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A repairable system with imperfect coverage and reboot
Applied Mathematics and Computation 246 (2014) 148–158
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A repairable system with imperfect coverage and reboot Jau-Chuan Ke ⇑, Tzu-Hsin Liu Department of Applied Statistics, National Taichung University of Science and Technology, Taichung, Taiwan, ROC
a r t i c l e
i n f o
Keywords: Availability Imperfect coverage Reboot delay Sensitivity analysis
a b s t r a c t In this paper, we examine a repairable system with imperfect coverage, reboot delay and one repair facility. The failure times and repair times of failed units are assumed to be exponential and general distributions, respectively. As a unit fails, it can be immediately detected, located and replaced with a coverage probability c by a standby, if one is available. We use a recursive method and the supplementary variable technique to develop the steady-state probabilities of down units at an arbitrary epoch. Then, an efficient algorithm is constructed to compute the steady-state availability. We adopt two repair time distributions, namely, exponential and gamma, to illustrate the method. Finally, we also perform a sensitivity analysis of the steady-state availability with respect to the system parameters for various repair time distributions. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Availability, or reliability, plays an important role when evaluating a system’s effectiveness. Availability/reliability has been widely studied because of its prevalence in power plants, production systems, manufacturing systems and industrial systems. Maintaining a high or required level of availability/reliability is often a fundamental requisite. Standby or redundancy is an efficient method which can be applied to improve system availability. The probability of successful recovery on the failure of a primary unit (or standby unit) is denoted by c. We assume that the coverage factor is the same for a primary unit failure as that for a standby failure. Quantity c, which includes the probabilities of successful detection and location of a failure, is known as the coverage factor or coverage probability (see [7]). The concept of the coverage factor and its effect on the reliability and availability model of a repairable system have been studied by a number of authors, including Trivedi [7], Wang and Chiu [9], Ke et al. [2,3] and Myers [6]. The status and trends of imperfect coverage models and the associated reliability analysis techniques were introduced in Amari et al. [1]. Ke et al. [5] investigated the statistical inferences of an availability system with imperfect coverage. Ke et al. [4] also studied a twounit redundant repairable system with detection delay and imperfect coverage. Recently, Wang et al. [11] investigated the reliability and sensitivity analysis of a repairable system with imperfect coverage under a service pressure condition. The concept of a reboot delay for a repairable system was introduced by Trivedi [7]. Wang et al. [10] completed a comparison of the reliability and availability between four systems with warm standby units, reboot delay and standby switching failures. Wang and Chen [8] performed a comparative analysis of the availability between three systems with general repair times, reboot delay and switching failures. Recently, Yen et al. [12] compared three different configurations with imperfect coverage and standby switching failures based on reliability and availability.
⇑ Corresponding author. http://dx.doi.org/10.1016/j.amc.2014.07.090 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
J.-C. Ke, T.-H. Liu / Applied Mathematics and Computation 246 (2014) 148–158
149
Notations M S K NðtÞ VðtÞ BðuÞ bðuÞ BðsÞ k
a c b Pi ðtÞ Pufi ðtÞ Pi Pufi i ðsÞ P b Að1Þ
number of primary units number of standby units minimum number of operating units number of failed units in the system remaining repair time for the failed units under repair distribution function of repair time probability density function of repair time Laplace transform of repair time failure rate of a primary unit failure rate of a standby unit coverage probability reboot delay rate probability of i failed units in the system at time t in the safe state, where i ¼ 0; 1; 2; . . . ; N probability of i failed units in the system at time t in the unsafe failure state, where i ¼ 0; 1; 2; . . . ; N 2 steady-state probability of i failed units in the system at time t in the safe state, where i ¼ 0; 1; 2; . . . ; N steady-state probability of i failed units in the system at time t in the unsafe failure state, where i ¼ 0; 1; 2; . . . ; N 2 Laplace transform of P i ðv Þ mean repair time steady-state availability
In recent years, the database in computer systems has become of utmost importance in our highly information-oriented society. In particular, a reliable database is the most indispensable instrument in on-line transaction processing systems, such as real-time systems used for bank accounts. The data in a computer system are frequently updated by adding or deleting files, and can be stored in other secondary media (treated as active units). However, data files in secondary media are sometimes damaged by errors due to noises, human errors and hardware faults. To ensure the safety of data, there must always be backup copies of all files in other places (treated as standby units), which are then available if files in the original secondary media are damaged. Because files are of different sizes, the time necessary to reconstruct a file is a random variable instead of a constant. A fault detecting server runs general purpose monitoring software to monitor data files in secondary media and backup copies. When data files in secondary media break down, the problem can be immediately detected by the fault-detecting server and replaced with a coverage probability c by a backup copy, if one is available. The database is in an unsafe failure state if any breakdown is not covered. The data files in secondary media (or backup copies) in the unsafe failure state can be cleared by a reboot. Reboot delay may take place at a rate for the data files in secondary media (or backup copies), which is exponentially distributed. The rest of the paper is organized as follows: Section 2 presents a brief description of the system; Section 3 presents a recursive method using the supplementary variable technique, treating the supplementary variable as the remaining repair time. The steady-state probability distribution of down units at an arbitrary epoch is developed and an efficient algorithm to find the steady-state availability is provided. Two simple examples for the two different repair time distributions, exponential and gamma, are given; in Section 4, numerical results are presented to illustrate the steady-state availability under different repair time distributions, and a sensitivity analysis is also conducted; and, finally, some conclusions are drawn in the last section.
2. Model description The repairable system includes M identical primary units, S standby units and one repairman/server. System failure is defined to be less than K units in an active operation, where K = 1, 2, . . ., M + S. Therefore, the system failure occurs if and only if the number of failed units in the system is larger or equal to N = M + S K + 1. The detailed descriptions of the assumptions are as follows: (i) The operating times of the primary units between breakdowns are independent, identically distributed (i.i.d.) variables having an exponential distribution with parameter k. (ii) If a primary unit fails, it is immediately replaced by a standby unit if one is available. Each standby unit fails independently of the state of all other units and has an exponential time-to-failure distribution with parameter a (0 < a < k).
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(iii) The failed unit is immediately repaired when the server is idle. When the server is busy, the arriving failure units have to queue up until the server is available. The repair sequence is according to the first come, first repaired discipline. (iv) The time to repair a failed unit is i.i.d. random variables having a distribution BðuÞðu P 0Þ, a probability density function bðuÞðu P 0Þ and mean repair time b. (v) When a unit fails, it can be immediately detected and located with a coverage probability c. (vi) Defining the unsafe failure state of the system as any one of the breakdowns is not covered. A primary unit fails in the unsafe failure state, which is cleared by a reboot. Reboot delay occurs at rate b for a primary unit (or standby unit) which is exponentially distributed. 3. Steady-state results 3.1. Steady-state probability We use the following supplementary variable: V remaining repair time for the failed units under repair. The state of the system at time t is given by: N(t) number of failed units in the system, and V(t) remaining repair time for the failed units under repair. Let us define:
Pi ðv ; tÞ ¼ PfNðtÞ ¼ i; v < VðtÞ 6 v þ dv ; in the safe stateg; Pi ðtÞ ¼
Z
1
Pi ðv ; tÞdv ;
v P 0;
i ¼ 0; 1; 2; . . . ; N;
i ¼ 1; 2; . . . ; N;
0
Pufi ðv ; tÞ ¼ PfNðtÞ ¼ i; v < VðtÞ 6 v þ dv ; in the unsafe failure stateg; Pufi ðtÞ ¼
Z 0
1
Pufi ðv ; tÞdv ;
v P 0;
i ¼ 0; 1; 2; . . . ; N 2;
i ¼ 0; 1; 2; . . . ; N 2:
Relating the state-transition rate diagram shown in Fig. 1, we obtain:
d P0 ðtÞ ¼ k0 P0 ðtÞ þ P1 ð0; tÞ; dt @ @ P1 ðv ; tÞ ¼ k1 P1 ðv ; tÞ þ k0 cP 0 ðv ; tÞ þ bðv ÞP2 ð0; tÞ þ bPuf0 ðv ; tÞ; @t @ v
ð1Þ
ð2Þ
@ @ Pi ðv ; tÞ ¼ ki Pi ðv ; tÞ þ ki1 cP i1 ðv ; tÞ þ bðv ÞPiþ1 ð0; tÞ þ bPufi1 ðv ; tÞ; @t @ v
2 6 i 6 N 2;
ð3Þ
@ @ PN ðv ; tÞ ¼ kN1 PN1 ðv ; tÞ; @t @ v
d Puf ðtÞ ¼ ki ð1 cÞPi ðtÞ bPufi ðtÞ; dt i
λ0 c
ð4Þ
0 6 i 6 N 2;
0
b (v )
b (v )
β uf0
λL−3c
λ1 (1 − c ) β uf1
λL−1
λL −2 c N-2
2
1
λ0 (1 − c )
λ2 c
λ1c
ð5Þ
b (v )
N
N-1
b (v )
b (v )
λN − 2 (1 − c )
b (v )
β
ufN-2
Fig. 1. The state transition diagram for the repairable system with imperfect coverage and reboot delay.
J.-C. Ke, T.-H. Liu / Applied Mathematics and Computation 246 (2014) 148–158
151
where
ki ¼
Mk þ ðS iÞa; 0 6 i 6 S 1 ðM þ S iÞk;
S6i6N
;
In steady-state, let
Pi ¼ lim Pi ðtÞ;
0 6 i 6 N;
t!1
Pi ðv Þ ¼ lim Pi ðv ; tÞ;
0 6 i 6 N;
t!1
Pufi ¼ lim P ufi ðtÞ;
0 6 i 6 N 2;
t!1
Pufi ðv Þ ¼ lim P ufi ðv ; tÞ; t!1
0 6 i 6 N 2:
We also define:
Pi ðv Þ ¼ bðv ÞPi ;
i ¼ 0; 1; 2; . . . ; N;
Pufi ðv Þ ¼ bðv ÞPufi ;
i ¼ 0; 1; 2; . . . ; N 2:
From Eqs. (1)–(5), we obtain the following steady-state equations:
P1 ð0Þ ¼ k0 P0 ;
ð6Þ
@ P1 ðv Þ ¼ k1 P1 ðv Þ þ k0 cP 0 ðv Þ þ bðv ÞP2 ð0Þ þ bPuf0 ðv Þ; @v
ð7Þ
@ Pi ðv Þ ¼ ki Pi ðv Þ þ ki1 cP i1 ðv Þ þ bðv ÞPiþ1 ð0Þ þ bPufi1 ðv Þ; 2 6 i 6 N 1; @v
ð8Þ
@ PN ðv Þ ¼ kN1 PN1 ðv Þ; @v
ð9Þ
0 ¼ ki ð1 cÞPi bP ufi ;
0 6 i 6 N 2:
ð10Þ
Further define:
¼ BðsÞ
Z
1
esv dBðv Þ;
0
i ðsÞ ¼ P
Z
1
esv Pi ðv Þdv ;
1 6 i 6 N;
0
i ð0Þ ¼ Pi ¼ P
Z
1
Pi ðv Þdv ;
1 6 i 6 N;
0
Z
1
esv
0
@ i ðsÞ Pi ð0Þ; Pi ðv Þdv ¼ sP @v
1 6 i 6 N:
Therefore, if the LST is taken of both sides of Eqs. (7)–(9), it is found that:
ðk1 sÞP 1 ðsÞ ¼ k0 BðsÞP 0 þ BðsÞP 2 ð0Þ P 1 ð0Þ; i ðsÞ ¼ ki1 BðsÞP ðki sÞP i1 þ BðsÞP iþ1 ð0Þ P i ð0Þ;
ð11Þ 2 6 i 6 N 1;
N ðsÞ ¼ kN1 BðsÞP sP N1 P N ð0Þ:
ð12Þ ð13Þ
From Eq. (10), P ufi can be expressed as:
Pufi ¼
ki ð1 cÞ Pi ; b
0 6 i 6 N 2:
ð14Þ
Using Eq. (6) in Eq. (11) and substituting s ¼ k1 and s ¼ 0, respectively, it implies that:
P2 ð0Þ ¼
1 Þ k0 ½1 Bðk P0 ; Bðk1 Þ
ð15Þ
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and
1 ð0Þ ¼ 1 P2 ð0Þ: P k1
ð16Þ
Setting s ¼ ki in Eq. (12), we get:
Piþ1 ð0Þ ¼
i ÞP i1 ð0Þ Pi ð0Þ ki1 Bðk ; Bðki Þ
26i6N1
ð17Þ
Now, setting s ¼ 0 in Eq. (12), we obtain:
i ð0Þ ¼ ki1 P i1 ð0Þ þ Piþ1 ð0Þ Pi ð0Þ ; P ki
i ¼ 2; 3; . . . ; N 1
ð18Þ
N ð0Þ, we differentiate Eq. (13) with respect to s and set s = 0. It yields: To find the unknown quantity P
N ð0Þ ¼ kN1 bP N1 ð0Þ; P
ð19Þ
where b ¼ ½dBðsÞ=ds s¼0 is the mean repair time. Therefore, P i ð0Þð1 6 i 6 NÞ and Puf ð0 6 i 6 N 2Þ can be expressed in terms of P0. Then, P0 can be determined by the nori
malizing condition:
P0 þ
N N2 X X i ð0Þ þ P Pufi ¼ 1: i¼1
ð20Þ
i¼0
1 ; P 2 ; . . . ; P N ; Puf ; Puf ; . . . ; P uf . Once P0 has been constructed, we obtain the steady-state probabilities: P 0 1 N2 3.2. Steady-state availability Let us denote by A(1) the steady-state availability of the repairable system with warm standby units. Since the system operable in states:
Að1Þ ¼ P0 þ
N1 N2 X X N ð0Þ; Pi ð0Þ þ Pufi ¼ 1 P i¼1
ð21Þ
i¼0
to demonstrate the working the recursive method, an algorithm for computing the steady-state availability is provided below. Algorithm for computing the steady-state availability Step Step Step Step Step Step Step
1. 2. 3. 4. 5. 6. 7.
Compute Compute Compute Compute Compute Compute Compute
P2(0) using Eq. (15) in terms of P0. 1 ð0Þ using Eq. (16) in terms of P0. P i ð0Þ (2 6 i 6 N 1) using Eqs. (17) and (18) in terms of P0. Pi+1(0) (2 6 i 6 N 1) and P N ð0Þ using Eq. (19) in terms of P0. P Pufi for each i = 0, 1, 2, . . ., N 2 using Eq. (14) in terms of P0. P0 using Eq. (20). the steady-state availability A(1) using Eq. (21).
3.3. Simple examples Based on the algorithm stated above, three simple examples are provided to illustrate the recursive method, in which the repair time distributions are exponential and gamma distributions. We consider four primary units, M = 4, and two warm standby units, S = 2. The minimum number of operating units is set to three, K = 3. Example 1: exponential repair time distribution We set the mean repair time, b = 1/l, where
¼ l=ðl þ sÞ. l is the repair rate. In this case, we have BðsÞ
Step 1. Compute P2 ð0Þ using Eq. (15) in terms of P 0 .
1 Þ þ aÞ k0 ½1 Bðk ð4k þ 2aÞ½1 Bð4k P0 ¼ P0 1Þ Bðk Bð4k þ aÞ ð4k þ 2aÞð4k þ aÞ ¼ P0 :
P2 ð0Þ ¼
l
1 ð0Þ using Eq. (16) in terms of P 0 . Step 2. Compute P
1 ð0Þ ¼ 1 P k1
P2 ð0Þ ¼
1 4k þ a
P2 ð0Þ ¼
4k þ 2a
l
P0 :
J.-C. Ke, T.-H. Liu / Applied Mathematics and Computation 246 (2014) 148–158
153
2 ð0Þ and P 3 ð0Þ using Eqs. (17) and (18) in terms of P0. Step 3. Compute P 3 ð0Þ, P 4 ð0Þ, P
P3 ð0Þ ¼
2 ÞP1 ð0Þ P2 ð0Þ ð4k þ aÞBð4kÞ P 1 ð0Þ 4kð4k þ 2aÞð4k þ aÞ P2 ð0Þ k1 Bðk ¼ ¼ P0 ; l2 Bðk2 Þ Bð4kÞ
P4 ð0Þ ¼
3 ÞP2 ð0Þ P3 ð0Þ 4kBð3kÞ P2 ð0Þ 12k2 ð4k þ 2aÞð4k þ aÞ P3 ð0Þ k2 Bðk ¼ ¼ P0 ; l3 Bðk3 Þ Bð3kÞ
2 ð0Þ ¼ k1 P1 ð0Þ þ P3 ð0Þ P2 ð0Þ ¼ ð4k þ aÞP1 ð0Þ þ P3 ð0Þ P2 ð0Þ ¼ ð4k þ 2aÞð4k þ aÞ P0 ; P k2 4k l2 3 ð0Þ ¼ k2 P2 ð0Þ þ P4 ð0Þ P3 ð0Þ ¼ 4kP2 ð0Þ þ P4 ð0Þ P3 ð0Þ ¼ 4kð4k þ 2aÞð4k þ aÞ P0 : P k3 3k l3 4 ð0Þ using Eq. (19) in terms of P0. Step 4. Compute P 2 4 ð0Þ ¼ k3 bP 3 ð0Þ ¼ 3k P 3 ð0Þ ¼ 12k ð4k þ 2aÞð4k þ aÞ P0 : P 4
l
l
Step 5. Compute P uf0 , P uf1 and Puf2 using Eq. (14) in terms of P0.
Puf0 ¼
k0 ð1 cÞ ð4k þ 2aÞð1 cÞ P0 ¼ P0 ; b b
Puf1 ¼
k1 ð1 cÞ ð4k þ aÞð1 cÞ ð4k þ 2aÞð4k þ aÞð1 cÞ P1 ¼ P1 ¼ P0 ; b b lb
Puf2 ¼
k2 ð1 cÞ 4kð1 cÞ 4kð4k þ 2aÞð4k þ aÞð1 cÞ P2 ¼ P2 ¼ P0 : b b l2 b
Step 6. Compute P0 using Eq. (20). i ð0Þ (i = 1, 2, 3, 4) and P uf (i = 0, 1, 2) can be expressed in terms of P0. Since From the above steps, P i P4 P2 P uf ¼ 1, we obtain P0. However, the expression of P0 is complicated, so we omit it for brevity. P0 þ Pi ð0Þ þ i¼1
i¼0
i
Step 7. Compute the steady-state availability A(1) using Eq. (21). i ð0Þ (i ¼ 1; 2; 3; 4) can be obtained from Steps 2 through 4, and P uf (i = 0, 1, 2) is Since P0 has been derived from Step 6, P i determined from Step 5. Then, the steady-state availability A(1) can be computed by:
Að1Þ ¼ P0 þ
3 2 X X 4 ð0Þ: P i ð0Þ þ Pufi ¼ 1 P i¼1
i¼0
Example 2: gamma repair time distribution The repair time has a gamma distribution with parameter r. We set the mean repair time, b = 1/l, where l is the repair rl r rate. In this case, we have BðsÞ ¼ ðsþr lÞ . Step 1. Compute P2(0) using Eq. (15) in terms of P0.
P2 ð0Þ ¼
1 Þ þ aÞ k0 ½1 Bðk ð4k þ 2aÞ½1 Bð4k ð4k þ 2aÞ½ðr l þ 4k þ aÞr ðr lÞr P0 ¼ P0 ¼ P0 : ðrlÞr Bðk1 Þ Bð4k þ aÞ
1 ð0Þ using Eq. (16) in terms of P0. Step 2. Compute P r r 1 ð0Þ ¼ 1 P 2 ð0Þ ¼ 1 P2 ð0Þ ¼ ð4k þ 2aÞ½ðr l þ 4k þ aÞ ðr lÞ P0 : P r k1 4k þ a ð4k þ aÞðr lÞ
2 ð0Þ and P 3 ð0Þ using Eqs. (17) and (18) in terms of P0. Step 3. Compute P3(0), P 4 ð0Þ, P
P3 ð0Þ ¼
2 ÞP1 ð0Þ P2 ð0Þ ð4k þ aÞBð4kÞ P 1 ð0Þ ð4k þ 2aÞ½ðr l þ 4k þ aÞr ðr lÞr ½ðrl þ 4kÞr ðrlÞr P2 ð0Þ k1 Bðk ¼ P0 ; ¼ 2Þ Bðk Bð4kÞ ðr lÞ2r
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1 Exp Gam(2.5) lognormal(0.5) lognormal(2) Weibull(0.5) Weibull(2)
0.9 0.8 0.7
Av
0.6 0.5 0.4 0.3 0.2 0.1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
λ Fig. 2. Availability for k and different repair times (a = 0.01, c = 0.8, l = 1 and b = 2.4).
P4 ð0Þ ¼ ¼
3 ÞP 2 ð0Þ P3 ð0Þ 4kBð3kÞ 2 ð0Þ P P3 ð0Þ k2 Bðk ¼ 3Þ Bðk Bð3kÞ ð4k þ 2aÞ½ðr l þ 4k þ aÞr ðr lÞr ½ðr l þ 4kÞr ðrlÞr ðr lÞ3r
½ðr l þ 3kÞr ðr lÞr P0 ;
2 ð0Þ ¼ k1 P1 ð0Þ þ P3 ð0Þ P 2 ð0Þ ¼ ð4k þ aÞP1 ð0Þ þ P3 ð0Þ P2 ð0Þ P k2 4k ð4k þ 2aÞ½ðr l þ 4k þ aÞr ðr lÞr ½ðr l þ 4kÞr ðrlÞr ¼ P0 ; 4kðrlÞ2r 3 ð0Þ ¼ k2 P2 ð0Þ þ P4 ð0Þ P 3 ð0Þ ¼ 4kP2 ð0Þ þ P4 ð0Þ P3 ð0Þ P k3 3k ð4k þ 2aÞ½ðr l þ 4k þ aÞr ðr lÞr ½ðr l þ 4kÞr ðrlÞr ¼ ½ðr l þ 3kÞr ðr lÞr P0 : 3kðrlÞ3r 4 ð0Þ using Eq. (19) in terms of P0. Step 4. Compute P r r r r 4 ð0Þ ¼ k3 bP 3 ð0Þ ¼ 3k P 3 ð0Þ ¼ ð4k þ 2aÞ½ðr l þ 4k þ aÞ ðr lÞ ½ðrl þ 4kÞ ðr lÞ ½ðr l þ 3kÞr ðrlÞr P0 : P 3r l lðrlÞ
Step 5. Compute Puf0 , P uf1 and P uf2 using Eq. (14) in terms of P0.
Puf0 ¼
k0 ð1 cÞ ð4k þ 2aÞð1 cÞ P0 ¼ P0 ; b b
Puf1 ¼
k1 ð1 cÞ ð4k þ aÞð1 cÞ ð4k þ 2aÞð1 cÞ½ðr l þ 4k þ aÞr ðrlÞr P1 ¼ P1 ¼ P0 ; b b bðrlÞr
Puf2 ¼
k2 ð1 cÞ 4kð1 cÞ ð4k þ 2aÞð1 cÞ½ðr l þ 4k þ aÞr ðrlÞr ½ðr l þ 4kÞr ðr lÞr P2 ¼ P2 ¼ P0 : b b bðr lÞ2r
Step 6. Compute P0 using Eq. (20). i ð0Þ (i = 1, 2, 3, 4) and Puf (i = 0, 1, 2) can be expressed in terms of P0. Since From the above steps, P i P4 P2 P uf ¼ 1, we obtain P0. However, the expression of P0 is complicated, so we omit it for brevity. P0 þ P i ð0Þ þ i¼1
i¼0
i
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1 0.98 0.96 0.94
Av
0.92 0.9 0.88 Exp Gam(2.5) lognormal(0.5) lognormal(2) Weibull(0.5) Weibull(2)
0.86 0.84 0.82 0.8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
μ Fig. 3. Availability for l and different repair times (k = 0.1, a = 0.01, c = 0.8 and b = 2.4).
Step 7. Compute the steady-state availability A(1) using Eq. (21). i ð0Þ (i = 1, 2, 3, 4) can be obtained from Steps 2 through 4, and P uf (i = 0, 1, 2) Since P0 has been derived from Step 6, P i is determined from Step 5. Then, the steady-state availability A(1) can be computed by:
Að1Þ ¼ P0 þ
3 2 X X 4 ð0Þ: P i ð0Þ þ Pufi ¼ 1 P i¼1
i¼0
4. Numerical analysis and discussion This section provides numerical experiments for a variety of repair time distributions, including exponential, gamma, lognormal and Weibull distributions. For convenience of computations, we consider M = 6, S = 3 and K = 4. The steady-state availability for the different repair time distributions with various system parameters are shown in Figs. 2–6. As Figs. 2 and 3 show, the steady-state availability decreases as the failure rate of primary unit k increases or the repair rate of failed unit l decreases for different repair distributions. As Fig. 4 shows, the coverage factor c rarely affects the steady-state system 1.015 Exp Gam(2.5) lognormal(0.5) lognormal(2) Weibull(0.5) Weibull(2)
1.01 1.005 1
Av
0.995 0.99 0.985 0.98 0.975 0.97 0
0.1
0.2
0.3
0.4
0.5 c
0.6
0.7
0.8
0.9
1
Fig. 4. Availability for various coverage factors and different repair times (k = 0.1, a = 0.01, l = 1 and b = 2.4).
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J.-C. Ke, T.-H. Liu / Applied Mathematics and Computation 246 (2014) 148–158
Fig. 5. Availability for k and l with exponential repair times (a = 0.01, c = 0.8 and b = 2.4).
1.01 1
Av
0.99 0.98 0.97 0.96 1 15 0.5
10
c
5 0
S
0
Fig. 6. Availability for different numbers of warm standby units and c with exponential repair times (k = 0.1, a = 0.01, l = 1 and b = 2.4).
Table 1 Sensitivity of steady-state system availability A(1) with respect to individual system parameters for different repair distributions (a = 0.01, l = 1, c = 0.9 and b = 2.4). k
Repair distribution
oA(1)/ok
oA(1)/ol
oA(1)/oc
0.1
Exponential Gamma (2.5) Lognormal (0.5) Lognormal (2) Weibull (0.5) Weibull (2)
0.5174 0.9063 0.7762 0.1267 0.0696 1.0136
0.0110 0.0187 0.0160 0.0028 0.0017 0.0208
0.0027 0.0044 0.0038 0.0007 0.0004 0.0049
0.3
Exponential Gamma (2.5) Lognormal (0.5) Lognormal (2) Weibull (0.5) Weibull (2)
1.5632 1.4978 1.5415 1.3939 0.7884 1.4710
0.2187 0.2400 0.2370 0.1404 0.0780 0.2424
0.0853 0.0670 0.0714 0.0789 0.0497 0.0628
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Table 2 Sensitivity of steady-state system availability A(1) with respect to individual system parameters for different repair distributions (k = 0.1, a = 0.01, c = 0.9 and b = 2.4).
l
Repair distribution
oA(1)/ok
oA(1)/ol
oA(1)/oc
0.2
Exponential Gamma (2.5) Lognormal (0.5) Lognormal (2) Weibull (0.5) Weibull (2)
2.3709 3.9124 3.4213 0.6197 0.3436 4.3036
0.2527 0.4035 0.3525 0.0678 0.0411 0.4425
0.0122 0.0190 0.0167 0.0034 0.0021 0.0206
1
Exponential Gamma (2.5) Lognormal (0.5) Lognormal (2) Weibull (0.5) Weibull (2)
0.5174 0.9063 0.7762 0.1267 0.0696 1.0136
0.0110 0.0187 0.0160 0.0028 0.0017 0.0208
0.0027 0.0044 0.0038 0.0007 0.0004 0.0049
Table 3 Sensitivity of steady-state system availability A(1) with respect to individual system parameters for different repair distributions (k = 0.1, a = 0.01, l = 1 and b = 2.4). c
Repair distribution
oA(1)/ok
oA(1)/ol
oA(1)/oc
0.7
Exponential Gamma (2.5) Lognormal (0.5) Lognormal (2) Weibull (0.5) Weibull (2)
0.4902 0.8617 0.7371 0.1196 0.0656 0.9646
0.0105 0.0179 0.0153 0.0026 0.0016 0.0200
0.0024 0.0040 0.0035 0.0006 0.0004 0.0044
0.9
Exponential Gamma (2.5) Lognormal (0.5) Lognormal (2) Weibull (0.5) Weibull (2)
0.5174 0.9063 0.7762 0.1267 0.0696 1.0136
0.0110 0.0187 0.0160 0.0028 0.0017 0.0208
0.0027 0.0044 0.0038 0.0007 0.0004 0.0049
availability for various repair distributions. For fixed system parameters, the Weibull repair times with smaller shape parameters always have greater steady-state system availability. Fig. 5 indicates similar tendencies to those shown in Figs. 2 and 3. Fig. 6 indicates that the number of warm standby units improves the steady-state system availability. A sensitivity analysis was conducted on the steady-state availability with respect to individual system parameters k, c and l by changing the specific value of the system parameter for the various repair time distributions. The numerical results of the sensitivity for different repair time distributions, as shown in Tables 1–3, indicate that the signs of sensitivity for parameters k and c are negative, which means that increasing these parameters will reduce the steady-state system availability. In contrast, l has a positive sign. Therefore, increasing this parameter will improve the steady-state system availability. Also, we find that k significantly affects the steady-state availability and that the effect of c is insensitive. 5. Conclusions In this paper, we investigated a repairable system with imperfect coverage and reboot delay. A step-by-step algorithm was constructed to compute the steady-state availability. We adopted two repair time distributions to illustrate the method. The numerical results indicated steady-state availability decreased as the failure rate of the primary unit or the number of warm standby units increased or the mean repair rate decreased. Moreover, the sensitivity analysis showed that the impact of the failure rate of the primary unit was of greater significance. References [1] S.V. Amari, A.F. Myers, A. Rauzy, K.S. Trivedi, Imperfect coverage models: status and trends, in: Handbook of Performability Engineering, Springer, Berlin, 2008, pp. 321–348. [2] J.-C. Ke, H.-I. Huang, C.-H. Lin, Parametric programming approach for a two-unit repairable system with imperfect coverage, reboot, and fuzzy parameters, IEEE Trans. Reliab. 57 (2008) 498–506. [3] J.C. Ke, S.L. Lee, Y.L. Hsu, Bayesian analysis for a redundant repairable system with imperfect coverage, Commun. Stat. – Simul. Comput. 37 (2008) 993– 1004. [4] J.C. Ke, S.L. Lee, M.Y. Ko, Two-unit redundant system with detection delay and imperfect coverage: confidence interval estimation, Qual. Technol. Quant. Manage. 8 (1) (2011) 1–14.
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[5] J.C. Ke, Z.L. Su, K.H. Wang, An availability system with general repair distribution: statistical inference, Commun. Stat. Theor. Methods 38 (2009) 1508– 1525. [6] A.F. Myers, K-out-of-n: G system reliability with imperfect fault coverage, IEEE Trans. Reliab. 56 (2007) 464–473. [7] K.S. Trivedi, Probability and Statistics with Reliability, Queuing and Computer Science Applications, second ed., John Wiley and Sons, New York, 2002. [8] K.H. Wang, Y.J. Chen, Comparative analysis of availability between three system with general repair times, reboot delay and switching failures, Appl. Math. Comput. 215 (2009) 384–394. [9] K.-H. Wang, L.-W. Chiu, Cost benefit analysis of availability systems with warm standby units and imperfect coverage, Appl. Math. Comput. 172 (2006) 1239–1256. [10] K.H. Wang, W.L. Don, J.B. Ke, Comparison of reliability between four systems with warm standby components and standby switching failures, Appl. Math. Comput. 183 (2006) 1310–1322. [11] K.H. Wang, T.C. Yen, J.J. Jian, Reliability and sensitivity analysis of a repairable system with imperfect coverage under service pressure condition, J. Manuf. Syst. 32 (2013) 357–363. [12] T.C. Yen, K.H. Wang, W.L. Chen, Comparative analysis of three systems with imperfect coverage and standby switching failures, in: The Fifth International Conference on Advances in Future Internet, Barcelona, Spain, (AFIN 2013), 2013.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A repairable system with imperfect coverage and reboot Jau-Chuan Ke ⇑, Tzu-Hsin Liu Department of Applied Statistics, National Taichung University of Science and Technology, Taichung, Taiwan, ROC
a r t i c l e
i n f o
Keywords: Availability Imperfect coverage Reboot delay Sensitivity analysis
a b s t r a c t In this paper, we examine a repairable system with imperfect coverage, reboot delay and one repair facility. The failure times and repair times of failed units are assumed to be exponential and general distributions, respectively. As a unit fails, it can be immediately detected, located and replaced with a coverage probability c by a standby, if one is available. We use a recursive method and the supplementary variable technique to develop the steady-state probabilities of down units at an arbitrary epoch. Then, an efficient algorithm is constructed to compute the steady-state availability. We adopt two repair time distributions, namely, exponential and gamma, to illustrate the method. Finally, we also perform a sensitivity analysis of the steady-state availability with respect to the system parameters for various repair time distributions. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Availability, or reliability, plays an important role when evaluating a system’s effectiveness. Availability/reliability has been widely studied because of its prevalence in power plants, production systems, manufacturing systems and industrial systems. Maintaining a high or required level of availability/reliability is often a fundamental requisite. Standby or redundancy is an efficient method which can be applied to improve system availability. The probability of successful recovery on the failure of a primary unit (or standby unit) is denoted by c. We assume that the coverage factor is the same for a primary unit failure as that for a standby failure. Quantity c, which includes the probabilities of successful detection and location of a failure, is known as the coverage factor or coverage probability (see [7]). The concept of the coverage factor and its effect on the reliability and availability model of a repairable system have been studied by a number of authors, including Trivedi [7], Wang and Chiu [9], Ke et al. [2,3] and Myers [6]. The status and trends of imperfect coverage models and the associated reliability analysis techniques were introduced in Amari et al. [1]. Ke et al. [5] investigated the statistical inferences of an availability system with imperfect coverage. Ke et al. [4] also studied a twounit redundant repairable system with detection delay and imperfect coverage. Recently, Wang et al. [11] investigated the reliability and sensitivity analysis of a repairable system with imperfect coverage under a service pressure condition. The concept of a reboot delay for a repairable system was introduced by Trivedi [7]. Wang et al. [10] completed a comparison of the reliability and availability between four systems with warm standby units, reboot delay and standby switching failures. Wang and Chen [8] performed a comparative analysis of the availability between three systems with general repair times, reboot delay and switching failures. Recently, Yen et al. [12] compared three different configurations with imperfect coverage and standby switching failures based on reliability and availability.
⇑ Corresponding author. http://dx.doi.org/10.1016/j.amc.2014.07.090 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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149
Notations M S K NðtÞ VðtÞ BðuÞ bðuÞ BðsÞ k
a c b Pi ðtÞ Pufi ðtÞ Pi Pufi i ðsÞ P b Að1Þ
number of primary units number of standby units minimum number of operating units number of failed units in the system remaining repair time for the failed units under repair distribution function of repair time probability density function of repair time Laplace transform of repair time failure rate of a primary unit failure rate of a standby unit coverage probability reboot delay rate probability of i failed units in the system at time t in the safe state, where i ¼ 0; 1; 2; . . . ; N probability of i failed units in the system at time t in the unsafe failure state, where i ¼ 0; 1; 2; . . . ; N 2 steady-state probability of i failed units in the system at time t in the safe state, where i ¼ 0; 1; 2; . . . ; N steady-state probability of i failed units in the system at time t in the unsafe failure state, where i ¼ 0; 1; 2; . . . ; N 2 Laplace transform of P i ðv Þ mean repair time steady-state availability
In recent years, the database in computer systems has become of utmost importance in our highly information-oriented society. In particular, a reliable database is the most indispensable instrument in on-line transaction processing systems, such as real-time systems used for bank accounts. The data in a computer system are frequently updated by adding or deleting files, and can be stored in other secondary media (treated as active units). However, data files in secondary media are sometimes damaged by errors due to noises, human errors and hardware faults. To ensure the safety of data, there must always be backup copies of all files in other places (treated as standby units), which are then available if files in the original secondary media are damaged. Because files are of different sizes, the time necessary to reconstruct a file is a random variable instead of a constant. A fault detecting server runs general purpose monitoring software to monitor data files in secondary media and backup copies. When data files in secondary media break down, the problem can be immediately detected by the fault-detecting server and replaced with a coverage probability c by a backup copy, if one is available. The database is in an unsafe failure state if any breakdown is not covered. The data files in secondary media (or backup copies) in the unsafe failure state can be cleared by a reboot. Reboot delay may take place at a rate for the data files in secondary media (or backup copies), which is exponentially distributed. The rest of the paper is organized as follows: Section 2 presents a brief description of the system; Section 3 presents a recursive method using the supplementary variable technique, treating the supplementary variable as the remaining repair time. The steady-state probability distribution of down units at an arbitrary epoch is developed and an efficient algorithm to find the steady-state availability is provided. Two simple examples for the two different repair time distributions, exponential and gamma, are given; in Section 4, numerical results are presented to illustrate the steady-state availability under different repair time distributions, and a sensitivity analysis is also conducted; and, finally, some conclusions are drawn in the last section.
2. Model description The repairable system includes M identical primary units, S standby units and one repairman/server. System failure is defined to be less than K units in an active operation, where K = 1, 2, . . ., M + S. Therefore, the system failure occurs if and only if the number of failed units in the system is larger or equal to N = M + S K + 1. The detailed descriptions of the assumptions are as follows: (i) The operating times of the primary units between breakdowns are independent, identically distributed (i.i.d.) variables having an exponential distribution with parameter k. (ii) If a primary unit fails, it is immediately replaced by a standby unit if one is available. Each standby unit fails independently of the state of all other units and has an exponential time-to-failure distribution with parameter a (0 < a < k).
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(iii) The failed unit is immediately repaired when the server is idle. When the server is busy, the arriving failure units have to queue up until the server is available. The repair sequence is according to the first come, first repaired discipline. (iv) The time to repair a failed unit is i.i.d. random variables having a distribution BðuÞðu P 0Þ, a probability density function bðuÞðu P 0Þ and mean repair time b. (v) When a unit fails, it can be immediately detected and located with a coverage probability c. (vi) Defining the unsafe failure state of the system as any one of the breakdowns is not covered. A primary unit fails in the unsafe failure state, which is cleared by a reboot. Reboot delay occurs at rate b for a primary unit (or standby unit) which is exponentially distributed. 3. Steady-state results 3.1. Steady-state probability We use the following supplementary variable: V remaining repair time for the failed units under repair. The state of the system at time t is given by: N(t) number of failed units in the system, and V(t) remaining repair time for the failed units under repair. Let us define:
Pi ðv ; tÞ ¼ PfNðtÞ ¼ i; v < VðtÞ 6 v þ dv ; in the safe stateg; Pi ðtÞ ¼
Z
1
Pi ðv ; tÞdv ;
v P 0;
i ¼ 0; 1; 2; . . . ; N;
i ¼ 1; 2; . . . ; N;
0
Pufi ðv ; tÞ ¼ PfNðtÞ ¼ i; v < VðtÞ 6 v þ dv ; in the unsafe failure stateg; Pufi ðtÞ ¼
Z 0
1
Pufi ðv ; tÞdv ;
v P 0;
i ¼ 0; 1; 2; . . . ; N 2;
i ¼ 0; 1; 2; . . . ; N 2:
Relating the state-transition rate diagram shown in Fig. 1, we obtain:
d P0 ðtÞ ¼ k0 P0 ðtÞ þ P1 ð0; tÞ; dt @ @ P1 ðv ; tÞ ¼ k1 P1 ðv ; tÞ þ k0 cP 0 ðv ; tÞ þ bðv ÞP2 ð0; tÞ þ bPuf0 ðv ; tÞ; @t @ v
ð1Þ
ð2Þ
@ @ Pi ðv ; tÞ ¼ ki Pi ðv ; tÞ þ ki1 cP i1 ðv ; tÞ þ bðv ÞPiþ1 ð0; tÞ þ bPufi1 ðv ; tÞ; @t @ v
2 6 i 6 N 2;
ð3Þ
@ @ PN ðv ; tÞ ¼ kN1 PN1 ðv ; tÞ; @t @ v
d Puf ðtÞ ¼ ki ð1 cÞPi ðtÞ bPufi ðtÞ; dt i
λ0 c
ð4Þ
0 6 i 6 N 2;
0
b (v )
b (v )
β uf0
λL−3c
λ1 (1 − c ) β uf1
λL−1
λL −2 c N-2
2
1
λ0 (1 − c )
λ2 c
λ1c
ð5Þ
b (v )
N
N-1
b (v )
b (v )
λN − 2 (1 − c )
b (v )
β
ufN-2
Fig. 1. The state transition diagram for the repairable system with imperfect coverage and reboot delay.
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151
where
ki ¼
Mk þ ðS iÞa; 0 6 i 6 S 1 ðM þ S iÞk;
S6i6N
;
In steady-state, let
Pi ¼ lim Pi ðtÞ;
0 6 i 6 N;
t!1
Pi ðv Þ ¼ lim Pi ðv ; tÞ;
0 6 i 6 N;
t!1
Pufi ¼ lim P ufi ðtÞ;
0 6 i 6 N 2;
t!1
Pufi ðv Þ ¼ lim P ufi ðv ; tÞ; t!1
0 6 i 6 N 2:
We also define:
Pi ðv Þ ¼ bðv ÞPi ;
i ¼ 0; 1; 2; . . . ; N;
Pufi ðv Þ ¼ bðv ÞPufi ;
i ¼ 0; 1; 2; . . . ; N 2:
From Eqs. (1)–(5), we obtain the following steady-state equations:
P1 ð0Þ ¼ k0 P0 ;
ð6Þ
@ P1 ðv Þ ¼ k1 P1 ðv Þ þ k0 cP 0 ðv Þ þ bðv ÞP2 ð0Þ þ bPuf0 ðv Þ; @v
ð7Þ
@ Pi ðv Þ ¼ ki Pi ðv Þ þ ki1 cP i1 ðv Þ þ bðv ÞPiþ1 ð0Þ þ bPufi1 ðv Þ; 2 6 i 6 N 1; @v
ð8Þ
@ PN ðv Þ ¼ kN1 PN1 ðv Þ; @v
ð9Þ
0 ¼ ki ð1 cÞPi bP ufi ;
0 6 i 6 N 2:
ð10Þ
Further define:
¼ BðsÞ
Z
1
esv dBðv Þ;
0
i ðsÞ ¼ P
Z
1
esv Pi ðv Þdv ;
1 6 i 6 N;
0
i ð0Þ ¼ Pi ¼ P
Z
1
Pi ðv Þdv ;
1 6 i 6 N;
0
Z
1
esv
0
@ i ðsÞ Pi ð0Þ; Pi ðv Þdv ¼ sP @v
1 6 i 6 N:
Therefore, if the LST is taken of both sides of Eqs. (7)–(9), it is found that:
ðk1 sÞP 1 ðsÞ ¼ k0 BðsÞP 0 þ BðsÞP 2 ð0Þ P 1 ð0Þ; i ðsÞ ¼ ki1 BðsÞP ðki sÞP i1 þ BðsÞP iþ1 ð0Þ P i ð0Þ;
ð11Þ 2 6 i 6 N 1;
N ðsÞ ¼ kN1 BðsÞP sP N1 P N ð0Þ:
ð12Þ ð13Þ
From Eq. (10), P ufi can be expressed as:
Pufi ¼
ki ð1 cÞ Pi ; b
0 6 i 6 N 2:
ð14Þ
Using Eq. (6) in Eq. (11) and substituting s ¼ k1 and s ¼ 0, respectively, it implies that:
P2 ð0Þ ¼
1 Þ k0 ½1 Bðk P0 ; Bðk1 Þ
ð15Þ
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and
1 ð0Þ ¼ 1 P2 ð0Þ: P k1
ð16Þ
Setting s ¼ ki in Eq. (12), we get:
Piþ1 ð0Þ ¼
i ÞP i1 ð0Þ Pi ð0Þ ki1 Bðk ; Bðki Þ
26i6N1
ð17Þ
Now, setting s ¼ 0 in Eq. (12), we obtain:
i ð0Þ ¼ ki1 P i1 ð0Þ þ Piþ1 ð0Þ Pi ð0Þ ; P ki
i ¼ 2; 3; . . . ; N 1
ð18Þ
N ð0Þ, we differentiate Eq. (13) with respect to s and set s = 0. It yields: To find the unknown quantity P
N ð0Þ ¼ kN1 bP N1 ð0Þ; P
ð19Þ
where b ¼ ½dBðsÞ=ds s¼0 is the mean repair time. Therefore, P i ð0Þð1 6 i 6 NÞ and Puf ð0 6 i 6 N 2Þ can be expressed in terms of P0. Then, P0 can be determined by the nori
malizing condition:
P0 þ
N N2 X X i ð0Þ þ P Pufi ¼ 1: i¼1
ð20Þ
i¼0
1 ; P 2 ; . . . ; P N ; Puf ; Puf ; . . . ; P uf . Once P0 has been constructed, we obtain the steady-state probabilities: P 0 1 N2 3.2. Steady-state availability Let us denote by A(1) the steady-state availability of the repairable system with warm standby units. Since the system operable in states:
Að1Þ ¼ P0 þ
N1 N2 X X N ð0Þ; Pi ð0Þ þ Pufi ¼ 1 P i¼1
ð21Þ
i¼0
to demonstrate the working the recursive method, an algorithm for computing the steady-state availability is provided below. Algorithm for computing the steady-state availability Step Step Step Step Step Step Step
1. 2. 3. 4. 5. 6. 7.
Compute Compute Compute Compute Compute Compute Compute
P2(0) using Eq. (15) in terms of P0. 1 ð0Þ using Eq. (16) in terms of P0. P i ð0Þ (2 6 i 6 N 1) using Eqs. (17) and (18) in terms of P0. Pi+1(0) (2 6 i 6 N 1) and P N ð0Þ using Eq. (19) in terms of P0. P Pufi for each i = 0, 1, 2, . . ., N 2 using Eq. (14) in terms of P0. P0 using Eq. (20). the steady-state availability A(1) using Eq. (21).
3.3. Simple examples Based on the algorithm stated above, three simple examples are provided to illustrate the recursive method, in which the repair time distributions are exponential and gamma distributions. We consider four primary units, M = 4, and two warm standby units, S = 2. The minimum number of operating units is set to three, K = 3. Example 1: exponential repair time distribution We set the mean repair time, b = 1/l, where
¼ l=ðl þ sÞ. l is the repair rate. In this case, we have BðsÞ
Step 1. Compute P2 ð0Þ using Eq. (15) in terms of P 0 .
1 Þ þ aÞ k0 ½1 Bðk ð4k þ 2aÞ½1 Bð4k P0 ¼ P0 1Þ Bðk Bð4k þ aÞ ð4k þ 2aÞð4k þ aÞ ¼ P0 :
P2 ð0Þ ¼
l
1 ð0Þ using Eq. (16) in terms of P 0 . Step 2. Compute P
1 ð0Þ ¼ 1 P k1
P2 ð0Þ ¼
1 4k þ a
P2 ð0Þ ¼
4k þ 2a
l
P0 :
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153
2 ð0Þ and P 3 ð0Þ using Eqs. (17) and (18) in terms of P0. Step 3. Compute P 3 ð0Þ, P 4 ð0Þ, P
P3 ð0Þ ¼
2 ÞP1 ð0Þ P2 ð0Þ ð4k þ aÞBð4kÞ P 1 ð0Þ 4kð4k þ 2aÞð4k þ aÞ P2 ð0Þ k1 Bðk ¼ ¼ P0 ; l2 Bðk2 Þ Bð4kÞ
P4 ð0Þ ¼
3 ÞP2 ð0Þ P3 ð0Þ 4kBð3kÞ P2 ð0Þ 12k2 ð4k þ 2aÞð4k þ aÞ P3 ð0Þ k2 Bðk ¼ ¼ P0 ; l3 Bðk3 Þ Bð3kÞ
2 ð0Þ ¼ k1 P1 ð0Þ þ P3 ð0Þ P2 ð0Þ ¼ ð4k þ aÞP1 ð0Þ þ P3 ð0Þ P2 ð0Þ ¼ ð4k þ 2aÞð4k þ aÞ P0 ; P k2 4k l2 3 ð0Þ ¼ k2 P2 ð0Þ þ P4 ð0Þ P3 ð0Þ ¼ 4kP2 ð0Þ þ P4 ð0Þ P3 ð0Þ ¼ 4kð4k þ 2aÞð4k þ aÞ P0 : P k3 3k l3 4 ð0Þ using Eq. (19) in terms of P0. Step 4. Compute P 2 4 ð0Þ ¼ k3 bP 3 ð0Þ ¼ 3k P 3 ð0Þ ¼ 12k ð4k þ 2aÞð4k þ aÞ P0 : P 4
l
l
Step 5. Compute P uf0 , P uf1 and Puf2 using Eq. (14) in terms of P0.
Puf0 ¼
k0 ð1 cÞ ð4k þ 2aÞð1 cÞ P0 ¼ P0 ; b b
Puf1 ¼
k1 ð1 cÞ ð4k þ aÞð1 cÞ ð4k þ 2aÞð4k þ aÞð1 cÞ P1 ¼ P1 ¼ P0 ; b b lb
Puf2 ¼
k2 ð1 cÞ 4kð1 cÞ 4kð4k þ 2aÞð4k þ aÞð1 cÞ P2 ¼ P2 ¼ P0 : b b l2 b
Step 6. Compute P0 using Eq. (20). i ð0Þ (i = 1, 2, 3, 4) and P uf (i = 0, 1, 2) can be expressed in terms of P0. Since From the above steps, P i P4 P2 P uf ¼ 1, we obtain P0. However, the expression of P0 is complicated, so we omit it for brevity. P0 þ Pi ð0Þ þ i¼1
i¼0
i
Step 7. Compute the steady-state availability A(1) using Eq. (21). i ð0Þ (i ¼ 1; 2; 3; 4) can be obtained from Steps 2 through 4, and P uf (i = 0, 1, 2) is Since P0 has been derived from Step 6, P i determined from Step 5. Then, the steady-state availability A(1) can be computed by:
Að1Þ ¼ P0 þ
3 2 X X 4 ð0Þ: P i ð0Þ þ Pufi ¼ 1 P i¼1
i¼0
Example 2: gamma repair time distribution The repair time has a gamma distribution with parameter r. We set the mean repair time, b = 1/l, where l is the repair rl r rate. In this case, we have BðsÞ ¼ ðsþr lÞ . Step 1. Compute P2(0) using Eq. (15) in terms of P0.
P2 ð0Þ ¼
1 Þ þ aÞ k0 ½1 Bðk ð4k þ 2aÞ½1 Bð4k ð4k þ 2aÞ½ðr l þ 4k þ aÞr ðr lÞr P0 ¼ P0 ¼ P0 : ðrlÞr Bðk1 Þ Bð4k þ aÞ
1 ð0Þ using Eq. (16) in terms of P0. Step 2. Compute P r r 1 ð0Þ ¼ 1 P 2 ð0Þ ¼ 1 P2 ð0Þ ¼ ð4k þ 2aÞ½ðr l þ 4k þ aÞ ðr lÞ P0 : P r k1 4k þ a ð4k þ aÞðr lÞ
2 ð0Þ and P 3 ð0Þ using Eqs. (17) and (18) in terms of P0. Step 3. Compute P3(0), P 4 ð0Þ, P
P3 ð0Þ ¼
2 ÞP1 ð0Þ P2 ð0Þ ð4k þ aÞBð4kÞ P 1 ð0Þ ð4k þ 2aÞ½ðr l þ 4k þ aÞr ðr lÞr ½ðrl þ 4kÞr ðrlÞr P2 ð0Þ k1 Bðk ¼ P0 ; ¼ 2Þ Bðk Bð4kÞ ðr lÞ2r
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1 Exp Gam(2.5) lognormal(0.5) lognormal(2) Weibull(0.5) Weibull(2)
0.9 0.8 0.7
Av
0.6 0.5 0.4 0.3 0.2 0.1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
λ Fig. 2. Availability for k and different repair times (a = 0.01, c = 0.8, l = 1 and b = 2.4).
P4 ð0Þ ¼ ¼
3 ÞP 2 ð0Þ P3 ð0Þ 4kBð3kÞ 2 ð0Þ P P3 ð0Þ k2 Bðk ¼ 3Þ Bðk Bð3kÞ ð4k þ 2aÞ½ðr l þ 4k þ aÞr ðr lÞr ½ðr l þ 4kÞr ðrlÞr ðr lÞ3r
½ðr l þ 3kÞr ðr lÞr P0 ;
2 ð0Þ ¼ k1 P1 ð0Þ þ P3 ð0Þ P 2 ð0Þ ¼ ð4k þ aÞP1 ð0Þ þ P3 ð0Þ P2 ð0Þ P k2 4k ð4k þ 2aÞ½ðr l þ 4k þ aÞr ðr lÞr ½ðr l þ 4kÞr ðrlÞr ¼ P0 ; 4kðrlÞ2r 3 ð0Þ ¼ k2 P2 ð0Þ þ P4 ð0Þ P 3 ð0Þ ¼ 4kP2 ð0Þ þ P4 ð0Þ P3 ð0Þ P k3 3k ð4k þ 2aÞ½ðr l þ 4k þ aÞr ðr lÞr ½ðr l þ 4kÞr ðrlÞr ¼ ½ðr l þ 3kÞr ðr lÞr P0 : 3kðrlÞ3r 4 ð0Þ using Eq. (19) in terms of P0. Step 4. Compute P r r r r 4 ð0Þ ¼ k3 bP 3 ð0Þ ¼ 3k P 3 ð0Þ ¼ ð4k þ 2aÞ½ðr l þ 4k þ aÞ ðr lÞ ½ðrl þ 4kÞ ðr lÞ ½ðr l þ 3kÞr ðrlÞr P0 : P 3r l lðrlÞ
Step 5. Compute Puf0 , P uf1 and P uf2 using Eq. (14) in terms of P0.
Puf0 ¼
k0 ð1 cÞ ð4k þ 2aÞð1 cÞ P0 ¼ P0 ; b b
Puf1 ¼
k1 ð1 cÞ ð4k þ aÞð1 cÞ ð4k þ 2aÞð1 cÞ½ðr l þ 4k þ aÞr ðrlÞr P1 ¼ P1 ¼ P0 ; b b bðrlÞr
Puf2 ¼
k2 ð1 cÞ 4kð1 cÞ ð4k þ 2aÞð1 cÞ½ðr l þ 4k þ aÞr ðrlÞr ½ðr l þ 4kÞr ðr lÞr P2 ¼ P2 ¼ P0 : b b bðr lÞ2r
Step 6. Compute P0 using Eq. (20). i ð0Þ (i = 1, 2, 3, 4) and Puf (i = 0, 1, 2) can be expressed in terms of P0. Since From the above steps, P i P4 P2 P uf ¼ 1, we obtain P0. However, the expression of P0 is complicated, so we omit it for brevity. P0 þ P i ð0Þ þ i¼1
i¼0
i
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1 0.98 0.96 0.94
Av
0.92 0.9 0.88 Exp Gam(2.5) lognormal(0.5) lognormal(2) Weibull(0.5) Weibull(2)
0.86 0.84 0.82 0.8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
μ Fig. 3. Availability for l and different repair times (k = 0.1, a = 0.01, c = 0.8 and b = 2.4).
Step 7. Compute the steady-state availability A(1) using Eq. (21). i ð0Þ (i = 1, 2, 3, 4) can be obtained from Steps 2 through 4, and P uf (i = 0, 1, 2) Since P0 has been derived from Step 6, P i is determined from Step 5. Then, the steady-state availability A(1) can be computed by:
Að1Þ ¼ P0 þ
3 2 X X 4 ð0Þ: P i ð0Þ þ Pufi ¼ 1 P i¼1
i¼0
4. Numerical analysis and discussion This section provides numerical experiments for a variety of repair time distributions, including exponential, gamma, lognormal and Weibull distributions. For convenience of computations, we consider M = 6, S = 3 and K = 4. The steady-state availability for the different repair time distributions with various system parameters are shown in Figs. 2–6. As Figs. 2 and 3 show, the steady-state availability decreases as the failure rate of primary unit k increases or the repair rate of failed unit l decreases for different repair distributions. As Fig. 4 shows, the coverage factor c rarely affects the steady-state system 1.015 Exp Gam(2.5) lognormal(0.5) lognormal(2) Weibull(0.5) Weibull(2)
1.01 1.005 1
Av
0.995 0.99 0.985 0.98 0.975 0.97 0
0.1
0.2
0.3
0.4
0.5 c
0.6
0.7
0.8
0.9
1
Fig. 4. Availability for various coverage factors and different repair times (k = 0.1, a = 0.01, l = 1 and b = 2.4).
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Fig. 5. Availability for k and l with exponential repair times (a = 0.01, c = 0.8 and b = 2.4).
1.01 1
Av
0.99 0.98 0.97 0.96 1 15 0.5
10
c
5 0
S
0
Fig. 6. Availability for different numbers of warm standby units and c with exponential repair times (k = 0.1, a = 0.01, l = 1 and b = 2.4).
Table 1 Sensitivity of steady-state system availability A(1) with respect to individual system parameters for different repair distributions (a = 0.01, l = 1, c = 0.9 and b = 2.4). k
Repair distribution
oA(1)/ok
oA(1)/ol
oA(1)/oc
0.1
Exponential Gamma (2.5) Lognormal (0.5) Lognormal (2) Weibull (0.5) Weibull (2)
0.5174 0.9063 0.7762 0.1267 0.0696 1.0136
0.0110 0.0187 0.0160 0.0028 0.0017 0.0208
0.0027 0.0044 0.0038 0.0007 0.0004 0.0049
0.3
Exponential Gamma (2.5) Lognormal (0.5) Lognormal (2) Weibull (0.5) Weibull (2)
1.5632 1.4978 1.5415 1.3939 0.7884 1.4710
0.2187 0.2400 0.2370 0.1404 0.0780 0.2424
0.0853 0.0670 0.0714 0.0789 0.0497 0.0628
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157
Table 2 Sensitivity of steady-state system availability A(1) with respect to individual system parameters for different repair distributions (k = 0.1, a = 0.01, c = 0.9 and b = 2.4).
l
Repair distribution
oA(1)/ok
oA(1)/ol
oA(1)/oc
0.2
Exponential Gamma (2.5) Lognormal (0.5) Lognormal (2) Weibull (0.5) Weibull (2)
2.3709 3.9124 3.4213 0.6197 0.3436 4.3036
0.2527 0.4035 0.3525 0.0678 0.0411 0.4425
0.0122 0.0190 0.0167 0.0034 0.0021 0.0206
1
Exponential Gamma (2.5) Lognormal (0.5) Lognormal (2) Weibull (0.5) Weibull (2)
0.5174 0.9063 0.7762 0.1267 0.0696 1.0136
0.0110 0.0187 0.0160 0.0028 0.0017 0.0208
0.0027 0.0044 0.0038 0.0007 0.0004 0.0049
Table 3 Sensitivity of steady-state system availability A(1) with respect to individual system parameters for different repair distributions (k = 0.1, a = 0.01, l = 1 and b = 2.4). c
Repair distribution
oA(1)/ok
oA(1)/ol
oA(1)/oc
0.7
Exponential Gamma (2.5) Lognormal (0.5) Lognormal (2) Weibull (0.5) Weibull (2)
0.4902 0.8617 0.7371 0.1196 0.0656 0.9646
0.0105 0.0179 0.0153 0.0026 0.0016 0.0200
0.0024 0.0040 0.0035 0.0006 0.0004 0.0044
0.9
Exponential Gamma (2.5) Lognormal (0.5) Lognormal (2) Weibull (0.5) Weibull (2)
0.5174 0.9063 0.7762 0.1267 0.0696 1.0136
0.0110 0.0187 0.0160 0.0028 0.0017 0.0208
0.0027 0.0044 0.0038 0.0007 0.0004 0.0049
availability for various repair distributions. For fixed system parameters, the Weibull repair times with smaller shape parameters always have greater steady-state system availability. Fig. 5 indicates similar tendencies to those shown in Figs. 2 and 3. Fig. 6 indicates that the number of warm standby units improves the steady-state system availability. A sensitivity analysis was conducted on the steady-state availability with respect to individual system parameters k, c and l by changing the specific value of the system parameter for the various repair time distributions. The numerical results of the sensitivity for different repair time distributions, as shown in Tables 1–3, indicate that the signs of sensitivity for parameters k and c are negative, which means that increasing these parameters will reduce the steady-state system availability. In contrast, l has a positive sign. Therefore, increasing this parameter will improve the steady-state system availability. Also, we find that k significantly affects the steady-state availability and that the effect of c is insensitive. 5. Conclusions In this paper, we investigated a repairable system with imperfect coverage and reboot delay. A step-by-step algorithm was constructed to compute the steady-state availability. We adopted two repair time distributions to illustrate the method. The numerical results indicated steady-state availability decreased as the failure rate of the primary unit or the number of warm standby units increased or the mean repair rate decreased. Moreover, the sensitivity analysis showed that the impact of the failure rate of the primary unit was of greater significance. References [1] S.V. Amari, A.F. Myers, A. Rauzy, K.S. Trivedi, Imperfect coverage models: status and trends, in: Handbook of Performability Engineering, Springer, Berlin, 2008, pp. 321–348. [2] J.-C. Ke, H.-I. Huang, C.-H. Lin, Parametric programming approach for a two-unit repairable system with imperfect coverage, reboot, and fuzzy parameters, IEEE Trans. Reliab. 57 (2008) 498–506. [3] J.C. Ke, S.L. Lee, Y.L. Hsu, Bayesian analysis for a redundant repairable system with imperfect coverage, Commun. Stat. – Simul. Comput. 37 (2008) 993– 1004. [4] J.C. Ke, S.L. Lee, M.Y. Ko, Two-unit redundant system with detection delay and imperfect coverage: confidence interval estimation, Qual. Technol. Quant. Manage. 8 (1) (2011) 1–14.
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