- Email: [email protected]
Reliability and sensitivity analysis of a repairable system with imperfect coverage under service pressure condition
Journal of Manufacturing Systems 32 (2013) 357–363
Contents lists available at SciVerse ScienceDirect
Journal of Manufacturing Systems journal homepage: www.elsevier.com/locate/jmansys
Technical paper
Reliability and sensitivity analysis of a repairable system with imperfect coverage under service pressure condition Kuo-Hsiung Wang a,∗ , Tseng-Chang Yen b , Jen-Ju Jian b a b
Department of Computer Science and Information Management, Providence University, Taichung 43301, Taiwan Department of Applied Mathematics, National Chung-Hsing University, Taichung 40227, Taiwan
a r t i c l e
i n f o
Article history: Received 30 May 2012 Received in revised form 10 September 2012 Accepted 30 January 2013 Available online 5 March 2013 Keywords: Imperfect coverage Mean time to system failure Reboot delay Service pressure condition Reliability Sensitivity analysis
a b s t r a c t This paper investigates reliability and sensitivity analysis of a repairable system with imperfect coverage under service pressure condition. Failure times and repair times of failed units are assumed to be exponentially distributed. As a unit fails, it may be immediately detected, located and replaced with a coverage probability c by a standby if one is available. When the repairmen are under the pressure of a long queue, the repairmen may increase the repair rate to reduce the queue length. We derive the explicit expressions for reliability function and mean time to system failure (MTTF). Various cases are analyzed to study the effects of different parameters on the system reliability and MTTF. We also accomplish sensitivity analysis and relative sensitivity analysis of the reliability characteristics with respect to system parameters. © 2013 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
1. Introduction Uncertainty is one of the important issues in management decisions. One of the most useful uncertainty measures is system reliability. The reliability of a system with standbys plays an important role in power plants, manufacturing systems, industrial systems and technical systems. Keeping a stable operating quality and a high level of reliability or availability is often a fundamental necessity. It maybe switches incompletely an existing spare module to a failed unit. When a failed unit is not detected, located and recovered, it needs time to be found and cleared. Therefore, we study the reliability of a system with multiple active units when covering a failed unit imperfectly. When the repairmen are under the pressure of a long queue, the repairmen may increase the repair rate to reduce the queue length. It may be impossible to switch in an existing spare module and then recover from a failure. Faults such as these are called to be not covered, and the probabilities of successful recovery on the failure of an active unit (or standby unit) is denoted by c. Quantity c which includes the probabilities of successful detection, location, and recovery from a failure is known as the coverage factor or coverage probability (see Trivedi [11]). A standby unit is called a ‘warm standby’ if its failure rate is nonzero and is less than the failure rate of an active unit. Active and warm
∗ Corresponding author. Fax: +886 4 26324045. E-mail address: [email protected] (K.-H. Wang).
standby units can be considered to be repairable. We continue with the assumption that the coverage factor is the same for active and standby unit failures. This paper differs from previous works in that: (i) the reliability problem with standby units has distinct characteristics which are different from the machine repair problem with standby units; (ii) it considers multiple imperfect coverage and reboot delay; and (iii) it considers the service pressure condition to prevent a long queue. The purpose of this article is to accomplish three objectives. The first objective is to develop the explicit expressions for reliability function, RY (t), and mean time to system failure, MTTF using Laplace transform techniques. The second objective is to perform sensitive analysis and relative sensitivity analysis of RY (t) and MTTF along with specified values of the system parameters. The third objective is to provide the numerical results to illustrate the sensitivity and the relative sensitivity of RY (t) and MTTF with respect to system parameters. Buzacott and Shanthikumar [2] reviewed queueing models that can be utilized to design manufacturing systems. The problem of machine failures and repairs in the context of queueing models has been investigated by several researchers. Govil and Fu [9] provided an excellent overview of the contributions of queueing models applied to manufacturing systems. In most papers, the queueing problems of the system discussed are more than the reliability problems of the system. Past work may be divided into two parts according to the system studied from the viewpoint of the queueing theory or from the viewpoint of the reliability. The major literature
0278-6125/$ – see front matter © 2013 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jmsy.2013.01.004
358
K.-H. Wang et al. / Journal of Manufacturing Systems 32 (2013) 357–363
we review is the viewpoint of the reliability system. Cao and Cheng [3] first introduced reliability concept into a queueing system with a repairable service station where the lifetime of the service station is exponentially distributed and its repair time has a general distribution. Wang and Sivazlian [16] presented the reliability characteristics of a system consisting of M operating machines, S warm standbys and R repairmen. They have established relations between system reliability and the number of spares, the number of repairmen, the failure rate and the repair rate. Meng [10] compared the mean time to system failure for four redundant series. He obtained a general ordering relationship between the MTTF of these four systems. Wang and Kuo [14] investigated the reliability and availability characteristics of four different series system configurations with mixed standby components. They have provided a systematic methodology to develop the MTTF and the steady-state availability of four configurations with mixed standby units. Galikowsky et al. [4] and Wang and Pearn [15] investigated the cost benefit analysis of series system with cold standby components and warm standby components, respectively. They developed the explicit expressions for the MTTF and the steady-state availability. Ke and Wang [6] extended Wang and Sivazlian’s [16] model by considering the balking and reneging in a repairable system. They provided the explicit expressions for the reliability characteristics of a repairable system with warm standby units plus balking and reneging. The concept of coverage factor and its effect on the reliability and availability model of a repairable system has been introduced by several authors such as Arnold [1], Trivedi [11], and Wang and Chiu [13]. The idea of imperfect coverage we discussed in this paper has been proposed by Trivedi [11]. Moreover, Wang and Chiu [13] analyzed the cost benefit analysis of availability systems with warm standby units and imperfect coverage. The concept of reboot delay and its effect on the reliability and availability model of a repairable system has been introduced by Trivedi [11]. Recently, Wang and Chen [12] investigated the reliability and availability analysis of a repairable system with standby switching failures. When the repairmen are under the pressure of a long queue, they may increase the repair rate to reduce the queue length. The concept of the service pressure coefficient was first introduced by Hiller and Lieberman [5]. Recently, Ke et al. [8] studied the reliability and sensitivity analysis of a system with multiple unreliable servers and standby switching failures. Ke et al. [7] first studied the reliability analysis of a system with standbys subjected to switching failure and presented a contour of the MTTF which is useful for the decision makers.
2. Description of the system and notations The repairable system includes M active units, S warm standby units and R reliable repairmen. The queue discipline of this repairable system is assumed to be FCFS (first-come first-serve). Each of the active units has an exponential time-to-failure distribution with parameter . Each of the warm standby units also has an exponential time-to-failure distribution with parameter ˛ (0 < ˛ < ). When a unit fails, it may be immediately detected, located and replaced with a coverage probability c by a standby if one is available. It is assumed that the replacing time is instantaneous. We define the unsafe failure state of the system as any one of the breakdowns is not covered. Active unit with failure in the unsafe failure state is refreshed by a reboot. Reboot delay takes place at rate ˇ for an active unit (or standby unit) which is exponentially distributed. The repair rate changes by the large number of failed units for waiting to repair. Therefore, we consider the service pressure coefficient a to improve the repair rate. Service pressure coefficient is a positive constant and indicates degree to which repair rate is affected by the number of failed units in the system. System failure is defined to be less than K active units, where K = 1, 2, . . ., M.
Therefore, if n denotes the number of failed units in the system, the system is failed if and only if n ≥ L = M + S − K + 1. In this paper, we consider two cases for system failure: Case 1, the system fails when all M + S units fail (i.e., K = 1); and Case 2, the system fails when the standby units are emptied (i.e., K = M). Notations M S R n s ˛ c ˇ a Pn∗ (s) ∗ Puf (s)
number of active units number of warm standby units number of repairmen number of failed units in the system Laplace transform variable failure rate of an active unit failure rate of a warm standby unit repair rate of a failed unit coverage probability reboot delay rate service pressure coefficient Laplace transform of Pn (t) Laplace transform of Pufn (t)
RY (t) MTTF
reliability function of the system mean time to system failure
n
3. Reliability characteristics At time t = 0, the system has just started operation with no failed units when the repairman is working. The reliability function under exponential failure time, exponential reboot time and exponential repair time can be developed through the birth and death process. Let n denote the number of failed units in the system. The mean failure rate n and the mean service rate n for this model are given by
n =
M + (S − n)˛,
0≤n≤S−1
(M + S − n),
S ≤ n ≤ M + S.
n =
1≤n≤R−1
n,
n(R + 1) a
R
R(n + 1)
(1)
,
(2)
R ≤ n ≤ M + S.
3.1. Differential-difference equations Let us define Pn (t) is the probability of exactly n failed units in the system at time t in the safe state and Pufn (t) is the probability of exactly n failed units in the system at time t in the unsafe failure state. From the state-transition-rate diagram shown in Fig. 2, we set up the differential difference equations as follows: dP0 (t) = −0 P0 (t) + 1 P1 (t) dt
(3)
dPn (t) = −(n + n )Pn (t) + n−1 cPn−1 (t) + ˇPufn (t) + n+1 Pn+1 (t), dt 1≤n≤L−2 dPufn (t) dt
= −ˇPufn (t) + n−1 (1 − c)Pn−1 (t),
(4)
1≤n≤L−1
(5)
dPL−1 (t) = −(L−1 + L−1 )PL−1 (t) + L−2 cPL−2 (t) + ˇPufL−1 (t), dt
(6)
dPL (t) = L−1 PL−1 (t). dt
(7)
Eqs. (3)–(7) can be written in matrix form as: ˙ P(t) = Q · P(t),
(8)
P(t) denotes the column vector where [P0 (t), Puf1 (t), P1 (t), . . . , PL−1 (t), PufL−1 (t), PL (t)]T . ˙ Note that the symbol T denotes the transpose. P(t) indicates the derivative of P(t) with respect to t and Q is the characteristic matrix
K.-H. Wang et al. / Journal of Manufacturing Systems 32 (2013) 357–363
359
of the system consisting of the system parameters , ˛, , c, ˇ and a.
we calculate the numerators det[Nn (s)] in (17). Then, substituting det[D(s)] and det[Nn (s)] into (17), we get
3.2. Laplace transform of Pn (t)
PL∗ (s) +
L−1
i
i=1
The Laplace transform of Pn (t) is defined as
∞
Pn∗ (s) =
e−st Pn (t) dt,
n = 0, 1, 2, . . . , L.
(9)
0
Recall that the Laplace transform formula is given by
∞
e−st
0
d Pn (t) dt = sPn ∗ (s) − Pn (0), dt
n = 0, 1, 2, . . . , L.
= P0 (0),
PL (t) +
1≤n≤L−2
1≤n≤L−1
∗ ∗ ∗ −L−2 cPL−2 (s) + (L−1 + L−1 + s)PL−1 (s) − ˇPuf
L−1
(13)
(s) = PL−1 (0), (14)
∗ −L−1 PL−1 (s) + sPL∗ (s) = PL (0).
(15)
Eqs. (11)–(15) can be written in matrix form as D(s)P ∗ (s) = P(0),
2
L−1
2
0]T .
At time t = 0, we let P(0) = [1, 0, 0, . . ., solve (16), we obtain the expressions for
L−1
Using Cramer’s rule to
D(−r) = A − rI, where A = D(0) is a (2L × 2L) matrix. We set det[D(−r)] to zero and find the corresponding distinct eigenvalues which may be real or complex. Suppose that there are i real distinct eigenvalues (excluding zero), say r1 , r2 , . . ., ri and j pairs of distinct conjugate complex eigenvalues, say (ri+1 , ri+1 ), (ri+2 , ri+2 ), . . ., (ri+j , ri+j ) where i and j satisfy i + 2j = 2L − 1. Next,
(18)
i
bl e
−rl t
+
j l=1
el − dl ul
dl e−ul t cos(vl t)
e−ul t sin(vl t) .
vl
(19)
Pufi (t)
= b0 .
It is noted if the above assumption is not valid, then the system may last as well as a repair is always effective. 3.3. Reliability function Let Y be the random variable and represent the time to failure of the system. Since PL (t) is the probability that the system has failed on or before time t. Thus the reliability function RY (t) is given by L−1
Pufi (t).
(20)
i=1
3.4. Mean time to system failure The mean time to system failure (MTTF) is always finite. From (20), we define the MTTF as follows:
∞
MTTF =
RY (t) dt = lim
s→0
= lim
0
= lim
s→0
∞
∞
RY (t)e−st dt
0
1 − PL (t) −
s→0
(17) where det[D(s)] denotes the determinant of matrix D(s) and det[Nn (s)] denotes the determinant obtained from replacing the nth column of D(s) by the initial vector P(0) = [1, 0, 0, . . ., 0]T . However, it is too complex to derive the explicit solutions Pn∗ (s) of (17). Therefore, we can use the computer software MAPLE to compute the solutions Pn∗ (s), where n = 0, uf1 , 1, uf2 , . . ., L − 2, ufL−1 , L − 1, L. We first consider the denominator det[D(s)] in (17). It is easy to see that s = 0 is a root of det[D(s)] = 0. Now let s = −r (r are unknown values), then we have
l=1
i=1
0
n = 0, uf1 , 1, uf2 , . . . , L − 2, ufL−1 , L − 1, L,
l=1
(16)
and the initial conditions P(0) is a column vector given by P ∗ (0) = ∗ (0), P ∗ (0), P ∗ (0), . . . , P ∗ (0), P ∗ ∗ (0), P ∗ (0)]T . (0), PL−1 [P0∗ (0), Puf L 1 L−2 uf uf
det[Nn (s)] = , det[D(s)]
PL (t) +
L−1
RY (t) = 1 − PL (t) −
where D(s) = sI − Q is a (2L × 2L) matrix, I is the identity matrix and P* (s) denotes a column vector given by P ∗ (s) = ∗ (s), P ∗ (s), P ∗ (s), . . . , P ∗ (s), P ∗ ∗ (s), P ∗ (s)]T , [P0∗ (s), Puf (s), PL−1 L 1 L−2 uf uf
j
Since, in this research, the system reliability is calculated from the assumption that the system fails eventually, we get
(12)
∗ ∗ (ˇ + s)Puf (s) − n−1 (1 − c)Pn−1 (s) = Pufn (0), n
i
l=1
+
lim
Pn∗ (s)
Pufi (t) = b0 +
i=0
t→∞
1
L−1
(11)
∗ ∗ ∗ (s)+(n + n + s)Pn∗ (s) − ˇPuf (s)−n+1 Pn+1 (s) = Pn (0), −n−1 cPn−1 n
1
dl s + el b0 bl + + , s s + rl s2 + (ri+j , ri+j )s + ri+j ri+j
where b0 , b1 , . . ., bi , d1 , d2 , . . ., dj and e1 , e2 , . . ., ej are unknown real numbers. Let ui and vi represent the real and imaginary parts of the complex eigenvalue ri+l , respectively. Using the inverse Laplace transform in (18) yields
(10)
Taking the Laplace transform on both sides of (3)–(7) and using (9) and (10), we can obtain (0 + s)P0∗ (s) − 1 P1∗ (s)
∗ Puf (s) =
L−1 i=1
Pufi (t) e
dt
1 ∗ − PL∗ (s) − Puf (s) . i s L−1
−st
(21)
i=1
4. Practical justification of the model A practical problem related to the computer system is presented for illustrative purposes. We consider a high-reliability Linux virtual cluster which provides the web service and is managed by the Linux Virtual Server (LVS) cluster management system (see Fig. 1). The LVS director receives requests from the Internet and forwards them to the real servers (treated as the active units). The high reliability of the virtual cluster can be provided by deploying several backup servers (treated as the standby units) in a virtual machine pool and a fault detecting server which running the generalpurpose monitoring software, mon, to monitor real servers and backup servers.
360
K.-H. Wang et al. / Journal of Manufacturing Systems 32 (2013) 357–363
Fig. 1. A high-availability Linux virtual cluster.
Initially both the real servers and backup servers are working. The real server fails independently of the state of the backup servers and vice versa. Let the time-to-failure of the real server and the time-to-failure of the backup servers be exponentially distributed with parameters and ˛ respectively. Service pressure coefficient a is a positive constant and indicates degree to which repair rate is affected by the number of failed units in the system. When a real server fails, it may be immediately detected by the fault detecting server, and replaced with a coverage probability c by the backup server if it is available. With probability 1 − c the
Fig. 3. System reliability for different failure rates . The system fails when all M + S units fail (K = 1).
parameters on RY (t). We choose M = 5, R = 2, K = 1 (the system fails when all M + S units fail) and consider the following seven cases.
Case
˛
c
ˇ
K
S
1 2 3 4 5 6 7
0.2, 0.3, 0.4, 0.5, 0.6 0.4 0.4 0.4 0.4 0.4 0.4
1.0 0.6, 0.8, 1.0, 1.2, 1.4 1.0 1.0 1.0 1.0 1.0
0.05 0.05 0, 0.2, 0.5, 0.8, 1.0 0.05 0.05 0.05 0.05
0.9 0.9 0.9 0.1, 0.3, 0.5, 0.7, 0.9 0.9 0.9 0.9
24.0 24.0 24.0 24.0 6.0, 12.0, 18.0, 24.0, 36.0 24.0 24.0
1 1 1 1 1 1, 2, 3, 4, 5 1
3 3 3 3 3 3 1, 2, 3, 4, 5
fault detecting server fails to cover the failure of the real server. The coverage factor is the same for the real server and backup server failures and is denoted by c. The Linux virtual cluster is in the unsafe failure state as any one of the breakdowns is not covered. A real server failure (or backup server failure) in the unsafe failure state can be cleared by a reboot. Reboot delay takes place at rate ˇ for a real server (or backup server) which is exponentially distributed. 5. Numerical results for RY (t) and MTTF In this section, we perform numerical experiments to investigate the effects of various system parameters on RY (t) and MTTF. We first analyze graphically to study the effects of various system
The effects of various system parameters on RY (t) are depicted in Figs. 3–9 for the above seven cases, respectively. It can be observed from Figs. 3 and 4 that the system reliability increases as decreases or increases. Recall that the reliability is a probability that the system is operational. The smaller failure rate of an active unit () or the larger repair rate of a failed unit () makes the system more reliable, which results in an increasing reliability. Fig. 5 shows that the system reliability increases as the service pressure coefficient a increases. Following the same reason as above, the larger service pressure coefficient enhances the repair rate with increasing reliability. It reveals from Figs. 6 and 7 that the coverage factor c and reboot delay ˇ rarely affect the system reliability. Thus, in order to understand this phenomenon, we conduct a sensitivity analysis for further details in the next section. We observe from Fig. 8 the
Fig. 2. State-transition-rate diagram for a repairable system with multiple imperfect coverage and service pressure condition.
K.-H. Wang et al. / Journal of Manufacturing Systems 32 (2013) 357–363
Fig. 4. System reliability for different service rates . The system fails when all M + S units fail (K = 1).
361
Fig. 6. System reliability for different cover probabilities c. The system fails when all M + S units fail (K = 1).
system reliability increases as K decreases. Specifically, the minimum number of operating units increases as K decreases, which is accounted for by the increase in the reliability. It appears from Fig. 9 that the system reliability is increased by adding the number of warm standby units. These results would be helpful for decision makers to manage the repairable system. Next, we provide the numerical results of MTTF to analyze the effects of various system parameters on MTTF shown in Tables 1 and 2. It should be noted from Table 1 that the impact of a becomes more significant for small values of . Moreover, an increment of 0.2 to a will cause an increment of at least 21% of the MTTF when = 0.2. Thus, it implies that the impact of service pressure coefficient should not be neglected for a repairable system when the active units have lower failure rate. We also observe from Table 1 that the MTTF can drastically decrease as increases. Table 2 shows that (i) an increment of 0.2 to a will cause an
Fig. 7. System reliability for different reboot delay rates ˇ. The system fails when all M + S units fail (K = 1).
increment of at least 15% of the MTTF for any value of c; and (ii) the MTTF rarely increases as the coverage factor c decreases. 6. Sensitivity analysis and relative sensitivity analysis In this section, we accomplish sensitivity analysis and relative sensitivity analysis on MTTF and RY (t) with respect to one of system parameters where = , ˛, , c, ˇ, a. We fix M = 5, S = 3, R = 2, = 0.4, Table 1 The MTTF for different values of and a (M = 5, S = 3, R = 2, K = 1, ˛ = 0.05, c = 0.9, = 1.0, ˇ = 24). a
0.2
Fig. 5. System reliability for different service pressure coefficients a. The system fails when all M + S units fail (K = 1).
0.0 0.2 0.4 0.6 0.8 1.0
4798.3 5821.6 7071.7 8597.7 10,462.5 12,742.7
0.3
0.4
0.5
0.6
430.7 511.0 607.9 725.0 866.7 1038.2
103.3 119.1 137.8 160.2 186.9 218.9
41.9 47.0 52.9 60.0 68.2 78.0
22.9 25.1 27.6 30.5 34.0 38.0
362
K.-H. Wang et al. / Journal of Manufacturing Systems 32 (2013) 357–363 Table 3 Sensitivity of MTTF with respect to different system parameters when K = 1 and K = 5 (M = 5, S = 3, R = 2, = 0.4, ˛ = 0.05, = 1.0, ˇ = 24, c = 0.9, a = 0.5).
K=1 K=5
˛
ˇ
c
a
−1760.15 −17.93
−89.80 −4.77
561.38 3.34
−0.06 −0.01
−10.64 −0.19
111.69 0.15
= 1.0, ˛ = 0.05, c = 0.9, ˇ = 24, a = 0.5 as a base case and consider two different types of system failure (i.e., K = 1 and K = 5). 6.1. Sensitivity analysis and relative sensitivity analysis for MTTF Differentiating (21) with respect to , we obtain =
∂MTTF = ∂
∞
0
∂RY (t) dt, ∂
(22)
where = , ˛, , c, ˇ, a. The relative sensitivity of MTTF is defined as the percentage change that resulting from the percentage change in one of system parameters , = Fig. 8. System reliability for different K values. The system fails when all M + S units fail (K = 1).
∂MTTF/MTTF = . MTTF ∂/
(23)
We first perform the sensitivity and the relative sensitivity of MTTF with respect to different system parameters , ˛, , a, c, ˇ, a, respectively. We will see how effect on the MTTF about system parameters. Numerical results of the sensitivity and the relative sensitivity of MTTF according to the base case (M = 5, S = 3, R = 2, = 0.4, ˛ = 0.05, = 1.0, ˇ = 24, c = 0.9, a = 0.5) when K = 1 and K = 5 are shown in Tables 3 and 4, respectively. The order of magnitude of the effect on MTTF about system parameters can be determined by the absolute value in Tables 3 and 4. Therefore, the order of magnitude of the sensitivity to the MTTF is > > a > ˛ > c > ˇ when K = 1, and > ˛ > > c > a > ˇ when K = 5. Moreover, the order of magnitude of the relative sensitivity to the MTTF is c > > > a > ˛ > ˇ when K = 1, and > > c > ˇ > ˛ > a when K = 5. 6.2. Sensitivity analysis and relative sensitivity analysis for RY (t) We perform the sensitivity analysis of changes in RY (t) with respect to one of system parameters where = , ˛, , c, ˇ, a. Differentiating (20) with respect to , we obtain ∂RY (t) ∂PL (t) ∂Pufi (t) =− − . ∂ ∂ ∂ L−1
(24)
i=1
Fig. 9. System reliability for different standby units S. The system fails when all M + S units fail (K = 1).
Table 2 The MTTF for different values of c and a (M = 5, S = 3, R = 2, K = 1, = 0.4, ˛ = 0.05, = 1.0, ˇ = 24). a
0.0 0.2 0.4 0.6 0.8 1.0
c 0.1
0.3
0.5
0.7
0.9
109.0 125.7 145.7 169.5 197.9 231.9
107.6 124.1 143.7 167.2 195.2 228.7
106.2 122.4 141.8 164.8 192.4 225.4
104.7 120.8 139.8 162.5 189.7 222.2
103.3 119.1 137.8 160.2 186.9 218.9
The relative sensitivity of RY (t) is defined as the percentage change that resulting from the percentage change in one of system parameters , =
∂RY (t)/RY (t) . ∂/
(25)
Numerical results are provided to illustrate the sensitivity of RY (t) with respect to system parameters. We again fix M = 5, S = 3, R = 2, = 0.4, = 1.0, ˛ = 0.05, c = 0.9, ˇ = 24, a = 0.5 as a base case. Figs. 10 and 11 show that the sensitivity analysis of RY (t) for the base case with respect to one of system parameters = , ˛, , c, ˇ, a for K = 1 and K = 5, respectively. We observe from Fig. 10 that Table 4 Relative sensitivity of MTTF with respect to different system parameters when K = 1 and K = 5 (M = 5, S = 3, R = 2, = 0.4, ˛ = 0.05, = 1.0, ˇ = 24, c = 0.9, a = 0.5).
K=1 K=5
˛
ˇ
c
a
−4.74016 −7.17009
−0.03023 −0.05654
3.77956 3.33783
−0.00913 −0.14377
−9.57743 −0.16892
0.37599 0.01747
K.-H. Wang et al. / Journal of Manufacturing Systems 32 (2013) 357–363
363
of , ˛, ˇ and c are negative; and (iii) the order of magnitude of the effect for K = 5 is > ˛ > > a ≈ ˇ ≈ c (i.e., a, ˇ and c are the least). 7. Conclusions In this paper, we investigate a repairable system with warm standby units and multiple imperfect coverage under the service pressure condition. We provide the explicit expressions for RY (t) and MTTF. The numerical results show that the impact of service pressure coefficient a should not be ignored for a repairable system when the active units have lower failure rate. Moreover, the sensitivity analysis and the relative sensitivity analysis indicate that the impact of service pressure coefficient a to RY (t) and MTTF is more significant than of c and ˇ. Overall, the order of the sensitivity to RY (t) is > > a > ˛ > ˇ ≈ c (K = 1). Moreover, the order of the sensitivity to MTTF is > > a > ˛ > c > ˇ (K = 1), and the order of the relative sensitivity to MTTF is c > > > a > ˛ > ˇ (K = 1). Fig. 10. Sensitivity of reliability with respect to different parameters when K = 1.
Fig. 11. Sensitivity of reliability with respect to different parameters when K = 5.
(i) the sensitivity of parameters and a have positive signs, which mean slightly increments of parameter’s value will improve RY (t); (ii) the sensitivity of parameters , ˛, ˇ and c have negative signs, which mean slightly increments of parameter’s value will abate RY (t); and (iii) the parameters ˇ and c rarely affect RY (t). From Fig. 10, we also find that the order of magnitude of the effect for K = 1 is > > a > ˛ > ˇ ≈ c (i.e., ˇ and c are the least). We see from Fig. 11 that (i) the impacts of and a are positive; (ii) the impacts
References [1] Arnold TF. The concept of coverage and its effect on the reliability model of a repairable system. IEEE Transactions on Computers 1973;C-22:251–4. [2] Buzacott JA, Shanthikumar JG. Design of manufacturing systems using queueing models. Queueing Systems 1992;12:135–213. [3] Cao J, Cheng K. Analysis of M/G/1 queueing system with repairable service station. Acta Mathematicae Applicate Sinica 1982;5:113–27. [4] Galikowsky C, Sivazlian BD, Chaovalitwongse P. Optimal redundancies for reliability and availability of series systems. Microelectronics and Reliability 1996;36:1537–46. [5] Hiller FS, Lieberman GJ. Introduction to operations research. New York, NY: McGraw-Hill; 2001. [6] Ke JC, Wang KH. The reliability analysis of balking and reneging in a repairable system with warm standbys. Quality and Reliability Engineering International 2002;18:467–78. [7] Ke JB, Lee WC, Ke JC. Reliability-based measure for a system with standbys subjected to switching failures. Engineering Computations: International Journal for Computer-Aided Engineering and Software 2008;25:694–706. [8] Ke JB, Lee WC, Wang KH. Reliability and sensitivity analysis of a system with multiple unreliable service stations and standby switching failures. Physica A: Statistical Mechanics and its Applications 2007;380:455–69. [9] Govil MG, Fu MC. Queueing theory in manufacturing: a survey. Journal of Manufacturing Systems 1999;18(3):214–40. [10] Meng FC. On comparison of MTBF between four redundant systems. Microelectronics and Reliability 1993;33:1987–90. [11] Trivedi KS. Probability and statistics with reliability queueing and computer science applications. 2nd ed. New York, NY: John Wiley and Sons; 2002. [12] Wang KH, Chen YJ. Comparative analysis of availability between three systems with general repair times, reboot delay and switching failures. Applied Mathematics and Computation 2009;215:384–94. [13] Wang KH, Chiu LW. Cost benefit analysis of availability systems with warm standby units and imperfect coverage. Applied Mathematics and Computation 2006;172:1239–56. [14] Wang KH, Kuo CC. Cost and probabilistic analysis of series systems with mixed standby components. Applied Mathematical Modelling 2000;24:957–67. [15] Wang KH, Pearn WL. Cost benefit analysis of series systems with warm standby components. Mathematical Methods of Operations Research 2003;58:247–58. [16] Wang KH, Sivazlian BD. Reliability of a system with warm standbys and repairmen. Microelectronics and Reliability 1989;29:849–60.
Contents lists available at SciVerse ScienceDirect
Journal of Manufacturing Systems journal homepage: www.elsevier.com/locate/jmansys
Technical paper
Reliability and sensitivity analysis of a repairable system with imperfect coverage under service pressure condition Kuo-Hsiung Wang a,∗ , Tseng-Chang Yen b , Jen-Ju Jian b a b
Department of Computer Science and Information Management, Providence University, Taichung 43301, Taiwan Department of Applied Mathematics, National Chung-Hsing University, Taichung 40227, Taiwan
a r t i c l e
i n f o
Article history: Received 30 May 2012 Received in revised form 10 September 2012 Accepted 30 January 2013 Available online 5 March 2013 Keywords: Imperfect coverage Mean time to system failure Reboot delay Service pressure condition Reliability Sensitivity analysis
a b s t r a c t This paper investigates reliability and sensitivity analysis of a repairable system with imperfect coverage under service pressure condition. Failure times and repair times of failed units are assumed to be exponentially distributed. As a unit fails, it may be immediately detected, located and replaced with a coverage probability c by a standby if one is available. When the repairmen are under the pressure of a long queue, the repairmen may increase the repair rate to reduce the queue length. We derive the explicit expressions for reliability function and mean time to system failure (MTTF). Various cases are analyzed to study the effects of different parameters on the system reliability and MTTF. We also accomplish sensitivity analysis and relative sensitivity analysis of the reliability characteristics with respect to system parameters. © 2013 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
1. Introduction Uncertainty is one of the important issues in management decisions. One of the most useful uncertainty measures is system reliability. The reliability of a system with standbys plays an important role in power plants, manufacturing systems, industrial systems and technical systems. Keeping a stable operating quality and a high level of reliability or availability is often a fundamental necessity. It maybe switches incompletely an existing spare module to a failed unit. When a failed unit is not detected, located and recovered, it needs time to be found and cleared. Therefore, we study the reliability of a system with multiple active units when covering a failed unit imperfectly. When the repairmen are under the pressure of a long queue, the repairmen may increase the repair rate to reduce the queue length. It may be impossible to switch in an existing spare module and then recover from a failure. Faults such as these are called to be not covered, and the probabilities of successful recovery on the failure of an active unit (or standby unit) is denoted by c. Quantity c which includes the probabilities of successful detection, location, and recovery from a failure is known as the coverage factor or coverage probability (see Trivedi [11]). A standby unit is called a ‘warm standby’ if its failure rate is nonzero and is less than the failure rate of an active unit. Active and warm
∗ Corresponding author. Fax: +886 4 26324045. E-mail address: [email protected] (K.-H. Wang).
standby units can be considered to be repairable. We continue with the assumption that the coverage factor is the same for active and standby unit failures. This paper differs from previous works in that: (i) the reliability problem with standby units has distinct characteristics which are different from the machine repair problem with standby units; (ii) it considers multiple imperfect coverage and reboot delay; and (iii) it considers the service pressure condition to prevent a long queue. The purpose of this article is to accomplish three objectives. The first objective is to develop the explicit expressions for reliability function, RY (t), and mean time to system failure, MTTF using Laplace transform techniques. The second objective is to perform sensitive analysis and relative sensitivity analysis of RY (t) and MTTF along with specified values of the system parameters. The third objective is to provide the numerical results to illustrate the sensitivity and the relative sensitivity of RY (t) and MTTF with respect to system parameters. Buzacott and Shanthikumar [2] reviewed queueing models that can be utilized to design manufacturing systems. The problem of machine failures and repairs in the context of queueing models has been investigated by several researchers. Govil and Fu [9] provided an excellent overview of the contributions of queueing models applied to manufacturing systems. In most papers, the queueing problems of the system discussed are more than the reliability problems of the system. Past work may be divided into two parts according to the system studied from the viewpoint of the queueing theory or from the viewpoint of the reliability. The major literature
0278-6125/$ – see front matter © 2013 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jmsy.2013.01.004
358
K.-H. Wang et al. / Journal of Manufacturing Systems 32 (2013) 357–363
we review is the viewpoint of the reliability system. Cao and Cheng [3] first introduced reliability concept into a queueing system with a repairable service station where the lifetime of the service station is exponentially distributed and its repair time has a general distribution. Wang and Sivazlian [16] presented the reliability characteristics of a system consisting of M operating machines, S warm standbys and R repairmen. They have established relations between system reliability and the number of spares, the number of repairmen, the failure rate and the repair rate. Meng [10] compared the mean time to system failure for four redundant series. He obtained a general ordering relationship between the MTTF of these four systems. Wang and Kuo [14] investigated the reliability and availability characteristics of four different series system configurations with mixed standby components. They have provided a systematic methodology to develop the MTTF and the steady-state availability of four configurations with mixed standby units. Galikowsky et al. [4] and Wang and Pearn [15] investigated the cost benefit analysis of series system with cold standby components and warm standby components, respectively. They developed the explicit expressions for the MTTF and the steady-state availability. Ke and Wang [6] extended Wang and Sivazlian’s [16] model by considering the balking and reneging in a repairable system. They provided the explicit expressions for the reliability characteristics of a repairable system with warm standby units plus balking and reneging. The concept of coverage factor and its effect on the reliability and availability model of a repairable system has been introduced by several authors such as Arnold [1], Trivedi [11], and Wang and Chiu [13]. The idea of imperfect coverage we discussed in this paper has been proposed by Trivedi [11]. Moreover, Wang and Chiu [13] analyzed the cost benefit analysis of availability systems with warm standby units and imperfect coverage. The concept of reboot delay and its effect on the reliability and availability model of a repairable system has been introduced by Trivedi [11]. Recently, Wang and Chen [12] investigated the reliability and availability analysis of a repairable system with standby switching failures. When the repairmen are under the pressure of a long queue, they may increase the repair rate to reduce the queue length. The concept of the service pressure coefficient was first introduced by Hiller and Lieberman [5]. Recently, Ke et al. [8] studied the reliability and sensitivity analysis of a system with multiple unreliable servers and standby switching failures. Ke et al. [7] first studied the reliability analysis of a system with standbys subjected to switching failure and presented a contour of the MTTF which is useful for the decision makers.
2. Description of the system and notations The repairable system includes M active units, S warm standby units and R reliable repairmen. The queue discipline of this repairable system is assumed to be FCFS (first-come first-serve). Each of the active units has an exponential time-to-failure distribution with parameter . Each of the warm standby units also has an exponential time-to-failure distribution with parameter ˛ (0 < ˛ < ). When a unit fails, it may be immediately detected, located and replaced with a coverage probability c by a standby if one is available. It is assumed that the replacing time is instantaneous. We define the unsafe failure state of the system as any one of the breakdowns is not covered. Active unit with failure in the unsafe failure state is refreshed by a reboot. Reboot delay takes place at rate ˇ for an active unit (or standby unit) which is exponentially distributed. The repair rate changes by the large number of failed units for waiting to repair. Therefore, we consider the service pressure coefficient a to improve the repair rate. Service pressure coefficient is a positive constant and indicates degree to which repair rate is affected by the number of failed units in the system. System failure is defined to be less than K active units, where K = 1, 2, . . ., M.
Therefore, if n denotes the number of failed units in the system, the system is failed if and only if n ≥ L = M + S − K + 1. In this paper, we consider two cases for system failure: Case 1, the system fails when all M + S units fail (i.e., K = 1); and Case 2, the system fails when the standby units are emptied (i.e., K = M). Notations M S R n s ˛ c ˇ a Pn∗ (s) ∗ Puf (s)
number of active units number of warm standby units number of repairmen number of failed units in the system Laplace transform variable failure rate of an active unit failure rate of a warm standby unit repair rate of a failed unit coverage probability reboot delay rate service pressure coefficient Laplace transform of Pn (t) Laplace transform of Pufn (t)
RY (t) MTTF
reliability function of the system mean time to system failure
n
3. Reliability characteristics At time t = 0, the system has just started operation with no failed units when the repairman is working. The reliability function under exponential failure time, exponential reboot time and exponential repair time can be developed through the birth and death process. Let n denote the number of failed units in the system. The mean failure rate n and the mean service rate n for this model are given by
n =
M + (S − n)˛,
0≤n≤S−1
(M + S − n),
S ≤ n ≤ M + S.
n =
1≤n≤R−1
n,
n(R + 1) a
R
R(n + 1)
(1)
,
(2)
R ≤ n ≤ M + S.
3.1. Differential-difference equations Let us define Pn (t) is the probability of exactly n failed units in the system at time t in the safe state and Pufn (t) is the probability of exactly n failed units in the system at time t in the unsafe failure state. From the state-transition-rate diagram shown in Fig. 2, we set up the differential difference equations as follows: dP0 (t) = −0 P0 (t) + 1 P1 (t) dt
(3)
dPn (t) = −(n + n )Pn (t) + n−1 cPn−1 (t) + ˇPufn (t) + n+1 Pn+1 (t), dt 1≤n≤L−2 dPufn (t) dt
= −ˇPufn (t) + n−1 (1 − c)Pn−1 (t),
(4)
1≤n≤L−1
(5)
dPL−1 (t) = −(L−1 + L−1 )PL−1 (t) + L−2 cPL−2 (t) + ˇPufL−1 (t), dt
(6)
dPL (t) = L−1 PL−1 (t). dt
(7)
Eqs. (3)–(7) can be written in matrix form as: ˙ P(t) = Q · P(t),
(8)
P(t) denotes the column vector where [P0 (t), Puf1 (t), P1 (t), . . . , PL−1 (t), PufL−1 (t), PL (t)]T . ˙ Note that the symbol T denotes the transpose. P(t) indicates the derivative of P(t) with respect to t and Q is the characteristic matrix
K.-H. Wang et al. / Journal of Manufacturing Systems 32 (2013) 357–363
359
of the system consisting of the system parameters , ˛, , c, ˇ and a.
we calculate the numerators det[Nn (s)] in (17). Then, substituting det[D(s)] and det[Nn (s)] into (17), we get
3.2. Laplace transform of Pn (t)
PL∗ (s) +
L−1
i
i=1
The Laplace transform of Pn (t) is defined as
∞
Pn∗ (s) =
e−st Pn (t) dt,
n = 0, 1, 2, . . . , L.
(9)
0
Recall that the Laplace transform formula is given by
∞
e−st
0
d Pn (t) dt = sPn ∗ (s) − Pn (0), dt
n = 0, 1, 2, . . . , L.
= P0 (0),
PL (t) +
1≤n≤L−2
1≤n≤L−1
∗ ∗ ∗ −L−2 cPL−2 (s) + (L−1 + L−1 + s)PL−1 (s) − ˇPuf
L−1
(13)
(s) = PL−1 (0), (14)
∗ −L−1 PL−1 (s) + sPL∗ (s) = PL (0).
(15)
Eqs. (11)–(15) can be written in matrix form as D(s)P ∗ (s) = P(0),
2
L−1
2
0]T .
At time t = 0, we let P(0) = [1, 0, 0, . . ., solve (16), we obtain the expressions for
L−1
Using Cramer’s rule to
D(−r) = A − rI, where A = D(0) is a (2L × 2L) matrix. We set det[D(−r)] to zero and find the corresponding distinct eigenvalues which may be real or complex. Suppose that there are i real distinct eigenvalues (excluding zero), say r1 , r2 , . . ., ri and j pairs of distinct conjugate complex eigenvalues, say (ri+1 , ri+1 ), (ri+2 , ri+2 ), . . ., (ri+j , ri+j ) where i and j satisfy i + 2j = 2L − 1. Next,
(18)
i
bl e
−rl t
+
j l=1
el − dl ul
dl e−ul t cos(vl t)
e−ul t sin(vl t) .
vl
(19)
Pufi (t)
= b0 .
It is noted if the above assumption is not valid, then the system may last as well as a repair is always effective. 3.3. Reliability function Let Y be the random variable and represent the time to failure of the system. Since PL (t) is the probability that the system has failed on or before time t. Thus the reliability function RY (t) is given by L−1
Pufi (t).
(20)
i=1
3.4. Mean time to system failure The mean time to system failure (MTTF) is always finite. From (20), we define the MTTF as follows:
∞
MTTF =
RY (t) dt = lim
s→0
= lim
0
= lim
s→0
∞
∞
RY (t)e−st dt
0
1 − PL (t) −
s→0
(17) where det[D(s)] denotes the determinant of matrix D(s) and det[Nn (s)] denotes the determinant obtained from replacing the nth column of D(s) by the initial vector P(0) = [1, 0, 0, . . ., 0]T . However, it is too complex to derive the explicit solutions Pn∗ (s) of (17). Therefore, we can use the computer software MAPLE to compute the solutions Pn∗ (s), where n = 0, uf1 , 1, uf2 , . . ., L − 2, ufL−1 , L − 1, L. We first consider the denominator det[D(s)] in (17). It is easy to see that s = 0 is a root of det[D(s)] = 0. Now let s = −r (r are unknown values), then we have
l=1
i=1
0
n = 0, uf1 , 1, uf2 , . . . , L − 2, ufL−1 , L − 1, L,
l=1
(16)
and the initial conditions P(0) is a column vector given by P ∗ (0) = ∗ (0), P ∗ (0), P ∗ (0), . . . , P ∗ (0), P ∗ ∗ (0), P ∗ (0)]T . (0), PL−1 [P0∗ (0), Puf L 1 L−2 uf uf
det[Nn (s)] = , det[D(s)]
PL (t) +
L−1
RY (t) = 1 − PL (t) −
where D(s) = sI − Q is a (2L × 2L) matrix, I is the identity matrix and P* (s) denotes a column vector given by P ∗ (s) = ∗ (s), P ∗ (s), P ∗ (s), . . . , P ∗ (s), P ∗ ∗ (s), P ∗ (s)]T , [P0∗ (s), Puf (s), PL−1 L 1 L−2 uf uf
j
Since, in this research, the system reliability is calculated from the assumption that the system fails eventually, we get
(12)
∗ ∗ (ˇ + s)Puf (s) − n−1 (1 − c)Pn−1 (s) = Pufn (0), n
i
l=1
+
lim
Pn∗ (s)
Pufi (t) = b0 +
i=0
t→∞
1
L−1
(11)
∗ ∗ ∗ (s)+(n + n + s)Pn∗ (s) − ˇPuf (s)−n+1 Pn+1 (s) = Pn (0), −n−1 cPn−1 n
1
dl s + el b0 bl + + , s s + rl s2 + (ri+j , ri+j )s + ri+j ri+j
where b0 , b1 , . . ., bi , d1 , d2 , . . ., dj and e1 , e2 , . . ., ej are unknown real numbers. Let ui and vi represent the real and imaginary parts of the complex eigenvalue ri+l , respectively. Using the inverse Laplace transform in (18) yields
(10)
Taking the Laplace transform on both sides of (3)–(7) and using (9) and (10), we can obtain (0 + s)P0∗ (s) − 1 P1∗ (s)
∗ Puf (s) =
L−1 i=1
Pufi (t) e
dt
1 ∗ − PL∗ (s) − Puf (s) . i s L−1
−st
(21)
i=1
4. Practical justification of the model A practical problem related to the computer system is presented for illustrative purposes. We consider a high-reliability Linux virtual cluster which provides the web service and is managed by the Linux Virtual Server (LVS) cluster management system (see Fig. 1). The LVS director receives requests from the Internet and forwards them to the real servers (treated as the active units). The high reliability of the virtual cluster can be provided by deploying several backup servers (treated as the standby units) in a virtual machine pool and a fault detecting server which running the generalpurpose monitoring software, mon, to monitor real servers and backup servers.
360
K.-H. Wang et al. / Journal of Manufacturing Systems 32 (2013) 357–363
Fig. 1. A high-availability Linux virtual cluster.
Initially both the real servers and backup servers are working. The real server fails independently of the state of the backup servers and vice versa. Let the time-to-failure of the real server and the time-to-failure of the backup servers be exponentially distributed with parameters and ˛ respectively. Service pressure coefficient a is a positive constant and indicates degree to which repair rate is affected by the number of failed units in the system. When a real server fails, it may be immediately detected by the fault detecting server, and replaced with a coverage probability c by the backup server if it is available. With probability 1 − c the
Fig. 3. System reliability for different failure rates . The system fails when all M + S units fail (K = 1).
parameters on RY (t). We choose M = 5, R = 2, K = 1 (the system fails when all M + S units fail) and consider the following seven cases.
Case
˛
c
ˇ
K
S
1 2 3 4 5 6 7
0.2, 0.3, 0.4, 0.5, 0.6 0.4 0.4 0.4 0.4 0.4 0.4
1.0 0.6, 0.8, 1.0, 1.2, 1.4 1.0 1.0 1.0 1.0 1.0
0.05 0.05 0, 0.2, 0.5, 0.8, 1.0 0.05 0.05 0.05 0.05
0.9 0.9 0.9 0.1, 0.3, 0.5, 0.7, 0.9 0.9 0.9 0.9
24.0 24.0 24.0 24.0 6.0, 12.0, 18.0, 24.0, 36.0 24.0 24.0
1 1 1 1 1 1, 2, 3, 4, 5 1
3 3 3 3 3 3 1, 2, 3, 4, 5
fault detecting server fails to cover the failure of the real server. The coverage factor is the same for the real server and backup server failures and is denoted by c. The Linux virtual cluster is in the unsafe failure state as any one of the breakdowns is not covered. A real server failure (or backup server failure) in the unsafe failure state can be cleared by a reboot. Reboot delay takes place at rate ˇ for a real server (or backup server) which is exponentially distributed. 5. Numerical results for RY (t) and MTTF In this section, we perform numerical experiments to investigate the effects of various system parameters on RY (t) and MTTF. We first analyze graphically to study the effects of various system
The effects of various system parameters on RY (t) are depicted in Figs. 3–9 for the above seven cases, respectively. It can be observed from Figs. 3 and 4 that the system reliability increases as decreases or increases. Recall that the reliability is a probability that the system is operational. The smaller failure rate of an active unit () or the larger repair rate of a failed unit () makes the system more reliable, which results in an increasing reliability. Fig. 5 shows that the system reliability increases as the service pressure coefficient a increases. Following the same reason as above, the larger service pressure coefficient enhances the repair rate with increasing reliability. It reveals from Figs. 6 and 7 that the coverage factor c and reboot delay ˇ rarely affect the system reliability. Thus, in order to understand this phenomenon, we conduct a sensitivity analysis for further details in the next section. We observe from Fig. 8 the
Fig. 2. State-transition-rate diagram for a repairable system with multiple imperfect coverage and service pressure condition.
K.-H. Wang et al. / Journal of Manufacturing Systems 32 (2013) 357–363
Fig. 4. System reliability for different service rates . The system fails when all M + S units fail (K = 1).
361
Fig. 6. System reliability for different cover probabilities c. The system fails when all M + S units fail (K = 1).
system reliability increases as K decreases. Specifically, the minimum number of operating units increases as K decreases, which is accounted for by the increase in the reliability. It appears from Fig. 9 that the system reliability is increased by adding the number of warm standby units. These results would be helpful for decision makers to manage the repairable system. Next, we provide the numerical results of MTTF to analyze the effects of various system parameters on MTTF shown in Tables 1 and 2. It should be noted from Table 1 that the impact of a becomes more significant for small values of . Moreover, an increment of 0.2 to a will cause an increment of at least 21% of the MTTF when = 0.2. Thus, it implies that the impact of service pressure coefficient should not be neglected for a repairable system when the active units have lower failure rate. We also observe from Table 1 that the MTTF can drastically decrease as increases. Table 2 shows that (i) an increment of 0.2 to a will cause an
Fig. 7. System reliability for different reboot delay rates ˇ. The system fails when all M + S units fail (K = 1).
increment of at least 15% of the MTTF for any value of c; and (ii) the MTTF rarely increases as the coverage factor c decreases. 6. Sensitivity analysis and relative sensitivity analysis In this section, we accomplish sensitivity analysis and relative sensitivity analysis on MTTF and RY (t) with respect to one of system parameters where = , ˛, , c, ˇ, a. We fix M = 5, S = 3, R = 2, = 0.4, Table 1 The MTTF for different values of and a (M = 5, S = 3, R = 2, K = 1, ˛ = 0.05, c = 0.9, = 1.0, ˇ = 24). a
0.2
Fig. 5. System reliability for different service pressure coefficients a. The system fails when all M + S units fail (K = 1).
0.0 0.2 0.4 0.6 0.8 1.0
4798.3 5821.6 7071.7 8597.7 10,462.5 12,742.7
0.3
0.4
0.5
0.6
430.7 511.0 607.9 725.0 866.7 1038.2
103.3 119.1 137.8 160.2 186.9 218.9
41.9 47.0 52.9 60.0 68.2 78.0
22.9 25.1 27.6 30.5 34.0 38.0
362
K.-H. Wang et al. / Journal of Manufacturing Systems 32 (2013) 357–363 Table 3 Sensitivity of MTTF with respect to different system parameters when K = 1 and K = 5 (M = 5, S = 3, R = 2, = 0.4, ˛ = 0.05, = 1.0, ˇ = 24, c = 0.9, a = 0.5).
K=1 K=5
˛
ˇ
c
a
−1760.15 −17.93
−89.80 −4.77
561.38 3.34
−0.06 −0.01
−10.64 −0.19
111.69 0.15
= 1.0, ˛ = 0.05, c = 0.9, ˇ = 24, a = 0.5 as a base case and consider two different types of system failure (i.e., K = 1 and K = 5). 6.1. Sensitivity analysis and relative sensitivity analysis for MTTF Differentiating (21) with respect to , we obtain =
∂MTTF = ∂
∞
0
∂RY (t) dt, ∂
(22)
where = , ˛, , c, ˇ, a. The relative sensitivity of MTTF is defined as the percentage change that resulting from the percentage change in one of system parameters , = Fig. 8. System reliability for different K values. The system fails when all M + S units fail (K = 1).
∂MTTF/MTTF = . MTTF ∂/
(23)
We first perform the sensitivity and the relative sensitivity of MTTF with respect to different system parameters , ˛, , a, c, ˇ, a, respectively. We will see how effect on the MTTF about system parameters. Numerical results of the sensitivity and the relative sensitivity of MTTF according to the base case (M = 5, S = 3, R = 2, = 0.4, ˛ = 0.05, = 1.0, ˇ = 24, c = 0.9, a = 0.5) when K = 1 and K = 5 are shown in Tables 3 and 4, respectively. The order of magnitude of the effect on MTTF about system parameters can be determined by the absolute value in Tables 3 and 4. Therefore, the order of magnitude of the sensitivity to the MTTF is > > a > ˛ > c > ˇ when K = 1, and > ˛ > > c > a > ˇ when K = 5. Moreover, the order of magnitude of the relative sensitivity to the MTTF is c > > > a > ˛ > ˇ when K = 1, and > > c > ˇ > ˛ > a when K = 5. 6.2. Sensitivity analysis and relative sensitivity analysis for RY (t) We perform the sensitivity analysis of changes in RY (t) with respect to one of system parameters where = , ˛, , c, ˇ, a. Differentiating (20) with respect to , we obtain ∂RY (t) ∂PL (t) ∂Pufi (t) =− − . ∂ ∂ ∂ L−1
(24)
i=1
Fig. 9. System reliability for different standby units S. The system fails when all M + S units fail (K = 1).
Table 2 The MTTF for different values of c and a (M = 5, S = 3, R = 2, K = 1, = 0.4, ˛ = 0.05, = 1.0, ˇ = 24). a
0.0 0.2 0.4 0.6 0.8 1.0
c 0.1
0.3
0.5
0.7
0.9
109.0 125.7 145.7 169.5 197.9 231.9
107.6 124.1 143.7 167.2 195.2 228.7
106.2 122.4 141.8 164.8 192.4 225.4
104.7 120.8 139.8 162.5 189.7 222.2
103.3 119.1 137.8 160.2 186.9 218.9
The relative sensitivity of RY (t) is defined as the percentage change that resulting from the percentage change in one of system parameters , =
∂RY (t)/RY (t) . ∂/
(25)
Numerical results are provided to illustrate the sensitivity of RY (t) with respect to system parameters. We again fix M = 5, S = 3, R = 2, = 0.4, = 1.0, ˛ = 0.05, c = 0.9, ˇ = 24, a = 0.5 as a base case. Figs. 10 and 11 show that the sensitivity analysis of RY (t) for the base case with respect to one of system parameters = , ˛, , c, ˇ, a for K = 1 and K = 5, respectively. We observe from Fig. 10 that Table 4 Relative sensitivity of MTTF with respect to different system parameters when K = 1 and K = 5 (M = 5, S = 3, R = 2, = 0.4, ˛ = 0.05, = 1.0, ˇ = 24, c = 0.9, a = 0.5).
K=1 K=5
˛
ˇ
c
a
−4.74016 −7.17009
−0.03023 −0.05654
3.77956 3.33783
−0.00913 −0.14377
−9.57743 −0.16892
0.37599 0.01747
K.-H. Wang et al. / Journal of Manufacturing Systems 32 (2013) 357–363
363
of , ˛, ˇ and c are negative; and (iii) the order of magnitude of the effect for K = 5 is > ˛ > > a ≈ ˇ ≈ c (i.e., a, ˇ and c are the least). 7. Conclusions In this paper, we investigate a repairable system with warm standby units and multiple imperfect coverage under the service pressure condition. We provide the explicit expressions for RY (t) and MTTF. The numerical results show that the impact of service pressure coefficient a should not be ignored for a repairable system when the active units have lower failure rate. Moreover, the sensitivity analysis and the relative sensitivity analysis indicate that the impact of service pressure coefficient a to RY (t) and MTTF is more significant than of c and ˇ. Overall, the order of the sensitivity to RY (t) is > > a > ˛ > ˇ ≈ c (K = 1). Moreover, the order of the sensitivity to MTTF is > > a > ˛ > c > ˇ (K = 1), and the order of the relative sensitivity to MTTF is c > > > a > ˛ > ˇ (K = 1). Fig. 10. Sensitivity of reliability with respect to different parameters when K = 1.
Fig. 11. Sensitivity of reliability with respect to different parameters when K = 5.
(i) the sensitivity of parameters and a have positive signs, which mean slightly increments of parameter’s value will improve RY (t); (ii) the sensitivity of parameters , ˛, ˇ and c have negative signs, which mean slightly increments of parameter’s value will abate RY (t); and (iii) the parameters ˇ and c rarely affect RY (t). From Fig. 10, we also find that the order of magnitude of the effect for K = 1 is > > a > ˛ > ˇ ≈ c (i.e., ˇ and c are the least). We see from Fig. 11 that (i) the impacts of and a are positive; (ii) the impacts
References [1] Arnold TF. The concept of coverage and its effect on the reliability model of a repairable system. IEEE Transactions on Computers 1973;C-22:251–4. [2] Buzacott JA, Shanthikumar JG. Design of manufacturing systems using queueing models. Queueing Systems 1992;12:135–213. [3] Cao J, Cheng K. Analysis of M/G/1 queueing system with repairable service station. Acta Mathematicae Applicate Sinica 1982;5:113–27. [4] Galikowsky C, Sivazlian BD, Chaovalitwongse P. Optimal redundancies for reliability and availability of series systems. Microelectronics and Reliability 1996;36:1537–46. [5] Hiller FS, Lieberman GJ. Introduction to operations research. New York, NY: McGraw-Hill; 2001. [6] Ke JC, Wang KH. The reliability analysis of balking and reneging in a repairable system with warm standbys. Quality and Reliability Engineering International 2002;18:467–78. [7] Ke JB, Lee WC, Ke JC. Reliability-based measure for a system with standbys subjected to switching failures. Engineering Computations: International Journal for Computer-Aided Engineering and Software 2008;25:694–706. [8] Ke JB, Lee WC, Wang KH. Reliability and sensitivity analysis of a system with multiple unreliable service stations and standby switching failures. Physica A: Statistical Mechanics and its Applications 2007;380:455–69. [9] Govil MG, Fu MC. Queueing theory in manufacturing: a survey. Journal of Manufacturing Systems 1999;18(3):214–40. [10] Meng FC. On comparison of MTBF between four redundant systems. Microelectronics and Reliability 1993;33:1987–90. [11] Trivedi KS. Probability and statistics with reliability queueing and computer science applications. 2nd ed. New York, NY: John Wiley and Sons; 2002. [12] Wang KH, Chen YJ. Comparative analysis of availability between three systems with general repair times, reboot delay and switching failures. Applied Mathematics and Computation 2009;215:384–94. [13] Wang KH, Chiu LW. Cost benefit analysis of availability systems with warm standby units and imperfect coverage. Applied Mathematics and Computation 2006;172:1239–56. [14] Wang KH, Kuo CC. Cost and probabilistic analysis of series systems with mixed standby components. Applied Mathematical Modelling 2000;24:957–67. [15] Wang KH, Pearn WL. Cost benefit analysis of series systems with warm standby components. Mathematical Methods of Operations Research 2003;58:247–58. [16] Wang KH, Sivazlian BD. Reliability of a system with warm standbys and repairmen. Microelectronics and Reliability 1989;29:849–60.