2. Assumptions and denotation The following assumptions are assumed to hold in what follows. A. At the beginning, both components are new, component 1 is working and component 2 is under cold standby. B. When both of the two components are in good condition, one is working and the other is under cold standby. When the working component fails, a repairman repairs the failed component immediately with probability p, or repairs it with a waiting time with probability 1 p. As soon as the working component fails, the standby one will start to work. Assume the switching is perfect. After the failed one has been repaired, it is either put in use the other one fails or put in standby if the other is working. If one fails while the other is still under repair, the failed one must wait for repair until the repair for the first failed is completed.
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J. Jia, S. Wu / Applied Mathematics and Computation 214 (2009) 133–141
C. The time interval from the completion of the (n 1)th repair to that of the nth repair of component i is called the nth cycle of component i, where i = 1, 2; n = 1, 2, . . .. Denote the working time and the repair time of component i in the nth ðiÞ cycle (i = 1, 2; n = 1, 2, . . .) as X ðiÞ n and Y n , respectively, the waiting time of component i (i = 1, 2) in the nth cycle as ðiÞ ðiÞ ; n ¼ 1; 2; . . .g, the cumulative distribution functions of X ðiÞ fZ ðiÞ n n , Y n and Z n , as Fn(x) Gn(x), and S(x), respectively. ðiÞ ðiÞ ; Y ; and Z (i = 1, 2, and n = 1, 2, . . .) are statistically independent. D. X ðiÞ n n n E. When a replacement is required, a brand new but identical component will be used to replace, and the replacement time is negligible. F. Denote the repair cost per unit time of two components as Cm, the working reward per unit time as Cw, the replacement cost as Cr.
2. Expected cost under replacement policy N Fig. 1 shows a typical scenario, given that the above-mentioned assumptions hold. In what follows, we consider a replacement policy N, where a replacement is carried out if the number of failures reaches N for component 1. Denote the time between the (n 1)th replacement and the nth replacement of the system as Tn. Obviously, {T1, T2, . . .} forms a renewal process. Let C(N) be the expected long-run cost per unit time of the system under the policy N. Because {T1, T2, . . .} is a renewal process, the interval time between two consecutive replacements is a renewal cycle. Then, according to renewal reward theorem, we can know that the long-run average cost per unit time is given by
CðNÞ ¼
Expected cost incurred in a cycle : Expected length of a cycle
ð1Þ
Let W be the length of a renewal cycle of the system, then ð1Þ
W ¼ X1 þ
N1 X ð1Þ ð1Þ ð2Þ ð1Þ ð1Þ ð2Þ ð1Þ ½maxfZ i þ Y i ; X i gIfAi g þ maxfY i ; X i gIfBi g i¼1
N2 X ð2Þ ð2Þ ð1Þ ð2Þ ð2Þ ð1Þ ð2Þ ð1Þ ½maxfZ i þ Y i ; X iþ1 gIfAi g þ maxfY i ; X iþ1 gIfBi g þ X N : þ i¼1
The expected length of a renewal cycle is ð1Þ
ð1Þ
EðWÞ ¼ E½X 1 þ E½X N þ
N1 X
ð1Þ
ð1Þ
ð2Þ
ð1Þ
ð1Þ
ð2Þ
ð1Þ
E½maxfZ i þ Y i ; X i gIfAi g þ maxfY i ; X i gIfBi g
i¼1
þ
N2 X
ð2Þ E½maxfZ i
ð2Þ
ð1Þ
ð2Þ
ð2Þ
ð1Þ
ð2Þ
þ Y i ; X iþ1 gIfAi g þ maxfY i ; X iþ1 gIfBi g:
ð2Þ
i¼1
Let C be the cost of a renewal cycle of the system under the policy N, then
C ¼ Cr þ Cm Cx
( N1 X
" N X
N2 X
i¼1 ð1Þ Xi
i¼1 ð1Þ
ð1Þ
Yi þ
ð2Þ
)
i¼1
þ
N1 X
ð2Þ
ð2Þ
ð1Þ
ð2Þ
ð2Þ
ð2Þ
ð1Þ
ð2Þ
Y i þ ½Y N1 IfAg þ ðX N Z N1 ÞIfAgIðAN1 Þ þ ½Y N1 IfBg þ X N IfBgIðBN1 Þ
ð2Þ Xi
# ð3Þ
;
i¼1 ð2Þ
ð1Þ
ð2Þ
ð2Þ
ð1Þ
ð2Þ
ð1Þ
ð2Þ
where A ¼ fX N Z N1 > Y N1 g, A ¼ fX N Z N1 < Y N1 g, B ¼ fX N Y N1 > 0g, and B ¼ fX N Y N1 < 0g.
Fig. 1. A possible progressive figure of the system.
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J. Jia, S. Wu / Applied Mathematics and Computation 214 (2009) 133–141
If X and Y are two independent non-negative random variables and their cumulative distribution functions are F(x) and G(x), respectively, we have following three lemmas. Denote E(C) as the expected value of C. By substituting the numerator and denominator of Eq. (1) with E(C) and E(W), respectively, we have
CðNÞ ¼
EðCÞ : EðWÞ
ð4Þ
Then the optimal replacement number can be obtained by minimising the value of C(N) in Eq. (4). Lemma 1
Z
EðmaxfX; YgÞ ¼ EX þ
1
Z0 1
¼ EY þ
FðxÞ½1 GðxÞdx
ð5Þ
GðxÞ½1 FðxÞdx:
ð6Þ
0
The proof of Lemma 1 is given in Appendix. Lemma 2
E½IfY > XgX þ E½If0 < Y < XgY ¼
Z
1
½1 FðxÞ½1 GðxÞdx:
ð7Þ
0
The proof of Lemma 2 is given in Appendix. Similarly, we have Lemma 3
E½ðY XÞIfY X > 0g ¼
Z
1
½1 GðxÞFðxÞdx:
ð8Þ
0
3. Special cases and discussion ðiÞ n1 t) and G(bn1t), respectively, where a > 1, 0 < b < 1. fY ðiÞ Denote the distributions of X ðiÞ n and Y n as F(a n ; n ¼ 1; 2; . . .g constitutes an increasing geometric process, whereas fX ðiÞ n ; n ¼ 1; 2; . . .g constitutes a decreasing geometric process. Then we have the following theorem. n1 n1 n1 Theorem 1. Assume F n ðtÞ ¼ Fðan1 tÞ ¼ 1 exp a k t , Gn ðtÞ ¼ Gðb tÞ ¼ 1 exp b l t , and SðtÞ ¼ 1 exp ct ,
(t P 0), respectively. Then the expected length of a renewal cycle is given by
EðWÞ ¼ k þ þ2
k þ ð2N 3Þð1 pÞc aN1 N2 X i¼1
l b
i1
þ
l b
N2
þ ð1 pÞk
N2
þ
pk2 b aN2 ðb
l
N2
k þ aN2 lÞ
þ
N2 X
! 1 l2 aN2 ðcaN2 þ kÞ ðaN2 l þ bN2 kÞð2bN2 ck þ lk þ aN2 lcÞ
2
ð1 pÞk2
i¼1
1
2 i1
ðai l þ b
i1
kÞð2b
ck þ lk þ ai lcÞ i1
þp
1 1 þ ai ðcai þ kÞ ai1 ðcai1 þ kÞ
þ
!
1 ðai1 l þ b !!
i1
i1
kÞð2b
ck þ lk þ ai1 lcÞ
kð1 þ aÞ klðai l þ ai1 l þ 2b kÞ i1 i1 ai ðb k þ ai lÞðb k þ ai1 lÞ
ð9Þ
and the expected cost is a cycle is
EðCÞ ¼ C m 2
N 2 X
l i1
i¼1
b
þ Cr Cx 2
þ
l b
N2
þ ð1 pÞ
N1 X k k þ N1 i1 a a i¼1
!
2l N2
b
aN1 lc2 N2
ðaN1 c þ kÞðb
k2 l
!
c lÞ ðaN1 c þ kÞðbN2 k þ aN1 lÞ
þ
!
kl N2
b
k þ aN1 l
p
ð10Þ
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J. Jia, S. Wu / Applied Mathematics and Computation 214 (2009) 133–141
and the expected long-run cost per unit time is given by
CðNÞ ¼
EðCÞ : EðWÞ
ð11Þ
If one sets a = 1 and b = 1, the above results E(W) and E(C) will be the situations where the components can be repaired as good as new.
4. Numerical examples 4.1. Parameter set 1 If we set a = 1.8, b = 0.98, k = 100, l = 10, c = 5, Cx = 500, Cm = 20, Cr = 5000, and p = 0.8, then the optimum number for a replacement will be N = 6, and the corresponding expected long-run cost per unit time is 433.41. The expected long-run cost per unit time is shown in Table 1, which corresponds to Fig. 1. 4.2. Parameter set 2 If we set a = 1.1, b = 0.98, k = 100, l = 1, c = 0.2, Cx = 500, Cm = 20, Cr = 5000, and p = 0.8, then the optimum number for a replacement will be N = 35, and the corresponding expected long-run cost per unit time is 491.85. The expected long-run cost per unit time is shown in Table 2, which corresponds to Fig. 2. Compare Figs. 2 and 3, we can find that the optimum replacement time becomes longer in the second situation. In both situations, we can easily find an optimum replacement time point. However, due to the complexity of Eq. (11), we are not able to prove that there exists a unique optimal value N.
Table 1 The expected long-run cost per unit time versus replacement times for parameter set 1. Times
Cost rate
Times
Cost rate
Times
Cost rate
Times
Cost rate
Times
Cost rate
1 2 3 4 5 6 7 8 9 10 11 12 13 14
88.74 107.92 279.47 389.09 428.42 433.41 424.96 411.2 395.37 378.99 362.83 347.25 332.41 318.37
15 16 17 18 19 20 21 22 23 24 25 26 27 28
305.11 292.59 280.78 269.62 259.06 249.07 239.6 230.61 222.07 213.94 206.21 198.84 191.8 185.08
29 30 31 32 33 34 35 36 37 38 39 40 41 42
178.66 172.52 166.64 161 155.6 150.42 145.44 140.66 136.06 131.63 127.38 123.27 119.32 115.51
43 44 45 46 47 48 49 50 51 52 53 54 55 56
111.84 108.29 104.87 101.56 98.37 95.28 92.29 89.4 86.6 83.89 81.27 78.73 76.27 73.88
57 58 59 60 61 62 63 64 65 66 67 68 69 70
71.56 69.32 67.14 65.02 62.97 60.97 59.04 57.15 55.33 53.55 51.82 50.14 48.5 46.91
Table 2 The expected long-run cost per unit time versus replacement times for parameter set 2. Times
Cost rate
Times
Cost rate
Times
Cost rate
Times
Cost rate
Times
Cost rate
1 2 3 4 5 6 7 8 9 10 11 12 13 14
17.37 18.96 37.01 58.53 83.54 111.78 142.75 175.68 209.62 243.55 276.47 307.51 336.03 361.63
15 16 17 18 19 20 21 22 23 24 25 26 27 28
384.14 403.59 420.14 434.04 445.6 455.12 462.91 469.24 474.35 478.46 481.74 484.34 486.39 487.99
29 30 31 32 33 34 35 36 37 38 39 40 41 42
489.22 490.15 490.84 491.32 491.63 491.8 491.85 491.79 491.65 491.43 491.13 490.77 490.36 489.88
43 44 45 46 47 48 49 50 51 52 53 54 55 56
489.36 488.78 488.15 487.48 486.76 485.99 485.18 484.33 483.43 482.5 481.52 480.5 479.44 478.35
57 58 59 60 61 62 63 64 65 66 67 68 69 70
477.22 476.05 474.85 473.61 472.34 471.03 469.7 468.33 466.93 465.5 464.05 462.56 461.05 459.5
Note: times in Tables 1 and 2 stands for replacement times; cost rate stands for the expected long-run cost per unit time.
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J. Jia, S. Wu / Applied Mathematics and Computation 214 (2009) 133–141
Fig. 2. The change of C(N) over N for parameter set 1.
Fig. 3. The change of C(N) over N for parameter set 2.
5. Conclusions Cold-standby systems are a category of important reliability structure in engineering. Searching an optimal replacement time point for such systems is of interest and important. This paper derived the expected long-run cost per unit time for a cold-standby system when time to repair is composed of two time periods: waiting time and real repair time. We also considered a special scenario where the working times and real repair times are geometric processes. Numerical examples are given to demonstrate the usefulness of the derived expression. Appendix. Proof of Lemma 1 Proof. Because X, Y are two independent random variables, therefore
EðmaxfX; YgÞ ¼
Z Z Z
maxfx; yg f ðxÞgðyÞdxdy ¼ D
Z
Z
yf ðxÞgðyÞdxdy þ x6y
y
yf ðxÞgðyÞdxdy þ
0
1
x
Z
Z Z
xf ðxÞgðyÞdxdy x>y
Z 1 xf ðxÞgðyÞdydx ¼ ygðyÞFðyÞdy þ xGðxÞf ðxÞdx 0 0 0 0 0 0 Z 1 Z 1 Z 1 Z 1 ¼ xFðxÞd½1 GðxÞ þ xGðxÞf ðxÞdx ¼ ½xf ðxÞ½1 GðxÞ þ xGðxÞf ðxÞdx þ FðxÞ½1 GðxÞdx 0 0 0 0 Z 1 ¼ EX þ FðxÞ½1 GðxÞdx ¼
1
Z
Z Z
1
139
J. Jia, S. Wu / Applied Mathematics and Computation 214 (2009) 133–141
where f ðxÞ ¼ dFðxÞ and gðyÞ ¼ dGðyÞ . dx dy
h
Proof of Lemma 2
Proof. As X and Y are two independent non-negative random variables,
E½IfY > XgX ¼
Z Z
xf ðxÞgðyÞdxdy ¼
Z
x
¼
Z
1
Z
0
x
x½1 GðxÞd½1 FðxÞ ¼
E½If0 < Y < XgY ¼
Z Z
yf ðxÞgðyÞdxdy ¼
xf ðxÞ½1 GðxÞdx
1
½1 GðxÞ xgðxÞ½1 FðxÞdx; 0
Z
0
1 0
Z
1 0
Z xf ðxÞgðyÞdy dx ¼
1
1
ygðyÞ½1 FðyÞdy ¼
Z
0
1
xgðxÞ½1 FðxÞdx
0
and
E½IfY > XgX þ E½If0 < Y < XgY ¼
Z
1
½1 FðxÞ½1 GðxÞdx:
0
Proof of Theorem Proof. According to the above theorems and formulas (2) and (3), we have ð1Þ
ð1Þ
EðWÞ ¼ E½X 1 þ E½X N þ
N 1 X
ð1Þ
ð1Þ
ð2Þ
ð1Þ
ð1Þ
ð2Þ
ð1Þ
E½maxfZ i þ Y i ; X i gIfAi g þ maxfY i ; X i gIfBi g
i¼1
þ
N2 X
ð2Þ
ð2Þ
ð1Þ
ð2Þ
ð2Þ
ð1Þ
ð2Þ
E½maxfZ i þ Y i ; X iþ1 gIfAi g þ maxfY i ; X iþ1 gIfBi g
i¼1 ð1Þ
ð1Þ
¼ E½X 1 þ E½X N þ
N1 X ð1Þ ð1Þ ð2Þ ð1Þ ð2Þ fE½maxfZ i þ Y i ; X i gð1 pÞ þ E½maxfY i ; X i gpg i¼1
þ
N2 X
ð2Þ
ð2Þ
ð1Þ
ð2Þ
ð1Þ
fE½maxfZ i þ Y i ; X iþ1 gð1 pÞg þ E½maxfY i ; X iþ1 gpg
i¼1
( ¼
ð1Þ E½X 1
þ
ð1Þ E½X N
þ
N 1 X
ð1Þ E½maxfZ i
þ
ð1Þ ð2Þ Y i ; X i g
N2 X
þ
i¼1
( þ
N1 X
ð1Þ ð2Þ E½maxfY i ; X i g
( ð1Þ
þ
N2 X
) ð2Þ ð1Þ E½maxfY i ; X iþ1 g
ð1Þ
N1 X
ð1Þ
ð1Þ
E½Z i þ Y i þ
i¼1 N1 X
ð1Þ
E½Y i þ
i¼1
¼kþ
þ
k aN1
k aN1
N2 Z X i¼1
p
N1 Z X i¼1
ð2Þ
E½Y i þ
N1 Z X i¼1
þ ð2N 3Þð1 pÞc þ 2
1
Hi ðtÞ½1 F i ðtÞdt þ 0
Gi ðtÞ½1 F i ðtÞdt þ
0
i¼1
N2 Z X i¼1
l b
i1
þ
ð2Þ
ð2Þ
E½Z i þ Y i þ
i¼1
1
N2 X
N2 X
l N2
b
þ
N1 Z X i¼1
N2 Z X i¼1
)
1
Hi ðtÞ½1 F iþ1 ðtÞdt ð1 pÞ
0
)
1
Gi ðtÞ½1 F iþ1 ðtÞdt p 0
1
½ð1 pÞHi ðtÞ þ pGi ðtÞ½1 F i ðtÞdt 0
1
½ð1 pÞHi ðtÞ þ pGi ðtÞ½1 F iþ1 ðtÞdt
0
i¼1
þ
N2 X i¼1
N2 Z X
¼kþ
ð1 pÞ
i¼1
¼ E½X 1 þ E½X N þ
þ
þ
ð2Þ ð1Þ Y i ; X iþ1 g
i¼1
i¼1
(
) ð2Þ E½maxfZ i
0
þ ð2N 3Þð1 pÞc þ 2
N2 X i¼1
1
½ð1 pÞHi ðtÞ þ pGðb
i1
l b
i1
þ
l N2
b
þ
Z
1
½ð1 pÞHN1 ðtÞ þ pGðb
N2
tÞ½1 FðaN2 tÞdt
0
tÞ½2 Fðai tÞ Fðai1 tÞdt
ð12Þ
140
J. Jia, S. Wu / Applied Mathematics and Computation 214 (2009) 133–141
and
# Z 1 Z 1 N2 N2 N1 þ ð1 pÞ ½1 R ðtÞ½1 Gðb tÞdt þ p ½1 Gðb tÞ½1 Fða tÞdt þ Cr N i1 0 0 i¼1 b i¼1 b " # N N1 X X k k Cx þ ai1 i¼1 ai1 i¼1 ( ) Z 1 Z 1 N2 X l l N2 N2 N1 ¼ Cm 2 þ N2 þ ð1 pÞ ½1 RN ðtÞ½1 Gðb tÞdt þ p ½1 Gðb tÞ½1 Fða tÞdt þ C r i1 0 0 b i¼1 b " # N 1 X k k ; Cx 2 þ ai1 aN1 i¼1
EL ¼ C m
" N1 X
l
þ i1
N2 X
l
ðiÞ
ðiÞ
ð1Þ
ð2Þ
where t P 0, Hi(t) and RN(x) represent the cumulative distribution functions of the random variables Z i þ Y i and X N Z N1 , respectively. Hence we have Hi (x) = S(t)*Gi(t), and RN(x) = FN(t)*[1 S(t)], where ‘‘*” indicates convolution, and
Hi ðtÞ ¼ SðtÞ Gðb
i1
" #) Z t( i1 b u tÞ ¼ 1 exp ðt uÞ d 1 exp
l
0
t ¼ 1 exp
l
c
(
cbi1 þ l
c
i1
exp
b
l
"
! t
exp
i1
2b
l
þ
1
! #)
c
t
;
N1 Z t a u 1 exp ðt uÞ d exp k c 0 N1 t k a t 1þ ; exp ¼ exp t exp k þ aN1 c c k c
RN ðtÞ ¼ FðaN1 tÞ ½1 SðtÞ ¼
where
Z
1
½ð1 pÞHN1 ðtÞ þ pGðb
0
N2
tÞ½1 FðaN2 tÞdt
"
# N2 1 l2 pk2 b þ ; ¼ ð1 pÞk N2 N2 N2 N2 N2 a ðca þ kÞ ðaN2 l þ b kÞð2b ck þ lk þ aN2 lcÞ aN2 ðb k þ aN2 lÞ 2
Z
1
½ð1 pÞHi ðtÞ þ pGðb
0
i1
tÞ½2 Fðai tÞ Fðai1 tÞdt
(
1 1 1 l2 ½ þ i1 i1 ai ðcai þ kÞ ai1 ðcai1 þ kÞ ðai l þ b kÞð2b ck þ lk þ ai lcÞ ) " # i1 1 kð1 þ aÞ klðai l þ ai1 l þ 2b kÞ i1 þp ; þ i1 i1 i1 ai ðai1 l þ b kÞð2b ck þ lk þ ai1 lcÞ ðb k þ ai lÞðb k þ ai1 lÞ
¼ ð1 pÞk2
Z
1
N2
½1 RN ðtÞ½1 Gðb
tÞdt
0
¼
Z
1
t 2 exp
c
0
¼
2l b
N2
( N2 ) N1 k a t b exp exp t dt t exp k þ aN1 c k c l
aN1 lc2 ðaN1
N2
c þ kÞðb
c lÞ
k2 l ðaN1
c þ kÞðbN2 k þ aN1 lÞ
and
Z
1
N2
½1 Gðb
0
N1
tÞ½1 Fða
tÞdt ¼
Z 0
1
( N2 ) N1 a t b kl exp t dt ¼ N2 : exp k l b k þ aN1 l
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