- Email: [email protected]
Multistate Multifailures System Analysis With Reworking Strategy and Imperfect Fault Coverage
CHAPTER 10
Multistate Multifailures System Analysis With Reworking Strategy and Imperfect Fault Coverage Monika Manglik*, Mangey Ram† *
Department of Mathematics, University of Petroleum & Energy Studies, Dehradun, India Department of Mathematics, Computer Science & Engineering, Graphic Era University, Dehradun, India
†
Abstract In designing a system, reliability prediction has an important role. When predicting the reliability of a system, the two key factors are considered: failure distribution of the component and configuration of the system. For the development of this rapidly developing field, this chapter provides an overview of a multicomponent repairable system and discusses the effect of the coverage factor on designed systems with various types of failures such as partial, catastrophic, and human failures. In this chapter, a Markov model has been developed for the system; also, the state transition probabilities have been determined for evaluating the various reliability measures. Supplementary variable techniques and Laplace transformations have been used for evaluating these reliability measures. Some specific cases along with a comparative study of different failure rates are also discussed. Finally, the practical utility of the model is discussed with the help of some graphical illustrations. Keywords: Imperfect fault coverage, Multistate system, Reworking strategy, System performance
1 INTRODUCTION Today, reliability has become the main concern due to the modern industrial process, and with an increasing level of sophistication, it comprises the highest number of manufacturing and industrial systems. Every system usually follows a failure-repair cycle. It means a system can exist in binary states: working (upstate) and failed state (downstate). These failures can be covered or uncovered. Because it directly impacts the system’s reliability, it is therefore important to demonstrate the proportion of recovered fault for accurately determining the coverage factor of the equipment and to analyze Advances in System Reliability Engineering https://doi.org/10.1016/B978-0-12-815906-4.00010-5
© 2019 Elsevier Inc. All rights reserved.
243
244
Advances in System Reliability Engineering
the impact of the parameter on the reliability of the overall system, according to Ram and Manglik [1]. In a fault-tolerant system, the proportion of handled faults and total faults is called the coverage factor. As far as the coverage probability is concerned, it is the probability of successful detection and recovery from a failure. Klauwer reinterpreted the definition of the coverage factor as a mathematical ratio [2]: Coverage factor ¼ all faults of a system can be exposed so that are covered= all possible faults a system can be exposed to coverage The coverage factor of the perfect coverage model is unity; any desired reliability of the system can be achieved by adding the replacements. However, it is not possible in the case of imperfect fault coverage. To improve the reliability of the system, not only is additional redundancy required but also improvement in the coverage factor. For an optimal level of redundancy, an accurate analysis is required; otherwise, due to imperfect fault coverage, the reliability of the system can be decreased with an increment in redundancy [3]. For the improvement of reliability of the systems, various authors have previously used the concept of coverage factor. Pham [4] examined a high voltage system with imperfect fault coverage by taking the failure rate of fault coverage as a constant. Akhtar [5] and Moustafa [6] analyzed the reliability of k-out-of-n:G system with imperfect fault coverage. The impact of fault detection and imperfect fault coverage on a repairable system has been considered by Trivedi [7]. Myers [8] studied the reliability of an k-out-of-n:G system with imperfect fault coverage. Ke et al. [9, 10] extended the model developed by Trivedi [7] with the help of asymptotic estimation and imperfect fault coverage; authors used Bayesian approach for predicting the performance measures of a repairable system with detection, imperfect coverage, and reboot. A comparative analysis for the availability of two systems with warm standby units has been given by Wang et al. [11]. In this study, authors assumed that the coverage factor of the active unit failure is different from that of the standby unit failure. Powell et al. [12] revealed that the coverage factor is used to compute the efficiency of their faulttolerance mechanisms. For estimating the effectiveness of a reconfiguration scheme in a fault-tolerant network, Reibman and Zaretsky [13] discussed a modeling approach in which a higher-level model and lower-level model represented the occurrence of failures in the network and the network reconfiguration system, respectively. Furthermore, coverage factor has been used by a number of authors in various reliability purposes, such as Cai et al.
Multistate Multifailures System Analysis
245
[14], Prabhudeva and Verma [15], and Kumar et al. [16]. Ram et al. [17] analyzed a parallel redundant system under two types of failures incorporating coverage factor. Cost analysis of three-state standby redundant electronic equipment has been done by Gupta and Sharma [18]. Manglik and Ram [19] analyzed a hydroelectric production power plant, and various reliability measures have also been discussed by the authors. Kumar and Ram [20] analyzed the system reliability measures under common cause failure. Ram and Manglik [21, 22] analyzed a multistate manufacturing system with common cause failure and waiting repair strategy; also, in addition to that, the authors analyzed the various reliability measures of an industrial system under standby modes and catastrophic failure. Ram and Manglik [23] proposed the mathematical modeling of a biometric system. Ram and Goyal [24] analyzed a flexible manufacturing system under the Copula-Coverage approach. Chopra and Ram [25] analyzed two nonidentical unit parallel systems with the concept of waiting repair. Goyal et al. [26] analyzed the sensitivity of three-unit series system under k-out-of-n redundancy, and Goyal and Ram [27] also studied a series-parallel system under warranty and preventive maintenance. Human performance plays an important role in the study of humanmachine interaction and system design [28]. Due to human error, the efficiency, security, and system concert is also reduced [29]. According to Dhillon and Yang [30] and Dhillon and Liu [31], human error is defined as the failure in execution of a particular task, due to which the planned operation or outcome is interrupted. Yang and Dhillon [32, 33] analyzed the availability of a repairable standby human-machine system and a general standby system with constant human error. To further the previously mentioned research, the authors designed a mathematical model for a standby complex system having two subsystems in the series by Markov process, and they obtained the state transition probability of each possible state of the system. The first subsystem has one main and one standby unit; after the failure of the main unit, the standby unit automatically takes the load. Further, the second subsystem has n units in series with partial failure property in its component. In this chapter, the effect of the coverage factor on reliability measures is discussed, and the chapter also analyzes how the fault coverage affects the expected profit of the system. The structure of the current chapter is as follows: Section 2 consists of the details related to the model including the nomenclature, system description, and assumptions. This section also describes the formulation and solution to the proposed model. In Section 3, the particular cases of reliability measures are discussed. Section 4 includes the results and discussion. Finally, in Section 5 the conclusion of the proposed analysis is discussed.
246
Advances in System Reliability Engineering
2 MATHEMATICAL MODEL DETAILS 2.1 Model Description and Assumptions In this chapter, the authors developed the mathematical model of a standby complex system with three types of failures by using the concept of coverage factor. The considered system consists of two subsystems A and B connected in series. Subsystem A consists of two units in which one is the main and the second one is in standby mode. On the other hand, subsystem B consists of n identical units in series. Initially, the system works with full capacity. After the failure of the main unit of subsystem A, the standby unit takes over its functioning immediately through the switchover device. Failure of both units of subsystem A results as a complete failure of the system. The partial failure of any jth unit (where j ¼ 1, …, n) of subsystem B brings the system to a reduced state from where the system can be automatically repaired (perfect coverage). Here, a reduced state means that the system state works less efficiently. Failure of both units of subsystem A, due to human failure and catastrophic failure, brings the system to complete failure mode. Furthermore, the authors have assumed that the system cannot be automatically recovered (imperfect fault coverage) due to catastrophic failure (a complete, sudden, often unexpected breakdown in any system). The configuration of the system and state transition diagram of the designed model are shown in Fig. 1A and B. Failure rates are inherently constant in general, whereas the repairs follow a general distribution. Markov process, supplementary variable technique, and Laplace transforms are used to evaluate the reliability measures.
2.2 Nomenclature The following nomenclature are used throughout the designing: λA λj λh , λ c P0, 0(t)/P0, j(t)/P1, 0(t)/P1, j(t)
The constant failure rate of the main and standby units of subsystem A. The constant partial failure rate of a jth unit of subsystem B. Failure rates of human failure and catastrophic failures, respectively. The probability at time t when the system is in full working condition/reduced state due to the partial failure of a jth unit of subsystem B/due to the failure of the main unit of subsystem A/due to the partial failure of a jth unit of subsystem B with the failure of the main unit of subsystem A, respectively.
Multistate Multifailures System Analysis
Subsystem A
247
Subsystem B
Main unit 1
2
n
3
Standby unit
(A)
fj (y)
S0 P0,0(t) (1–c)lc lA
P0,j(t)
clj S7 Pc(0,t)
(1–c)lc
S1
S3
lA
(1–c)lc
Pc(w,t) (1–c)lc
S4
clj
P1,0(t)
P1,j(t) fj (y)
lA m(x)
lA
lh S6
S2
P2,0(0,t)
PF(0,t)
P2,0(x,t)
PF(z,t) lh
S5 P2,j(0,t)
fj (y)
P2,j(y,t) Y (z)
(B) Fig. 1 (A) System configuration. (B) State transition diagram of the proposed model.
μ(x)/ψ(z)/φj(y)
P2,0(x, t) PF(z, t)
Repair rates of the system from failed state due to standby unit failure/human error/jth unit partial failure of subsystem B with the failure of the standby unit of subsystem A. The probability of state that both the units of subsystem A have failed and the system is under repair, elapsed repair time is x; t. The probability of state that the system has failed due to human error and under repair, elapsed repair time is z; t.
248
Advances in System Reliability Engineering
P2, j(y, t)
Pc(w, t) C Ep(t) K1, K2 S
The probability of state that the system has failed due to the failure of subsystem A with subsystem B is degraded and under repair, elapsed repair time is y; t. The probability of state that the system has failed due to catastrophic failure. Coverage factor of the system. Expected profit during up time. Revenue cost and service cost per unit time, respectively. Laplace parameter.
2.3 State Description and Transition Diagram The state description of the system is given as: State
State description
S0 S1
The system is working with full capacity. The system is in a good state after the failure of the main unit of subsystem A. It is in failed state after the failure of the standby unit subsystem A. It is in a reduced state due to the partial failure of any jth unit subsystem B. It is in a reduced state due to the failure of the main unit of subsystem A and the partial failure of any jth unit of subsystem B. It is in failed state due to the failure of both units of subsystem A and the partial failure of any jth unit of subsystem B. The system has stopped working due to human failure. Failed state due to catastrophic failure.
S2 S3 S4 S5 S6 S7
The assumptions associated with the model are mentioned here: (i) Primarily, the system is working with full capacity. (ii) The three states of the system are good, reduced, and failed states. (iii) Catastrophic failure, human failure, and failure of a standby unit of subsystem A results the complete failure of the system. (iv) The system can be automatically repaired in case of covered faults (partially failed units of subsystem B). (v) Repairing is not possible after the sudden (catastrophic) failure (uncovered fault). (vi) After repairing the system, it works with full capacity.
Multistate Multifailures System Analysis
249
2.4 Formulation and Solution of the Mathematical Model The following set of differential equations govern the present mathematical model: ð∞ ∂ + λA + cλj + λh + ð1 c Þλc P0, 0 ðtÞ ¼ P2, 0 ðx, t ÞμðxÞdx ∂t 0 ð∞ ð∞ P2, j ðy, tÞϕj ðyÞdy + PF ðz, t Þψ ðzÞdz + ϕj ðyÞP0, j ðt Þ (1) +
0
0
∂ + λA + cλj + λh + ð1 c Þλc P1, 0 ðtÞ ¼ φj ðyÞP1, j ðtÞ + λA P0, 0 ðtÞ ∂t ∂ ∂ + + μðxÞ P2, 0 ðx, t Þ ¼ 0 ∂t ∂x ∂ + λA + φj ðyÞ + ð1 c Þλc P0, j ðt Þ ¼ cλj P0, 0 ðtÞ ∂t ∂ + λA + φj ðyÞ + ð1 c Þλc P1, j ðtÞ ¼ cλj P0, 0 ðt Þ + λA P0, j ðtÞ ∂t ∂ ∂ + + φj ðyÞ P2, j ðy, tÞ ¼ 0 ∂t ∂y ∂ ∂ + + ψ ðzÞ PF ðz, tÞ ¼ 0 ∂t ∂z ∂ ∂ + Pc ðw, t Þ ¼ 0 ∂t ∂w
(2) (3) (4) (5) (6) (7) (8)
Boundary conditions: P2, 0 ð0, tÞ ¼ λA P1, 0 ðtÞ
(9)
P2, j ð0, tÞ ¼ λA P1, j ðt Þ
(10)
PF ð0, tÞ ¼ λh ½P0, 0 ðtÞ + P1, 0 ðt Þ Pc ð0, t Þ ¼ ð1 c Þλc P0, 0 ðtÞ + P1, 0 ðtÞ + P0, j ðtÞ + P1, j ðtÞ P0 ðt Þ ¼ 1, t ¼ 0 P0 ðt Þ ¼ 0, t > 0
(11) (12) (13)
250
Advances in System Reliability Engineering
Taking the Laplace transformation of Eqs. (1)–(12), we get: ð∞ s + λA + cλj + λh + ð1 c Þλc P 0, 0 ðsÞ ¼ 1 + P 2, 0 ðx, sÞμðxÞdx 0 ð∞ ð∞ + P 2, j ðy, sÞϕj ðyÞdy + P F ðz, sÞψ ðzÞdz + ϕj ðyÞP 0, j ðsÞ 0
(14)
0
s + λA + cλj + λh + ð1 c Þλc P 1, 0 ðsÞ ¼ φj ðyÞP 1, j ðsÞ + λA P 0, 0 ðsÞ ∂ s + + μðxÞ P 2, 0 ðx, sÞ ¼ 0 ∂x h i s + λA + ϕj ðyÞ + ð1 c Þλc P 0, j ðsÞ ¼ cλj P 0, 0 ðsÞ h i s + λA + ϕj ðyÞ + ð1 c Þλc P 1, j ðsÞ ¼ cλj P 0, 0 ðsÞ + λA P 0, j ðsÞ ∂ s + + φj ðyÞ P 2, j ðy, sÞ ¼ 0 ∂y ∂ s+ + ψ ðzÞ P F ðz, sÞ ¼ 0 ∂z ∂ s+ P c ðw, sÞ ¼ 0 ∂w
(15) (16) (17) (18) (19) (20) (21)
P 2, 0 ð0, sÞ ¼ λA P 1, 0 ðsÞ
(22)
P 2, j ð0, sÞ ¼ λA P 1, j ðsÞ
(23)
P F ð0, sÞ ¼ λh ½P 0, 0 ðsÞ + P 1, 0 ðsÞ P c ð0, sÞ ¼ ð1 c Þλc P 0, 0 ðsÞ + P 1, 0 ðsÞ + P 0, j ðsÞ + P 1, j ðsÞ
(24) (25)
Solving Eqs. (14)–(21) with the help of Eqs. (13) and (22)–(25), one may get various state probabilities: P 0, 0 ðsÞ ¼
1 d2
cλj 1 λA P 1, 0 ðsÞ ¼ 1+ + λA P 0, 0 ðsÞ φ ðyÞ ð s + d1 Þ ðs + d1 Þ ðs + dÞ j 1 cλj P 0, 0 ðsÞ ð s + d1 Þ cλj P 0, 0 ðsÞ λA P 1, j ðsÞ ¼ 1+ ðs + d1 Þ ð s + d1 Þ P0, j ðsÞ ¼
(26) (27) (28) (29)
Multistate Multifailures System Analysis
251
λA φj ðyÞ cλj λA 1 Sμ ðsÞ 1+ P 2, 0 ðsÞ ¼ ðs + dÞ ðs + d1 Þ ðs + d1 Þ s
1 2 1 Sμ ðsÞ P 0, 0 ðsÞ (30) λA + ðs + dÞ s λA cλj λA 1 Sφ ðsÞ 1+ P 2, j ðsÞ ¼ P 0, 0 ðsÞ (31) ðs + d1 Þ ð s + d1 Þ s cλj λh λA 1 Sψ ðsÞ P F ðsÞ ¼ λh + φ ðyÞ 1+ ðs + d Þ j ðs + d1 Þ ð s + d1 Þ s λA λh 1 Sψ ðsÞ P 0, 0 ðsÞ (32) + ðs + d Þ s
3 cλj 1 λA 6 1 + ðs + dÞ φj ðyÞ ðs + d1 Þ 1 + ðs + d1 Þ + λA + 7 1 7 6 P c ðsÞ ¼ ð1 c Þλc P 0, 0 ðsÞ6 7 5 4 1 s cλj λA cλj + 1+ ðs + d1 Þ ð s + d1 Þ ðs + d1 Þ (33) 2
Ð∞ Ðx Where Sμ ðsÞ ¼ 0 μðxÞ exp sx 0 μðxÞdx dx, Ð∞ Ðx Sψ ðsÞ ¼ 0 ψ ðzÞ exp sz 0 ψ ðzÞdz dx, n n oo Ð∞ Ðx Sφ ðsÞ ¼ 0 φj ðyÞ exp sy 0 φj ðyÞdy dy, d ¼ λA + cλj + λh + (1 c)λc, d1 ¼ λA + (1 c)λc + φj(y), and
3
cλj λA λA 2 7 6 ðs + d Þ ðs + d Þ ðs + d Þ 1 + ðs + d Þ Sμ ðsÞ ðs + d Þ Sμ ðsÞ 7 6 1 1 7 6 7 6 cλj λA 7 6 d2 ¼ 6 λA 7 1+ Sϕ ðsÞ λh Sψ ðsÞ 7 6 ð s + d Þ ð s + d Þ 1 1 7 6 7 6 4 cλ cλ λ λA 1 j j 5 h Sψ ðsÞϕj ðyÞ 1+ λA λh Sψ ðsÞ ϕj ðyÞ ðs + d Þ ðs + d1 Þ ðs + d1 Þ ðs + d1 Þ ðs + d Þ 2
λA ϕj ðyÞ
P up ðsÞ ¼ P 0, 0 ðsÞ + P 1, 0 ðsÞ + P 0, j ðsÞ + P 1, j ðsÞ 3 2 cλj 1 λA 1+ 1+ + λA φ ðyÞ 6 7 ðs + d1 Þ ðs + d1 Þ ðs + dÞ j 6 7 ¼6 7P 0, 0 ðsÞ 4 5 cλj 1 λA cλj + 1+ + ðs + d1 Þ ð s + d1 Þ ðs + d1 Þ (34)
252
Advances in System Reliability Engineering
P down ðsÞ ¼ P 2, 0 ðsÞ + P F ðsÞ + P 2, j ðsÞ + P c ðsÞ 3 2 λ φ ðyÞ cλ 1 S μ ðs Þ λA 1 A j j 2 1 S μ ðs Þ + 1 + λ A 7 6 ðs + d Þ ðs + d Þ s s ðs + d1 Þ ðs + d Þ 1 7 6 6 7 7 6 cλ 6 1 Sψ ðsÞ λA λh 1 Sψ ðsÞ 7 λh λA j 7 6 +λh+ ð Þ + φ 1+ y 7 6 s s ðs + d Þ j ðs + d 1 Þ ðs + d 1 Þ ðs + d Þ 7 6 7 6 7 6 λ cλ 1 S φ ðs Þ 7 6 λA j A 7P 0, 0 ðsÞ ¼6 + 7 6 ðs + d Þ 1 + ðs + d Þ s 1 1 7 6 7 6 3 2 7 6 cλj λA 1 7 6 7 6 1 + + λ 1 + ð y Þ φ 7 6 j A 7 6 1 ð s + d Þ ð s + d Þ ð s + d Þ 7 6 1 1 7 6 7 6 7 6 + ð1 c Þλc 6 7 5 4 s cλj 5 4 λA 1 cλj + 1+ + ðs + d1 Þ ðs + d 1 Þ ðs + d1 Þ (35)
3 PARTICULAR CASES 3.1 Availability Analysis Consider the values of various parameters as λA ¼ 0.020, λc ¼ 0.025, λh ¼ 0.030, λj ¼ 0.035, φj(y) ¼ 1, c ¼ 0.1, 0.2, 0.3, 0.4, 0.5. Using all these values in Eq. (34), then after taking the inverse Laplace transform and varying time unit t from 0 to 15, we obtain Table 1 and correspondingly Fig. 2, which represents the behavior of availability of the system with respect to time. Table 1 Availability versus time Availability Pup(t) Time (t)
c 5 0.1
c 5 0.2
c 5 0.3
c 5 0.4
c 5 0.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1.00000 0.96267 0.93788 0.91771 0.89935 0.88179 0.86470 0.84795 0.83149 0.81531 0.79941 0.78378 0.76843 0.75334 0.73852 0.72397
1.00000 0.96837 0.94878 0.93338 0.91946 0.90695 0.89284 0.87973 0.86669 0.85373 0.84085 0.82805 0.81536 0.80278 0.79031 0.77797
1.00000 0.97409 0.95971 0.94915 0.93973 0.93056 0.92134 0.91199 0.90251 0.89291 0.88322 0.87344 0.86361 0.85375 0.84387 0.83398
1.00000 0.97981 0.97069 0.96501 0.96017 0.95532 0.95019 0.94474 0.93896 0.93288 0.92655 0.91999 0.91323 0.90631 0.89924 0.89205
1.00000 0.98555 0.98171 0.98096 0.98078 0.98034 0.97942 0.97798 0.97604 0.97366 0.97087 0.96772 0.96425 0.96050 0.95650 0.95228
Multistate Multifailures System Analysis
253
1.00
c = 0.5
Availability of the system
0.95
0.90
c = 0.4
0.85 c = 0.3 0.80 c = 0.2 0.75 c = 0.1 0.70 0
2
4
6
8 10 Time unit
12
14
16
Fig. 2 Availability versus time.
3.2 Reliability Analysis The time-dependent system reliability discussed by Yang and Dhillon [33] without repair can be obtained by inverting Eq. (34). Taking the same set of the parameter as mentioned in Section 3.1 and varying time unit t from 0 to 15, one can obtain the system’s reliability without repair as shown in Table 2 and Fig. 3, respectively.
3.3 Mean Time to Failure (MTTF) Analysis MTTF of a system is given as the average time between the failures of a system. By considering all the repairs equal to zero in Eq. (34), as s tends to zero, we can obtain the MTTF as:
1+ MTTF ¼
cλj λA + + λA + cλj + λh + ð1 c Þλc λA + ð1 c Þλc λA + cλj + λh + ð1 c Þλc
cλj 1 +
λA λA + ð1 c Þλc λA + ð1 c Þλc
(36)
Setting λA ¼ 0.020, λc ¼ 0.025, λh ¼ 0.030, λj ¼ 0.035, c ¼ 0.1, 0.2, 0.3, 0.4, 0.5 and varying λA, λc, λh, λj one by one, respectively, as 0.1, 0.2, 0.3, 0.4, 0.05, 0.6, 0.7, 0.8, 0.9 in Eq. (36), one may obtain the variation of MTTF with respect to failure rates as mentioned in Table 3 and Fig. 4.
254
Advances in System Reliability Engineering
Table 2 Reliability versus time Reliability R(t) Time (t)
c 5 0.1
c 5 0.2
c 5 0.3
c 5 0.4
c 5 0.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1.00000 0.95198 0.90591 0.86174 0.81945 0.77897 0.74026 0.70328 0.66797 0.63427 0.60213 0.57150 0.54232 0.51454 0.48809 0.46293
1.00000 0.95767 0.91672 0.87716 0.83897 0.80216 0.76672 0.73263 0.69986 0.66840 0.63821 0.60927 0.58154 0.55498 0.52957 0.50526
1.00000 0.96338 0.92758 0.89266 0.85865 0.82559 0.79351 0.76241 0.73231 0.70320 0.67510 0.64797 0.62183 0.59664 0.57240 0.54909
1.00000 0.96910 0.93849 0.90826 0.87850 0.84927 0.82064 0.79264 0.76532 0.73870 0.71280 0.68764 0.66323 0.63956 0.61665 0.59447
1.00000 0.97483 0.94944 0.92395 0.89851 0.87320 0.84811 0.82333 0.79890 0.77490 0.75135 0.72830 0.70577 0.68378 0.66235 0.64149
1.0
Reliability of the system
0.9
0.8
0.7 c = 0.5 0.6
c = 0.4
0.5
c = 0.3 c = 0.2 c = 0.1
0.4 0
2
4
Fig. 3 Reliability versus time.
6
8 10 Time unit
12
14
16
Table 3 MTTF as a function of failure rates Failure rates c 5 0.1 of λA , λc , λc λh λj λA λh, λj
λA
λc
λh
λj
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
11.47911 7.27926 5.33499 4.21111 3.47851 2.96308 2.58071 2.28575 2.05130
9.48894 5.41135 3.78201 2.90567 2.35862 1.98473 1.71307 1.50678 1.34480
10.70445 6.14765 4.30875 3.31597 2.69481 2.26957 1.96020 1.72504 1.54025
27.46913 11.94738 33.47107 7.42688 37.72189 5.40608 40.88888 4.25274 43.33910 3.50580 45.29085 2.98234 46.88208 2.59502 48.20415 2.29680 49.32000 2.06009
11.03520 7.13613 5.26544 4.17018 3.45156 2.94405 2.56654 2.27480 2.04258
8.42365 4.80686 3.35948 2.58107 2.09519 1.76313 1.52186 1.33864 1.19478
9.18114 5.22260 3.64527 2.79888 2.27122 1.91086 1.64915 1.45047 1.29450
22.10592 25.71725 28.67382 31.13810 33.22318 35.01005 36.55828 37.91256 39.10714
MTTF c 5 0.2
c 5 0.3 λA
c 5 0.4
λc
λh
λj
10.82837 6.18056 4.32240 3.32179 2.69674 2.26937 1.95879 1.72290 1.53768
12.46262 7.21765 5.07702 3.91512 3.18582 2.68549 2.32095 2.04353 1.82534
33.14661 40.86428 45.75862 49.13777 51.61051 53.49821 54.98644 56.18983 57.18299
c 5 0.5
λA
λc
λh
λj
λA
λc
λh
λj
12.44194 7.57917 5.47875 4.29508 3.53348 3.00183 2.60948 2.30795 2.06895
12.54064 7.19011 5.03675 3.87376 3.14609 2.64809 2.28596 2.01081 1.79471
1.79471 8.46944 5.97672 4.61718 3.76138 3.17317 2.74402 2.41711 2.15979
39.32620 48.38263 53.64559 57.08440 59.50697 61.30566 62.69391 63.79779 64.69656
12.96489 7.73632 5.55303 4.33815 3.56154 3.02154 2.62408 2.31920 2.07788
14.82247 8.56940 6.02274 4.63958 3.77141 3.17609 2.74264 2.41304 2.15400
16.94411 9.95313 7.04407 5.45050 4.44481 3.75238 3.24660 2.86095 2.55719
46.23507 56.43331 61.95352 65.41281 67.78349 69.50947 70.82221 71.85425 72.68691
256
Advances in System Reliability Engineering
75 70 65 60 55 50 MTTF
45 40 35 30 25 20
aj 15 AJ 36 10
ak AK 37
5
al AL 38
am AM 39
an AN 40
ao AO 41
ap AP 42
0 0.2
0.4 0.6 Variation in failure rates
aq AQ 43 0.8
ar AR 44 1.0
Fig. 4 MTTF as a function of failure rates.
3.4 Expected Profit To maintain the product reliability, cost control is very important. Undoubtedly, only the reliability will not guarantee the viability of the product or system. Similarly, when the relating systems’ reliabilities are too low, the arbitrary cost-cutting can be harmful to profit [19]. The expected profit during the interval [0, t) is given as: ðt Ep ðtÞ ¼ K1 Pup ðtÞdt tK 2 (37) 0
Using the availability equation, expected profit for the same values of the parameters is given by setting K1 ¼ 1 and K2 ¼ 0.1, 0.3, 0.6, respectively, and one gets Table 4 and Fig. 5, respectively.
3.5 Busy Period Analysis or Mean Time to Repair (MTTR) MTTR is the average time taken by a system to recover from any failure. By taking all values of all repair rates to zero in Eq. (35), as s tends to zero, one can obtain the MTTR as: MTTR ¼
λh λA + cλj + λh + ð1 c Þλc
(38)
Table 4 Expected profit versus time Expected profit Ep(t) c 5 0.1
c 5 0.2
c 5 0.3
K2 5 0.1
K2 5 0.3
K2 5 0.6
K2 5 0.1
K2 5 0.3
K2 5 0.6
K2 5 0.1
K2 5 0.3
K2 5 0.6
0 1 2 3 4 5 6 7 8 9
0 0.87969 1.72938 2.55696 3.36540 4.15593 4.92914 5.68545 6.42515 7.14853
0 0.67969 1.32938 1.95696 2.56540 3.15593 3.72914 4.28545 4.82515 5.34853
0 0.37969 0.72938 1.05696 1.36540 1.65593 1.92914 2.18545 2.42515 2.64853
0 0.88259 1.74062 2.58151 3.40786 4.22060 5.02004 5.80632 6.57953 7.33974
0 0.68259 1.34062 1.98151 2.60786 3.22060 3.82004 4.40632 4.97953 5.53974
0 0.38259 0.74062 1.08151 1.40786 1.72060 2.02004 2.30632 2.57953 2.83974
0 0.88550 1.75188 2.60615 3.45055 4.28570 5.11166 5.92834 6.73561 7.53333
0 0.68550 1.35188 2.00615 2.65055 3.28570 3.91166 4.52834 5.13561 5.73333
0 0.38550 0.75188 1.10615 1.45055 1.78570 2.11166 2.42834 2.73561 3.03333
c 5 0.4
c 5 0.5
K2 5 0.3
K2 5 0.6
K2 5 0.1
K2 5 0.3
K2 5 0.6
0 0.88840 1.76318 2.63089 3.49346 4.35123 5.20402 6.05152 6.89340 7.72934
0 0.68840 1.36318 2.03089 2.69346 3.35123 4.00402 4.65152 5.29340 5.92934
0 0.38840 0.76318 1.13089 1.48346 1.85123 2.20402 2.55152 2.89340 3.22934
0 0.89132 1.77450 2.65573 3.53660 4.41720 5.29713 6.17587 7.05293 7.92782
0 0.69132 1.37450 2.05573 2.73660 3.41720 4.09713 4.77587 5.45293 6.12782
0 0.39132 0.77450 1.15573 1.53660 1.91720 2.29713 2.67587 3.05293 3.42782
257
K2 5 0.1
Multistate Multifailures System Analysis
Time (t)
258
Advances in System Reliability Engineering
9 8 7
Expected Profit
6 5 4 3 2 1 0 0
2
4
6
8
10
Time unit
Fig. 5 Expected versus time.
Setting λA ¼ 0.020, λc ¼ 0.025, λh ¼ 0.030, λj ¼ 0.035, c ¼ 0.1, 0.2, 0.3, 0.4, 0.5 and varying λA, λc, λh, λj one by one, respectively, as 0.1, 0.2, 0.3, 0.4, 0.05, 0.6, 0.7, 0.8, 0.9 in Eq. (38), the variation of mean time to repair with respect to failure rates may be easily obtained, as seen in Table 5 and Fig. 6.
4 RESULTS AND DISCUSSION In this chapter, the authors have analyzed a multistate, repairable system under the concept of covered faults with three types of failure. Also, various reliability measures by using Markov model and supplementary variable techniques have been discussed by the authors. The availability of any system lies between 0 and 1. Fig. 2 shows the impact of coverage factor on the availability of the system. It can be easily observed from the figure that, with an increase in time, the availability of the system decreases, whereas an increment in coverage factor results in an increase in availability of the system. Fig. 3 shows the behavior of reliability of the system with rest to time. From the graph, it can be observed that the reliability of the system decreases
Table 5 MTTR as a function of failure rates MTTR Failure rates of λA, λc, λh, λj
λA
λc
λh
λj
λA
λc
λh
λj
λA
λc
λh
λj
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.19230 0.11718 0.08426 0.06578 0.05395 0.04573 0.03968 0.03504 0.03138
0.20905 0.12847 0.09273 0.07255 0.05958 0.05054 0.04389 0.03878 0.03474
0.68493 0.81300 0.86705 0.89686 0.91575 0.92879 0.93833 0.94562 0.95137
0.36363 0.32432 0.29268 0.26666 0.24489 0.22641 0.21052 0.19672 0.18461
0.19108 0.11673 0.08403 0.06564 0.05385 0.04566 0.03963 0.03500 0.03134
0.21897 0.13824 0.10101 0.07957 0.06564 0.05586 0.04862 0.04304 0.03861
0.68027 0.80971 0.86455 0.89485 0.91407 0.92735 0.93708 0.94451 0.95036
0.33333 0.27272 0.23076 0.20000 0.17647 0.15789 0.14285 0.13043 0.12000
0.18987 0.11627 0.08379 0.06550 0.05376 0.04559 0.03957 0.03496 0.03131
0.22988 0.14962 0.11090 0.08810 0.07308 0.06243 0.05449 0.04834 0.04344
0.67567 0.80645 0.86206 0.89285 0.91240 0.92592 0.93582 0.94339 0.94936
0.30769 0.23529 0.19047 0.16000 0.13793 0.12121 0.10810 0.09756 0.08888
c 5 0.1
c 5 0.2
c 5 0.3
c 5 0.4
c 5 0.5
λA
λc
λh
λj
λA
λc
λh
λj
0.18867 0.11583 0.08356 0.06535 0.05366 0.04552 0.03952 0.03492 0.03128
0.24193 0.16304 0.12295 0.09868 0.08241 0.07075 0.06198 0.05514 0.04966
0.67114 0.80321 0.85959 0.89086 0.91074 0.92449 0.93457 0.94228 0.94836
0.28571 0.20689 0.16216 0.13333 0.11320 0.09836 0.08695 0.07792 0.07058
0.18750 0.11538 0.08333 0.06521 0.05357 0.04545 0.03947 0.03488 0.03125
0.25531 0.17910 0.13793 0.11214 0.09448 0.08163 0.07185 0.06417 0.05797
0.66666 0.80000 0.85714 0.88888 0.90909 0.92307 0.93333 0.94117 0.94736
0.26666 0.18461 0.14117 0.11428 0.09600 0.08275 0.07272 0.06486 0.05853
260
Advances in System Reliability Engineering
1.0 bk
bj
bm
bn
bo
bl
BL
BM 65
BN 66
BO 67
bi
0.8
bh
bg
MTTR
0.6
0.4
BG
0.2
59
BH BI 60
BJ 61
BK
62
63
64
0.0 0.2
0.4 0.6 Variation in failure rates
0.8
1.0
Fig. 6 MTTR as a function of failure rates.
constantly as the time increases, whereas the reliability of the system increases with the increment in coverage factor. Fig. 4 is the study of MTTF of the system with respect to various failures. From the figure, it can be observed that mean time to failure of the system increases due to the partial failure of the jth unit of subsystem B, and decreases due to the failure of the main and standby units of subsystem A, human failure, and catastrophic failure. From the graph, it can be concluded that, at the time of system operation, both the failures (human and catastrophic) are controlled comparative to partial failure of any jth unit of subsystem B. It can also be seen that mean time to failure of the system increases with the increment in coverage factor. Table 4 is obtained by varying service cost, taking the revenue cost per unit time at value 1 with coverage factor values at 0.1, 0.2, 0.3, 0.4, 0.5. From the figure, it can be observed that expected profit decreases with increase in service cost. This analysis shows that maximum profit can be attained by minimizing the service cost. Also, it can be seen that an increment in coverage factor results in an increase in expected profit. Mean time to repair of the system with respect to failure rates at different values of coverage factor has been shown in Fig. 6. It can be easily observed
Multistate Multifailures System Analysis
261
from the figure that it increases with increment in human failure rate, whereas it decreases with an increment in unit failure rate and catastrophic failure rate. Hence, the study shows that the system takes more recovery time due to human failure in comparison to unit failure and catastrophic failure. Also, from Fig. 6, we determined that MTTR decreases in each case with an increment in coverage factor.
5 CONCLUSION In this work, the authors have investigated a two-unit standby system with multiple failures, including the concept of fault coverage, and derived the explicit expressions as a function of time and coverage factor for the system performance measures. The numerical illustrations of these reliability measures show the effect of coverage factor, revealing that the performance of the system is improved significantly. The availability and reliability of the system and reduction in cost can be improved by introducing the concept of coverage factor. The results can be applied to the analysis, design, and operation of various systems subjected to fault coverage involving different reliability measures. Future work will be open to involving the performance analysis of various systems using the concept of coverage.
APPENDIX 1 At any time t, if the system is in state Si, then the probability of the system to be in that state is defined as: the probability that the system is in state Si at time t and remains there in the interval (t, t + Δt) or/and if it is in some other state at time t then it should transit to the state Si in the interval (t, t + Δt) provided transition exist between the states and Δt ! 0) Accordingly, Eqs. (39)–(46) are interpreted as: The probability of the system to be in state S0 in the interval (t, t + Δt) is given by:
P0, 0 ðt + Δt Þ ¼ð1 λA ΔtÞ 1 cλj Δt ð1 λh ΔtÞð1 ð1 c Þλc Δt ÞP0 ðtÞ ð∞ P2, 0 ðx, tÞμðxÞΔtdx + 0 ð∞ + P2, j ðy, tÞϕj ðyÞΔtdy 0 ð∞ + PF ðz, t Þψ ðzÞΔtdz + ϕj ðyÞΔtP 0, j ðtÞ 0
262
Advances in System Reliability Engineering
P0, 0 ðt + Δt Þ P0, 0 ðtÞ ) lim + λA + cλj + λh + ð1 c Þλc P0, 0 ðt Þ Δt!0 Δt ð∞ ð∞ ð∞ P2, j ðy, t Þϕj ðyÞdy + PF ðz, tÞψ ðzÞdz ¼ P2, 0 ðx, t ÞμðxÞdx + 0 0 0 ∂ + λA + cλj + λh + ð1 c Þλc P0, 0 ðtÞ + ϕj ðyÞP0, j ðt Þ ) ∂t ð∞ ð∞ ð∞ ¼ P2, 0 ðx, t ÞμðxÞdx + P2, j ðy, t Þϕj ðyÞdy + PF ðz, tÞψ ðzÞdz 0
0
(39)
0
+ ϕj ðyÞP0, j ðt Þ For state S1,
P1, 0 ðt + Δt Þ ¼ ð1 λA ΔtÞ 1 cλj Δt ð1 λh Δt Þð1 ð1 c Þλc ΔtP 1, 0 ðtÞ P1, 0 ðt + Δt Þ P1, 0 ðt Þ + ϕj ðyÞΔtP 1, j ðt Þ + λA ΔtP 0, 0 ðtÞ ) lim Δt!0 Δt
+ λA + cλj + λh + ð1 c Þλc P1, 0 ðtÞ ¼ φj ðyÞP1, j ðt Þ + λA P0, 0 ðtÞ ∂ ) +λA + cλj + λh+ð1 c Þλc P1, 0 ðt Þ¼ φj ðyÞP1, j ðtÞ+λA P0, 0 ðtÞ ∂t (40) For state S2, P2, 0 ðx + Δx, t + Δt Þ ¼ f1 μðxÞΔt gP2, 0 ðx, tÞ P2, 0 ðx+ Δx, t +ΔtÞ P2, 0 ðx, tÞ ) lim +μðxÞP2, 0 ðx, tÞ Δt Δx!0 Δt!0
∂ ∂ ¼0) + + μðxÞ P2, 0 ðx, tÞ ¼ 0 ∂t ∂x
(41)
For state S3,
P0, j ðt + Δt Þ ¼ ð1 λA ΔtÞ 1 φj ðyÞΔt ð1 ð1 c Þλc ΔtÞP0, j ðtÞ P0, j ðt+ΔtÞP0, j ðt Þ +λA P0, j ðt Þ+φj ðyÞP0, j ðtÞ Δt!0 Δt + ð1 c Þλc P0, j ðtÞ ¼ cλj P0, 0 ðtÞ ∂ ) + λA + φj ðyÞ + ð1 c Þλc P0, j ðtÞ ¼ cλj P0, 0 ðtÞ ∂t
+cλj ΔtP 0, 0 ðt Þ ) lim
ð42Þ
263
Multistate Multifailures System Analysis
For state S4,
P1, j ðt + ΔtÞ ¼ ð1 λA ΔtÞ 1 φj ðyÞΔt ð1 ð1 c Þλc ΔtÞP1, j ðtÞ P0, j ðt + Δt Þ P0, j ðtÞ Δt + λA P0, j ðtÞ + φj ðyÞP0, j ðtÞ + ð1 c Þλc P0, j ðt Þ ¼ cλj P0, 0 ðtÞ ∂ + λA + φj ðyÞ + ð1 c Þλc + λA P0, j ðtÞ ) ∂t + cλj ΔtP 0, 0 ðt Þ + λA ΔtP 0, j ðtÞ ) lim
Δt!0
P0, j ðtÞ ¼ cλj P0, 0 ðt Þ + λA P0, j ðtÞ
ð43Þ
For state S5,
n o P2, j ðy + Δy, t + ΔtÞ ¼ 1 φj ðyÞΔt P2, j ðy, t Þ ) lim Δy!0
P2, j ðy + Δy, t + Δt Þ P2, j ðy, t Þ +φj ðyÞP2, j ðy, t Þ Δt
Δt!0
∂ ∂ ¼0) + + φj ðyÞ P2, j ðy, tÞ ¼ 0 ∂t ∂y
(44)
For state S6, PF ðz + Δz, t + Δt Þ ¼ f1 ψ ðzÞΔtgPF ðz, tÞ PF ðz + Δy, t + Δt Þ PF ðy, tÞ + ψ ðzÞPF ðz, t Þ ¼ 0 ) lim Δt Δz!0 Δt!0
)
∂ ∂ + + ψ ðzÞ PF ðz, tÞ ¼ 0 ∂t ∂z
(45)
For state S7, Pc ðw + Δw, t + Δt Þ ¼ 0 Pc ðw + Δw, t + Δt Þ Pc ðw, t Þ ¼0 ) lim Δt Δw!0 Δt!0 ∂ ∂ ) + Pc ðz, tÞ ¼ 0 ∂t ∂w
(46)
264
Advances in System Reliability Engineering
APPENDIX 2 Boundary conditions of the system are obtained corresponding to transitions between the states where the transition from a state with elapsed repair time exists [32]. From Fig. 1B, the boundary conditions are: P2, 0 ð0, tÞ ¼ λA P1, 0 ðtÞ
(47)
P2, j ð0, tÞ ¼ λA P1, j ðtÞ
(48)
P2, j ð0, tÞ ¼ λh ½P0, 0 ðtÞ + P1, 0 ðtÞ Pc ð0, tÞ ¼ ð1 c Þλc P0, 0 ðtÞ + P1, 0 ðtÞ + P0, j ðtÞ + P1, j ðt Þ
(49) (50)
When the system is perfectly good, that is, in initial state S0, then P0 ð0Þ ¼ 1
(51)
and other state probabilities are zero at t ¼ 0.
REFERENCES [1] M. Ram, M. Manglik, Performance evaluation of a multi-state system covering imperfect fault coverage, Commun. Stat. Simul. Comput. 45 (9) (2016) 3259–3280. [2] L.A.T. Klauwer, Application of Formal Methods to Fault Injection and Coverage Factor Calculation (Master Thesis in Computer Science), Chalmers Tekniska Hogskolas INSA’de Lyon, 2006. [3] T.F. Arnold, The concept of coverage and its effect on the reliability model of a repairable system, IEEE Trans. Reliab. C-22 (1973) 251–254. [4] H. Pham, Reliability analysis of a high voltage system with dependent failures and imperfect coverage, Reliab. Eng. Syst. Saf. 37 (1992) 25–28. [5] S. Akhtar, Reliability of k-out-of-n:G system with imperfect fault coverage, IEEE Trans. Reliab. 43 (1994) 101–106. [6] M. Moustafa, Reliability analysis of k-out-of-n:G systems with dependent failures and imperfect coverage, Reliab. Eng. Syst. Saf. 58 (1997) 15–17. [7] K.S. Trivedi, Probability and Statistics With reliability. Queuing and computer Science Applications, John Wiley & sons, Inc., New York, 2002. [8] A. Myers, k-out-of-n:G system reliability with imperfect fault coverage, IEEE Trans. Reliab. 56 (2007) 464–473. [9] J.C. Ke, S.L. Lee, Y.L. Hsu, On a repairable system with detection, imperfect coverage and reboot: Bayesian approach, Simul. Model. Pract. Theory 16 (2008) 353–367. [10] J.C. Ke, Z.-L. Su, K.-H. Wang, Y.L. Hsu, Simulation inferences for an availability system with general repair distribution and imperfect fault coverage, Simul. Model. Pract. Theory 18 (3) (2010) 338–347. [11] K.-H. Wang, T.-C. Yen, Y.-C. Fang, Comparison of availability between two systems with warm standby units and different imperfect coverage, Qual. Technol. Quant. Manag. 9 (3) (2012) 265–282. [12] D. Powell, E. Martins, J. Arlat, Y. Crouzet, Estimators for fault tolerance coverage evaluation, IEEE Trans. Comput. 44 (2) (1995) 261–274.
Multistate Multifailures System Analysis
265
[13] A. Reibman, H. Zaretsky, in: Modeling fault coverage and reliability in a fault-tolerant network, Global Telecommunications Conference, vol. 2, 1990, pp. 689–692. [14] K.-Y. Cai, C.-Y. Wen, M.-L. Zhang, Fuzzy reliability modeling of gracefully degradable computing systems, Reliab. Eng. Syst. Saf. 33 (1991) 141–157. [15] S. Prabhudeva, A.K. Verma, Coverage modeling and reliability analysis using multistate function, Int. J. Autom. Comput. 4 (4) (2007) 380–387. [16] K. Kumar, J. Singh, P. Kumar, Fuzzy reliability and fuzzy availability of the serial process in butter- oil processing plant, J. Math. Stat. 5 (1) (2009) 65–71. [17] M. Ram, R.G. Varshney, S.B. Singh, Performance improvement of a parallel redundant system with coverage factor, J. Eng. Sci. Technol. 8 (3) (2013) 350–358. [18] P.P. Gupta, R.K. Sharma, Cost analysis of a three-state standby redundant electronic equipment, Microelectron. Reliab. 25 (6) (1985) 1029–1033. [19] M. Manglik, M. Ram, Behavioural analysis of a hydroelectric production power plant under reworking scheme, Int. J. Prod. Res. 53 (2) (2015) 648–664. [20] A. Kumar, M. Ram, System reliability measures in the presence of common cause failures, Int. J. Ind. Syst. Eng. 24 (1) (2016) 44–61. [21] M. Ram, M. Manglik, An analysis to multi-state manufacturing system with common cause failure and waiting repair strategy, Cogn. Eng. 3 (1) (2016) 1266185. [22] M. Ram, M. Manglik, Reliability measures analysis of an industrial system under standby modes and catastrophic failure, Int. J. Oper. Res. Inf. Syst. 7 (3) (2016) 36–56. [23] M. Ram, M. Manglik, Stochastic biometric system modelling with rework strategy, Int. J. Math. Eng. Manag. Sci. 1 (1) (2016) 1–17. [24] M. Ram, N. Goyal, Stochastic design exploration with rework of flexible manufacturing system under copula-coverage approach, Int. J. Reliab. Qual. Saf. Eng. 25 (2) (2017) 1850007. [25] G. Chopra, M. Ram, Stochastic analysis of two non-identical unit parallel system incorporating waiting time, Int. J. Qual. Reliab. Manag. 34 (6) (2017) 817–832. [26] N. Goyal, M. Ram, S. Amoli, A. Suyal, Sensitivity analysis of a three-unit series system under k-out-of-n redundancy, Int. J. Qual. Reliab. Manag. 34 (6) (2017) 770–784. [27] N. Goyal, M. Ram, Series-parallel system study under warranty and preventive maintenance, in: M. Ram, J.P. Davim (Eds.), Mathematics Applied to Engineering, Elsevier, 2017, pp. 97–113, ISBN: 978-0-12-810998-4. [28] B.S. Dhillon, Human errors: a review, Microelectron. Reliab. 29 (3) (1989) 299–304. [29] M. Ram, S.B. Singh, V.V. Singh, Stochastic analysis of a standby system with waiting repair strategy, IEEE Trans. Syst. Man Cybern. Syst. Hum. 43 (3) (2013) 698–707. [30] B.S. Dhillon, N. Yang, Probabilistic analysis of a maintainable system with human error, J. Qual. Maint. Eng. 1 (2) (1995) 50–59. [31] B.S. Dhillon, Y. Liu, Human error in maintenance: a review, J. Qual. Maint. Eng. 12 (1) (2006) 21–36. [32] N. Yang, B.S. Dhillon, Availability analysis of a repairable standby human-machine system, Microelectron. Reliab. 35 (11) (1995) 1401–1413. [33] N. Yang, B.S. Dhillon, Stochastic analysis of a general standby system with constant human error and arbitrary system repair rates, Microelectron. Reliab. 35 (7) (1995) 1037–1045.
Multistate Multifailures System Analysis With Reworking Strategy and Imperfect Fault Coverage Monika Manglik*, Mangey Ram† *
Department of Mathematics, University of Petroleum & Energy Studies, Dehradun, India Department of Mathematics, Computer Science & Engineering, Graphic Era University, Dehradun, India
†
Abstract In designing a system, reliability prediction has an important role. When predicting the reliability of a system, the two key factors are considered: failure distribution of the component and configuration of the system. For the development of this rapidly developing field, this chapter provides an overview of a multicomponent repairable system and discusses the effect of the coverage factor on designed systems with various types of failures such as partial, catastrophic, and human failures. In this chapter, a Markov model has been developed for the system; also, the state transition probabilities have been determined for evaluating the various reliability measures. Supplementary variable techniques and Laplace transformations have been used for evaluating these reliability measures. Some specific cases along with a comparative study of different failure rates are also discussed. Finally, the practical utility of the model is discussed with the help of some graphical illustrations. Keywords: Imperfect fault coverage, Multistate system, Reworking strategy, System performance
1 INTRODUCTION Today, reliability has become the main concern due to the modern industrial process, and with an increasing level of sophistication, it comprises the highest number of manufacturing and industrial systems. Every system usually follows a failure-repair cycle. It means a system can exist in binary states: working (upstate) and failed state (downstate). These failures can be covered or uncovered. Because it directly impacts the system’s reliability, it is therefore important to demonstrate the proportion of recovered fault for accurately determining the coverage factor of the equipment and to analyze Advances in System Reliability Engineering https://doi.org/10.1016/B978-0-12-815906-4.00010-5
© 2019 Elsevier Inc. All rights reserved.
243
244
Advances in System Reliability Engineering
the impact of the parameter on the reliability of the overall system, according to Ram and Manglik [1]. In a fault-tolerant system, the proportion of handled faults and total faults is called the coverage factor. As far as the coverage probability is concerned, it is the probability of successful detection and recovery from a failure. Klauwer reinterpreted the definition of the coverage factor as a mathematical ratio [2]: Coverage factor ¼ all faults of a system can be exposed so that are covered= all possible faults a system can be exposed to coverage The coverage factor of the perfect coverage model is unity; any desired reliability of the system can be achieved by adding the replacements. However, it is not possible in the case of imperfect fault coverage. To improve the reliability of the system, not only is additional redundancy required but also improvement in the coverage factor. For an optimal level of redundancy, an accurate analysis is required; otherwise, due to imperfect fault coverage, the reliability of the system can be decreased with an increment in redundancy [3]. For the improvement of reliability of the systems, various authors have previously used the concept of coverage factor. Pham [4] examined a high voltage system with imperfect fault coverage by taking the failure rate of fault coverage as a constant. Akhtar [5] and Moustafa [6] analyzed the reliability of k-out-of-n:G system with imperfect fault coverage. The impact of fault detection and imperfect fault coverage on a repairable system has been considered by Trivedi [7]. Myers [8] studied the reliability of an k-out-of-n:G system with imperfect fault coverage. Ke et al. [9, 10] extended the model developed by Trivedi [7] with the help of asymptotic estimation and imperfect fault coverage; authors used Bayesian approach for predicting the performance measures of a repairable system with detection, imperfect coverage, and reboot. A comparative analysis for the availability of two systems with warm standby units has been given by Wang et al. [11]. In this study, authors assumed that the coverage factor of the active unit failure is different from that of the standby unit failure. Powell et al. [12] revealed that the coverage factor is used to compute the efficiency of their faulttolerance mechanisms. For estimating the effectiveness of a reconfiguration scheme in a fault-tolerant network, Reibman and Zaretsky [13] discussed a modeling approach in which a higher-level model and lower-level model represented the occurrence of failures in the network and the network reconfiguration system, respectively. Furthermore, coverage factor has been used by a number of authors in various reliability purposes, such as Cai et al.
Multistate Multifailures System Analysis
245
[14], Prabhudeva and Verma [15], and Kumar et al. [16]. Ram et al. [17] analyzed a parallel redundant system under two types of failures incorporating coverage factor. Cost analysis of three-state standby redundant electronic equipment has been done by Gupta and Sharma [18]. Manglik and Ram [19] analyzed a hydroelectric production power plant, and various reliability measures have also been discussed by the authors. Kumar and Ram [20] analyzed the system reliability measures under common cause failure. Ram and Manglik [21, 22] analyzed a multistate manufacturing system with common cause failure and waiting repair strategy; also, in addition to that, the authors analyzed the various reliability measures of an industrial system under standby modes and catastrophic failure. Ram and Manglik [23] proposed the mathematical modeling of a biometric system. Ram and Goyal [24] analyzed a flexible manufacturing system under the Copula-Coverage approach. Chopra and Ram [25] analyzed two nonidentical unit parallel systems with the concept of waiting repair. Goyal et al. [26] analyzed the sensitivity of three-unit series system under k-out-of-n redundancy, and Goyal and Ram [27] also studied a series-parallel system under warranty and preventive maintenance. Human performance plays an important role in the study of humanmachine interaction and system design [28]. Due to human error, the efficiency, security, and system concert is also reduced [29]. According to Dhillon and Yang [30] and Dhillon and Liu [31], human error is defined as the failure in execution of a particular task, due to which the planned operation or outcome is interrupted. Yang and Dhillon [32, 33] analyzed the availability of a repairable standby human-machine system and a general standby system with constant human error. To further the previously mentioned research, the authors designed a mathematical model for a standby complex system having two subsystems in the series by Markov process, and they obtained the state transition probability of each possible state of the system. The first subsystem has one main and one standby unit; after the failure of the main unit, the standby unit automatically takes the load. Further, the second subsystem has n units in series with partial failure property in its component. In this chapter, the effect of the coverage factor on reliability measures is discussed, and the chapter also analyzes how the fault coverage affects the expected profit of the system. The structure of the current chapter is as follows: Section 2 consists of the details related to the model including the nomenclature, system description, and assumptions. This section also describes the formulation and solution to the proposed model. In Section 3, the particular cases of reliability measures are discussed. Section 4 includes the results and discussion. Finally, in Section 5 the conclusion of the proposed analysis is discussed.
246
Advances in System Reliability Engineering
2 MATHEMATICAL MODEL DETAILS 2.1 Model Description and Assumptions In this chapter, the authors developed the mathematical model of a standby complex system with three types of failures by using the concept of coverage factor. The considered system consists of two subsystems A and B connected in series. Subsystem A consists of two units in which one is the main and the second one is in standby mode. On the other hand, subsystem B consists of n identical units in series. Initially, the system works with full capacity. After the failure of the main unit of subsystem A, the standby unit takes over its functioning immediately through the switchover device. Failure of both units of subsystem A results as a complete failure of the system. The partial failure of any jth unit (where j ¼ 1, …, n) of subsystem B brings the system to a reduced state from where the system can be automatically repaired (perfect coverage). Here, a reduced state means that the system state works less efficiently. Failure of both units of subsystem A, due to human failure and catastrophic failure, brings the system to complete failure mode. Furthermore, the authors have assumed that the system cannot be automatically recovered (imperfect fault coverage) due to catastrophic failure (a complete, sudden, often unexpected breakdown in any system). The configuration of the system and state transition diagram of the designed model are shown in Fig. 1A and B. Failure rates are inherently constant in general, whereas the repairs follow a general distribution. Markov process, supplementary variable technique, and Laplace transforms are used to evaluate the reliability measures.
2.2 Nomenclature The following nomenclature are used throughout the designing: λA λj λh , λ c P0, 0(t)/P0, j(t)/P1, 0(t)/P1, j(t)
The constant failure rate of the main and standby units of subsystem A. The constant partial failure rate of a jth unit of subsystem B. Failure rates of human failure and catastrophic failures, respectively. The probability at time t when the system is in full working condition/reduced state due to the partial failure of a jth unit of subsystem B/due to the failure of the main unit of subsystem A/due to the partial failure of a jth unit of subsystem B with the failure of the main unit of subsystem A, respectively.
Multistate Multifailures System Analysis
Subsystem A
247
Subsystem B
Main unit 1
2
n
3
Standby unit
(A)
fj (y)
S0 P0,0(t) (1–c)lc lA
P0,j(t)
clj S7 Pc(0,t)
(1–c)lc
S1
S3
lA
(1–c)lc
Pc(w,t) (1–c)lc
S4
clj
P1,0(t)
P1,j(t) fj (y)
lA m(x)
lA
lh S6
S2
P2,0(0,t)
PF(0,t)
P2,0(x,t)
PF(z,t) lh
S5 P2,j(0,t)
fj (y)
P2,j(y,t) Y (z)
(B) Fig. 1 (A) System configuration. (B) State transition diagram of the proposed model.
μ(x)/ψ(z)/φj(y)
P2,0(x, t) PF(z, t)
Repair rates of the system from failed state due to standby unit failure/human error/jth unit partial failure of subsystem B with the failure of the standby unit of subsystem A. The probability of state that both the units of subsystem A have failed and the system is under repair, elapsed repair time is x; t. The probability of state that the system has failed due to human error and under repair, elapsed repair time is z; t.
248
Advances in System Reliability Engineering
P2, j(y, t)
Pc(w, t) C Ep(t) K1, K2 S
The probability of state that the system has failed due to the failure of subsystem A with subsystem B is degraded and under repair, elapsed repair time is y; t. The probability of state that the system has failed due to catastrophic failure. Coverage factor of the system. Expected profit during up time. Revenue cost and service cost per unit time, respectively. Laplace parameter.
2.3 State Description and Transition Diagram The state description of the system is given as: State
State description
S0 S1
The system is working with full capacity. The system is in a good state after the failure of the main unit of subsystem A. It is in failed state after the failure of the standby unit subsystem A. It is in a reduced state due to the partial failure of any jth unit subsystem B. It is in a reduced state due to the failure of the main unit of subsystem A and the partial failure of any jth unit of subsystem B. It is in failed state due to the failure of both units of subsystem A and the partial failure of any jth unit of subsystem B. The system has stopped working due to human failure. Failed state due to catastrophic failure.
S2 S3 S4 S5 S6 S7
The assumptions associated with the model are mentioned here: (i) Primarily, the system is working with full capacity. (ii) The three states of the system are good, reduced, and failed states. (iii) Catastrophic failure, human failure, and failure of a standby unit of subsystem A results the complete failure of the system. (iv) The system can be automatically repaired in case of covered faults (partially failed units of subsystem B). (v) Repairing is not possible after the sudden (catastrophic) failure (uncovered fault). (vi) After repairing the system, it works with full capacity.
Multistate Multifailures System Analysis
249
2.4 Formulation and Solution of the Mathematical Model The following set of differential equations govern the present mathematical model: ð∞ ∂ + λA + cλj + λh + ð1 c Þλc P0, 0 ðtÞ ¼ P2, 0 ðx, t ÞμðxÞdx ∂t 0 ð∞ ð∞ P2, j ðy, tÞϕj ðyÞdy + PF ðz, t Þψ ðzÞdz + ϕj ðyÞP0, j ðt Þ (1) +
0
0
∂ + λA + cλj + λh + ð1 c Þλc P1, 0 ðtÞ ¼ φj ðyÞP1, j ðtÞ + λA P0, 0 ðtÞ ∂t ∂ ∂ + + μðxÞ P2, 0 ðx, t Þ ¼ 0 ∂t ∂x ∂ + λA + φj ðyÞ + ð1 c Þλc P0, j ðt Þ ¼ cλj P0, 0 ðtÞ ∂t ∂ + λA + φj ðyÞ + ð1 c Þλc P1, j ðtÞ ¼ cλj P0, 0 ðt Þ + λA P0, j ðtÞ ∂t ∂ ∂ + + φj ðyÞ P2, j ðy, tÞ ¼ 0 ∂t ∂y ∂ ∂ + + ψ ðzÞ PF ðz, tÞ ¼ 0 ∂t ∂z ∂ ∂ + Pc ðw, t Þ ¼ 0 ∂t ∂w
(2) (3) (4) (5) (6) (7) (8)
Boundary conditions: P2, 0 ð0, tÞ ¼ λA P1, 0 ðtÞ
(9)
P2, j ð0, tÞ ¼ λA P1, j ðt Þ
(10)
PF ð0, tÞ ¼ λh ½P0, 0 ðtÞ + P1, 0 ðt Þ Pc ð0, t Þ ¼ ð1 c Þλc P0, 0 ðtÞ + P1, 0 ðtÞ + P0, j ðtÞ + P1, j ðtÞ P0 ðt Þ ¼ 1, t ¼ 0 P0 ðt Þ ¼ 0, t > 0
(11) (12) (13)
250
Advances in System Reliability Engineering
Taking the Laplace transformation of Eqs. (1)–(12), we get: ð∞ s + λA + cλj + λh + ð1 c Þλc P 0, 0 ðsÞ ¼ 1 + P 2, 0 ðx, sÞμðxÞdx 0 ð∞ ð∞ + P 2, j ðy, sÞϕj ðyÞdy + P F ðz, sÞψ ðzÞdz + ϕj ðyÞP 0, j ðsÞ 0
(14)
0
s + λA + cλj + λh + ð1 c Þλc P 1, 0 ðsÞ ¼ φj ðyÞP 1, j ðsÞ + λA P 0, 0 ðsÞ ∂ s + + μðxÞ P 2, 0 ðx, sÞ ¼ 0 ∂x h i s + λA + ϕj ðyÞ + ð1 c Þλc P 0, j ðsÞ ¼ cλj P 0, 0 ðsÞ h i s + λA + ϕj ðyÞ + ð1 c Þλc P 1, j ðsÞ ¼ cλj P 0, 0 ðsÞ + λA P 0, j ðsÞ ∂ s + + φj ðyÞ P 2, j ðy, sÞ ¼ 0 ∂y ∂ s+ + ψ ðzÞ P F ðz, sÞ ¼ 0 ∂z ∂ s+ P c ðw, sÞ ¼ 0 ∂w
(15) (16) (17) (18) (19) (20) (21)
P 2, 0 ð0, sÞ ¼ λA P 1, 0 ðsÞ
(22)
P 2, j ð0, sÞ ¼ λA P 1, j ðsÞ
(23)
P F ð0, sÞ ¼ λh ½P 0, 0 ðsÞ + P 1, 0 ðsÞ P c ð0, sÞ ¼ ð1 c Þλc P 0, 0 ðsÞ + P 1, 0 ðsÞ + P 0, j ðsÞ + P 1, j ðsÞ
(24) (25)
Solving Eqs. (14)–(21) with the help of Eqs. (13) and (22)–(25), one may get various state probabilities: P 0, 0 ðsÞ ¼
1 d2
cλj 1 λA P 1, 0 ðsÞ ¼ 1+ + λA P 0, 0 ðsÞ φ ðyÞ ð s + d1 Þ ðs + d1 Þ ðs + dÞ j 1 cλj P 0, 0 ðsÞ ð s + d1 Þ cλj P 0, 0 ðsÞ λA P 1, j ðsÞ ¼ 1+ ðs + d1 Þ ð s + d1 Þ P0, j ðsÞ ¼
(26) (27) (28) (29)
Multistate Multifailures System Analysis
251
λA φj ðyÞ cλj λA 1 Sμ ðsÞ 1+ P 2, 0 ðsÞ ¼ ðs + dÞ ðs + d1 Þ ðs + d1 Þ s
1 2 1 Sμ ðsÞ P 0, 0 ðsÞ (30) λA + ðs + dÞ s λA cλj λA 1 Sφ ðsÞ 1+ P 2, j ðsÞ ¼ P 0, 0 ðsÞ (31) ðs + d1 Þ ð s + d1 Þ s cλj λh λA 1 Sψ ðsÞ P F ðsÞ ¼ λh + φ ðyÞ 1+ ðs + d Þ j ðs + d1 Þ ð s + d1 Þ s λA λh 1 Sψ ðsÞ P 0, 0 ðsÞ (32) + ðs + d Þ s
3 cλj 1 λA 6 1 + ðs + dÞ φj ðyÞ ðs + d1 Þ 1 + ðs + d1 Þ + λA + 7 1 7 6 P c ðsÞ ¼ ð1 c Þλc P 0, 0 ðsÞ6 7 5 4 1 s cλj λA cλj + 1+ ðs + d1 Þ ð s + d1 Þ ðs + d1 Þ (33) 2
Ð∞ Ðx Where Sμ ðsÞ ¼ 0 μðxÞ exp sx 0 μðxÞdx dx, Ð∞ Ðx Sψ ðsÞ ¼ 0 ψ ðzÞ exp sz 0 ψ ðzÞdz dx, n n oo Ð∞ Ðx Sφ ðsÞ ¼ 0 φj ðyÞ exp sy 0 φj ðyÞdy dy, d ¼ λA + cλj + λh + (1 c)λc, d1 ¼ λA + (1 c)λc + φj(y), and
3
cλj λA λA 2 7 6 ðs + d Þ ðs + d Þ ðs + d Þ 1 + ðs + d Þ Sμ ðsÞ ðs + d Þ Sμ ðsÞ 7 6 1 1 7 6 7 6 cλj λA 7 6 d2 ¼ 6 λA 7 1+ Sϕ ðsÞ λh Sψ ðsÞ 7 6 ð s + d Þ ð s + d Þ 1 1 7 6 7 6 4 cλ cλ λ λA 1 j j 5 h Sψ ðsÞϕj ðyÞ 1+ λA λh Sψ ðsÞ ϕj ðyÞ ðs + d Þ ðs + d1 Þ ðs + d1 Þ ðs + d1 Þ ðs + d Þ 2
λA ϕj ðyÞ
P up ðsÞ ¼ P 0, 0 ðsÞ + P 1, 0 ðsÞ + P 0, j ðsÞ + P 1, j ðsÞ 3 2 cλj 1 λA 1+ 1+ + λA φ ðyÞ 6 7 ðs + d1 Þ ðs + d1 Þ ðs + dÞ j 6 7 ¼6 7P 0, 0 ðsÞ 4 5 cλj 1 λA cλj + 1+ + ðs + d1 Þ ð s + d1 Þ ðs + d1 Þ (34)
252
Advances in System Reliability Engineering
P down ðsÞ ¼ P 2, 0 ðsÞ + P F ðsÞ + P 2, j ðsÞ + P c ðsÞ 3 2 λ φ ðyÞ cλ 1 S μ ðs Þ λA 1 A j j 2 1 S μ ðs Þ + 1 + λ A 7 6 ðs + d Þ ðs + d Þ s s ðs + d1 Þ ðs + d Þ 1 7 6 6 7 7 6 cλ 6 1 Sψ ðsÞ λA λh 1 Sψ ðsÞ 7 λh λA j 7 6 +λh+ ð Þ + φ 1+ y 7 6 s s ðs + d Þ j ðs + d 1 Þ ðs + d 1 Þ ðs + d Þ 7 6 7 6 7 6 λ cλ 1 S φ ðs Þ 7 6 λA j A 7P 0, 0 ðsÞ ¼6 + 7 6 ðs + d Þ 1 + ðs + d Þ s 1 1 7 6 7 6 3 2 7 6 cλj λA 1 7 6 7 6 1 + + λ 1 + ð y Þ φ 7 6 j A 7 6 1 ð s + d Þ ð s + d Þ ð s + d Þ 7 6 1 1 7 6 7 6 7 6 + ð1 c Þλc 6 7 5 4 s cλj 5 4 λA 1 cλj + 1+ + ðs + d1 Þ ðs + d 1 Þ ðs + d1 Þ (35)
3 PARTICULAR CASES 3.1 Availability Analysis Consider the values of various parameters as λA ¼ 0.020, λc ¼ 0.025, λh ¼ 0.030, λj ¼ 0.035, φj(y) ¼ 1, c ¼ 0.1, 0.2, 0.3, 0.4, 0.5. Using all these values in Eq. (34), then after taking the inverse Laplace transform and varying time unit t from 0 to 15, we obtain Table 1 and correspondingly Fig. 2, which represents the behavior of availability of the system with respect to time. Table 1 Availability versus time Availability Pup(t) Time (t)
c 5 0.1
c 5 0.2
c 5 0.3
c 5 0.4
c 5 0.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1.00000 0.96267 0.93788 0.91771 0.89935 0.88179 0.86470 0.84795 0.83149 0.81531 0.79941 0.78378 0.76843 0.75334 0.73852 0.72397
1.00000 0.96837 0.94878 0.93338 0.91946 0.90695 0.89284 0.87973 0.86669 0.85373 0.84085 0.82805 0.81536 0.80278 0.79031 0.77797
1.00000 0.97409 0.95971 0.94915 0.93973 0.93056 0.92134 0.91199 0.90251 0.89291 0.88322 0.87344 0.86361 0.85375 0.84387 0.83398
1.00000 0.97981 0.97069 0.96501 0.96017 0.95532 0.95019 0.94474 0.93896 0.93288 0.92655 0.91999 0.91323 0.90631 0.89924 0.89205
1.00000 0.98555 0.98171 0.98096 0.98078 0.98034 0.97942 0.97798 0.97604 0.97366 0.97087 0.96772 0.96425 0.96050 0.95650 0.95228
Multistate Multifailures System Analysis
253
1.00
c = 0.5
Availability of the system
0.95
0.90
c = 0.4
0.85 c = 0.3 0.80 c = 0.2 0.75 c = 0.1 0.70 0
2
4
6
8 10 Time unit
12
14
16
Fig. 2 Availability versus time.
3.2 Reliability Analysis The time-dependent system reliability discussed by Yang and Dhillon [33] without repair can be obtained by inverting Eq. (34). Taking the same set of the parameter as mentioned in Section 3.1 and varying time unit t from 0 to 15, one can obtain the system’s reliability without repair as shown in Table 2 and Fig. 3, respectively.
3.3 Mean Time to Failure (MTTF) Analysis MTTF of a system is given as the average time between the failures of a system. By considering all the repairs equal to zero in Eq. (34), as s tends to zero, we can obtain the MTTF as:
1+ MTTF ¼
cλj λA + + λA + cλj + λh + ð1 c Þλc λA + ð1 c Þλc λA + cλj + λh + ð1 c Þλc
cλj 1 +
λA λA + ð1 c Þλc λA + ð1 c Þλc
(36)
Setting λA ¼ 0.020, λc ¼ 0.025, λh ¼ 0.030, λj ¼ 0.035, c ¼ 0.1, 0.2, 0.3, 0.4, 0.5 and varying λA, λc, λh, λj one by one, respectively, as 0.1, 0.2, 0.3, 0.4, 0.05, 0.6, 0.7, 0.8, 0.9 in Eq. (36), one may obtain the variation of MTTF with respect to failure rates as mentioned in Table 3 and Fig. 4.
254
Advances in System Reliability Engineering
Table 2 Reliability versus time Reliability R(t) Time (t)
c 5 0.1
c 5 0.2
c 5 0.3
c 5 0.4
c 5 0.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1.00000 0.95198 0.90591 0.86174 0.81945 0.77897 0.74026 0.70328 0.66797 0.63427 0.60213 0.57150 0.54232 0.51454 0.48809 0.46293
1.00000 0.95767 0.91672 0.87716 0.83897 0.80216 0.76672 0.73263 0.69986 0.66840 0.63821 0.60927 0.58154 0.55498 0.52957 0.50526
1.00000 0.96338 0.92758 0.89266 0.85865 0.82559 0.79351 0.76241 0.73231 0.70320 0.67510 0.64797 0.62183 0.59664 0.57240 0.54909
1.00000 0.96910 0.93849 0.90826 0.87850 0.84927 0.82064 0.79264 0.76532 0.73870 0.71280 0.68764 0.66323 0.63956 0.61665 0.59447
1.00000 0.97483 0.94944 0.92395 0.89851 0.87320 0.84811 0.82333 0.79890 0.77490 0.75135 0.72830 0.70577 0.68378 0.66235 0.64149
1.0
Reliability of the system
0.9
0.8
0.7 c = 0.5 0.6
c = 0.4
0.5
c = 0.3 c = 0.2 c = 0.1
0.4 0
2
4
Fig. 3 Reliability versus time.
6
8 10 Time unit
12
14
16
Table 3 MTTF as a function of failure rates Failure rates c 5 0.1 of λA , λc , λc λh λj λA λh, λj
λA
λc
λh
λj
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
11.47911 7.27926 5.33499 4.21111 3.47851 2.96308 2.58071 2.28575 2.05130
9.48894 5.41135 3.78201 2.90567 2.35862 1.98473 1.71307 1.50678 1.34480
10.70445 6.14765 4.30875 3.31597 2.69481 2.26957 1.96020 1.72504 1.54025
27.46913 11.94738 33.47107 7.42688 37.72189 5.40608 40.88888 4.25274 43.33910 3.50580 45.29085 2.98234 46.88208 2.59502 48.20415 2.29680 49.32000 2.06009
11.03520 7.13613 5.26544 4.17018 3.45156 2.94405 2.56654 2.27480 2.04258
8.42365 4.80686 3.35948 2.58107 2.09519 1.76313 1.52186 1.33864 1.19478
9.18114 5.22260 3.64527 2.79888 2.27122 1.91086 1.64915 1.45047 1.29450
22.10592 25.71725 28.67382 31.13810 33.22318 35.01005 36.55828 37.91256 39.10714
MTTF c 5 0.2
c 5 0.3 λA
c 5 0.4
λc
λh
λj
10.82837 6.18056 4.32240 3.32179 2.69674 2.26937 1.95879 1.72290 1.53768
12.46262 7.21765 5.07702 3.91512 3.18582 2.68549 2.32095 2.04353 1.82534
33.14661 40.86428 45.75862 49.13777 51.61051 53.49821 54.98644 56.18983 57.18299
c 5 0.5
λA
λc
λh
λj
λA
λc
λh
λj
12.44194 7.57917 5.47875 4.29508 3.53348 3.00183 2.60948 2.30795 2.06895
12.54064 7.19011 5.03675 3.87376 3.14609 2.64809 2.28596 2.01081 1.79471
1.79471 8.46944 5.97672 4.61718 3.76138 3.17317 2.74402 2.41711 2.15979
39.32620 48.38263 53.64559 57.08440 59.50697 61.30566 62.69391 63.79779 64.69656
12.96489 7.73632 5.55303 4.33815 3.56154 3.02154 2.62408 2.31920 2.07788
14.82247 8.56940 6.02274 4.63958 3.77141 3.17609 2.74264 2.41304 2.15400
16.94411 9.95313 7.04407 5.45050 4.44481 3.75238 3.24660 2.86095 2.55719
46.23507 56.43331 61.95352 65.41281 67.78349 69.50947 70.82221 71.85425 72.68691
256
Advances in System Reliability Engineering
75 70 65 60 55 50 MTTF
45 40 35 30 25 20
aj 15 AJ 36 10
ak AK 37
5
al AL 38
am AM 39
an AN 40
ao AO 41
ap AP 42
0 0.2
0.4 0.6 Variation in failure rates
aq AQ 43 0.8
ar AR 44 1.0
Fig. 4 MTTF as a function of failure rates.
3.4 Expected Profit To maintain the product reliability, cost control is very important. Undoubtedly, only the reliability will not guarantee the viability of the product or system. Similarly, when the relating systems’ reliabilities are too low, the arbitrary cost-cutting can be harmful to profit [19]. The expected profit during the interval [0, t) is given as: ðt Ep ðtÞ ¼ K1 Pup ðtÞdt tK 2 (37) 0
Using the availability equation, expected profit for the same values of the parameters is given by setting K1 ¼ 1 and K2 ¼ 0.1, 0.3, 0.6, respectively, and one gets Table 4 and Fig. 5, respectively.
3.5 Busy Period Analysis or Mean Time to Repair (MTTR) MTTR is the average time taken by a system to recover from any failure. By taking all values of all repair rates to zero in Eq. (35), as s tends to zero, one can obtain the MTTR as: MTTR ¼
λh λA + cλj + λh + ð1 c Þλc
(38)
Table 4 Expected profit versus time Expected profit Ep(t) c 5 0.1
c 5 0.2
c 5 0.3
K2 5 0.1
K2 5 0.3
K2 5 0.6
K2 5 0.1
K2 5 0.3
K2 5 0.6
K2 5 0.1
K2 5 0.3
K2 5 0.6
0 1 2 3 4 5 6 7 8 9
0 0.87969 1.72938 2.55696 3.36540 4.15593 4.92914 5.68545 6.42515 7.14853
0 0.67969 1.32938 1.95696 2.56540 3.15593 3.72914 4.28545 4.82515 5.34853
0 0.37969 0.72938 1.05696 1.36540 1.65593 1.92914 2.18545 2.42515 2.64853
0 0.88259 1.74062 2.58151 3.40786 4.22060 5.02004 5.80632 6.57953 7.33974
0 0.68259 1.34062 1.98151 2.60786 3.22060 3.82004 4.40632 4.97953 5.53974
0 0.38259 0.74062 1.08151 1.40786 1.72060 2.02004 2.30632 2.57953 2.83974
0 0.88550 1.75188 2.60615 3.45055 4.28570 5.11166 5.92834 6.73561 7.53333
0 0.68550 1.35188 2.00615 2.65055 3.28570 3.91166 4.52834 5.13561 5.73333
0 0.38550 0.75188 1.10615 1.45055 1.78570 2.11166 2.42834 2.73561 3.03333
c 5 0.4
c 5 0.5
K2 5 0.3
K2 5 0.6
K2 5 0.1
K2 5 0.3
K2 5 0.6
0 0.88840 1.76318 2.63089 3.49346 4.35123 5.20402 6.05152 6.89340 7.72934
0 0.68840 1.36318 2.03089 2.69346 3.35123 4.00402 4.65152 5.29340 5.92934
0 0.38840 0.76318 1.13089 1.48346 1.85123 2.20402 2.55152 2.89340 3.22934
0 0.89132 1.77450 2.65573 3.53660 4.41720 5.29713 6.17587 7.05293 7.92782
0 0.69132 1.37450 2.05573 2.73660 3.41720 4.09713 4.77587 5.45293 6.12782
0 0.39132 0.77450 1.15573 1.53660 1.91720 2.29713 2.67587 3.05293 3.42782
257
K2 5 0.1
Multistate Multifailures System Analysis
Time (t)
258
Advances in System Reliability Engineering
9 8 7
Expected Profit
6 5 4 3 2 1 0 0
2
4
6
8
10
Time unit
Fig. 5 Expected versus time.
Setting λA ¼ 0.020, λc ¼ 0.025, λh ¼ 0.030, λj ¼ 0.035, c ¼ 0.1, 0.2, 0.3, 0.4, 0.5 and varying λA, λc, λh, λj one by one, respectively, as 0.1, 0.2, 0.3, 0.4, 0.05, 0.6, 0.7, 0.8, 0.9 in Eq. (38), the variation of mean time to repair with respect to failure rates may be easily obtained, as seen in Table 5 and Fig. 6.
4 RESULTS AND DISCUSSION In this chapter, the authors have analyzed a multistate, repairable system under the concept of covered faults with three types of failure. Also, various reliability measures by using Markov model and supplementary variable techniques have been discussed by the authors. The availability of any system lies between 0 and 1. Fig. 2 shows the impact of coverage factor on the availability of the system. It can be easily observed from the figure that, with an increase in time, the availability of the system decreases, whereas an increment in coverage factor results in an increase in availability of the system. Fig. 3 shows the behavior of reliability of the system with rest to time. From the graph, it can be observed that the reliability of the system decreases
Table 5 MTTR as a function of failure rates MTTR Failure rates of λA, λc, λh, λj
λA
λc
λh
λj
λA
λc
λh
λj
λA
λc
λh
λj
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.19230 0.11718 0.08426 0.06578 0.05395 0.04573 0.03968 0.03504 0.03138
0.20905 0.12847 0.09273 0.07255 0.05958 0.05054 0.04389 0.03878 0.03474
0.68493 0.81300 0.86705 0.89686 0.91575 0.92879 0.93833 0.94562 0.95137
0.36363 0.32432 0.29268 0.26666 0.24489 0.22641 0.21052 0.19672 0.18461
0.19108 0.11673 0.08403 0.06564 0.05385 0.04566 0.03963 0.03500 0.03134
0.21897 0.13824 0.10101 0.07957 0.06564 0.05586 0.04862 0.04304 0.03861
0.68027 0.80971 0.86455 0.89485 0.91407 0.92735 0.93708 0.94451 0.95036
0.33333 0.27272 0.23076 0.20000 0.17647 0.15789 0.14285 0.13043 0.12000
0.18987 0.11627 0.08379 0.06550 0.05376 0.04559 0.03957 0.03496 0.03131
0.22988 0.14962 0.11090 0.08810 0.07308 0.06243 0.05449 0.04834 0.04344
0.67567 0.80645 0.86206 0.89285 0.91240 0.92592 0.93582 0.94339 0.94936
0.30769 0.23529 0.19047 0.16000 0.13793 0.12121 0.10810 0.09756 0.08888
c 5 0.1
c 5 0.2
c 5 0.3
c 5 0.4
c 5 0.5
λA
λc
λh
λj
λA
λc
λh
λj
0.18867 0.11583 0.08356 0.06535 0.05366 0.04552 0.03952 0.03492 0.03128
0.24193 0.16304 0.12295 0.09868 0.08241 0.07075 0.06198 0.05514 0.04966
0.67114 0.80321 0.85959 0.89086 0.91074 0.92449 0.93457 0.94228 0.94836
0.28571 0.20689 0.16216 0.13333 0.11320 0.09836 0.08695 0.07792 0.07058
0.18750 0.11538 0.08333 0.06521 0.05357 0.04545 0.03947 0.03488 0.03125
0.25531 0.17910 0.13793 0.11214 0.09448 0.08163 0.07185 0.06417 0.05797
0.66666 0.80000 0.85714 0.88888 0.90909 0.92307 0.93333 0.94117 0.94736
0.26666 0.18461 0.14117 0.11428 0.09600 0.08275 0.07272 0.06486 0.05853
260
Advances in System Reliability Engineering
1.0 bk
bj
bm
bn
bo
bl
BL
BM 65
BN 66
BO 67
bi
0.8
bh
bg
MTTR
0.6
0.4
BG
0.2
59
BH BI 60
BJ 61
BK
62
63
64
0.0 0.2
0.4 0.6 Variation in failure rates
0.8
1.0
Fig. 6 MTTR as a function of failure rates.
constantly as the time increases, whereas the reliability of the system increases with the increment in coverage factor. Fig. 4 is the study of MTTF of the system with respect to various failures. From the figure, it can be observed that mean time to failure of the system increases due to the partial failure of the jth unit of subsystem B, and decreases due to the failure of the main and standby units of subsystem A, human failure, and catastrophic failure. From the graph, it can be concluded that, at the time of system operation, both the failures (human and catastrophic) are controlled comparative to partial failure of any jth unit of subsystem B. It can also be seen that mean time to failure of the system increases with the increment in coverage factor. Table 4 is obtained by varying service cost, taking the revenue cost per unit time at value 1 with coverage factor values at 0.1, 0.2, 0.3, 0.4, 0.5. From the figure, it can be observed that expected profit decreases with increase in service cost. This analysis shows that maximum profit can be attained by minimizing the service cost. Also, it can be seen that an increment in coverage factor results in an increase in expected profit. Mean time to repair of the system with respect to failure rates at different values of coverage factor has been shown in Fig. 6. It can be easily observed
Multistate Multifailures System Analysis
261
from the figure that it increases with increment in human failure rate, whereas it decreases with an increment in unit failure rate and catastrophic failure rate. Hence, the study shows that the system takes more recovery time due to human failure in comparison to unit failure and catastrophic failure. Also, from Fig. 6, we determined that MTTR decreases in each case with an increment in coverage factor.
5 CONCLUSION In this work, the authors have investigated a two-unit standby system with multiple failures, including the concept of fault coverage, and derived the explicit expressions as a function of time and coverage factor for the system performance measures. The numerical illustrations of these reliability measures show the effect of coverage factor, revealing that the performance of the system is improved significantly. The availability and reliability of the system and reduction in cost can be improved by introducing the concept of coverage factor. The results can be applied to the analysis, design, and operation of various systems subjected to fault coverage involving different reliability measures. Future work will be open to involving the performance analysis of various systems using the concept of coverage.
APPENDIX 1 At any time t, if the system is in state Si, then the probability of the system to be in that state is defined as: the probability that the system is in state Si at time t and remains there in the interval (t, t + Δt) or/and if it is in some other state at time t then it should transit to the state Si in the interval (t, t + Δt) provided transition exist between the states and Δt ! 0) Accordingly, Eqs. (39)–(46) are interpreted as: The probability of the system to be in state S0 in the interval (t, t + Δt) is given by:
P0, 0 ðt + Δt Þ ¼ð1 λA ΔtÞ 1 cλj Δt ð1 λh ΔtÞð1 ð1 c Þλc Δt ÞP0 ðtÞ ð∞ P2, 0 ðx, tÞμðxÞΔtdx + 0 ð∞ + P2, j ðy, tÞϕj ðyÞΔtdy 0 ð∞ + PF ðz, t Þψ ðzÞΔtdz + ϕj ðyÞΔtP 0, j ðtÞ 0
262
Advances in System Reliability Engineering
P0, 0 ðt + Δt Þ P0, 0 ðtÞ ) lim + λA + cλj + λh + ð1 c Þλc P0, 0 ðt Þ Δt!0 Δt ð∞ ð∞ ð∞ P2, j ðy, t Þϕj ðyÞdy + PF ðz, tÞψ ðzÞdz ¼ P2, 0 ðx, t ÞμðxÞdx + 0 0 0 ∂ + λA + cλj + λh + ð1 c Þλc P0, 0 ðtÞ + ϕj ðyÞP0, j ðt Þ ) ∂t ð∞ ð∞ ð∞ ¼ P2, 0 ðx, t ÞμðxÞdx + P2, j ðy, t Þϕj ðyÞdy + PF ðz, tÞψ ðzÞdz 0
0
(39)
0
+ ϕj ðyÞP0, j ðt Þ For state S1,
P1, 0 ðt + Δt Þ ¼ ð1 λA ΔtÞ 1 cλj Δt ð1 λh Δt Þð1 ð1 c Þλc ΔtP 1, 0 ðtÞ P1, 0 ðt + Δt Þ P1, 0 ðt Þ + ϕj ðyÞΔtP 1, j ðt Þ + λA ΔtP 0, 0 ðtÞ ) lim Δt!0 Δt
+ λA + cλj + λh + ð1 c Þλc P1, 0 ðtÞ ¼ φj ðyÞP1, j ðt Þ + λA P0, 0 ðtÞ ∂ ) +λA + cλj + λh+ð1 c Þλc P1, 0 ðt Þ¼ φj ðyÞP1, j ðtÞ+λA P0, 0 ðtÞ ∂t (40) For state S2, P2, 0 ðx + Δx, t + Δt Þ ¼ f1 μðxÞΔt gP2, 0 ðx, tÞ P2, 0 ðx+ Δx, t +ΔtÞ P2, 0 ðx, tÞ ) lim +μðxÞP2, 0 ðx, tÞ Δt Δx!0 Δt!0
∂ ∂ ¼0) + + μðxÞ P2, 0 ðx, tÞ ¼ 0 ∂t ∂x
(41)
For state S3,
P0, j ðt + Δt Þ ¼ ð1 λA ΔtÞ 1 φj ðyÞΔt ð1 ð1 c Þλc ΔtÞP0, j ðtÞ P0, j ðt+ΔtÞP0, j ðt Þ +λA P0, j ðt Þ+φj ðyÞP0, j ðtÞ Δt!0 Δt + ð1 c Þλc P0, j ðtÞ ¼ cλj P0, 0 ðtÞ ∂ ) + λA + φj ðyÞ + ð1 c Þλc P0, j ðtÞ ¼ cλj P0, 0 ðtÞ ∂t
+cλj ΔtP 0, 0 ðt Þ ) lim
ð42Þ
263
Multistate Multifailures System Analysis
For state S4,
P1, j ðt + ΔtÞ ¼ ð1 λA ΔtÞ 1 φj ðyÞΔt ð1 ð1 c Þλc ΔtÞP1, j ðtÞ P0, j ðt + Δt Þ P0, j ðtÞ Δt + λA P0, j ðtÞ + φj ðyÞP0, j ðtÞ + ð1 c Þλc P0, j ðt Þ ¼ cλj P0, 0 ðtÞ ∂ + λA + φj ðyÞ + ð1 c Þλc + λA P0, j ðtÞ ) ∂t + cλj ΔtP 0, 0 ðt Þ + λA ΔtP 0, j ðtÞ ) lim
Δt!0
P0, j ðtÞ ¼ cλj P0, 0 ðt Þ + λA P0, j ðtÞ
ð43Þ
For state S5,
n o P2, j ðy + Δy, t + ΔtÞ ¼ 1 φj ðyÞΔt P2, j ðy, t Þ ) lim Δy!0
P2, j ðy + Δy, t + Δt Þ P2, j ðy, t Þ +φj ðyÞP2, j ðy, t Þ Δt
Δt!0
∂ ∂ ¼0) + + φj ðyÞ P2, j ðy, tÞ ¼ 0 ∂t ∂y
(44)
For state S6, PF ðz + Δz, t + Δt Þ ¼ f1 ψ ðzÞΔtgPF ðz, tÞ PF ðz + Δy, t + Δt Þ PF ðy, tÞ + ψ ðzÞPF ðz, t Þ ¼ 0 ) lim Δt Δz!0 Δt!0
)
∂ ∂ + + ψ ðzÞ PF ðz, tÞ ¼ 0 ∂t ∂z
(45)
For state S7, Pc ðw + Δw, t + Δt Þ ¼ 0 Pc ðw + Δw, t + Δt Þ Pc ðw, t Þ ¼0 ) lim Δt Δw!0 Δt!0 ∂ ∂ ) + Pc ðz, tÞ ¼ 0 ∂t ∂w
(46)
264
Advances in System Reliability Engineering
APPENDIX 2 Boundary conditions of the system are obtained corresponding to transitions between the states where the transition from a state with elapsed repair time exists [32]. From Fig. 1B, the boundary conditions are: P2, 0 ð0, tÞ ¼ λA P1, 0 ðtÞ
(47)
P2, j ð0, tÞ ¼ λA P1, j ðtÞ
(48)
P2, j ð0, tÞ ¼ λh ½P0, 0 ðtÞ + P1, 0 ðtÞ Pc ð0, tÞ ¼ ð1 c Þλc P0, 0 ðtÞ + P1, 0 ðtÞ + P0, j ðtÞ + P1, j ðt Þ
(49) (50)
When the system is perfectly good, that is, in initial state S0, then P0 ð0Þ ¼ 1
(51)
and other state probabilities are zero at t ¼ 0.
REFERENCES [1] M. Ram, M. Manglik, Performance evaluation of a multi-state system covering imperfect fault coverage, Commun. Stat. Simul. Comput. 45 (9) (2016) 3259–3280. [2] L.A.T. Klauwer, Application of Formal Methods to Fault Injection and Coverage Factor Calculation (Master Thesis in Computer Science), Chalmers Tekniska Hogskolas INSA’de Lyon, 2006. [3] T.F. Arnold, The concept of coverage and its effect on the reliability model of a repairable system, IEEE Trans. Reliab. C-22 (1973) 251–254. [4] H. Pham, Reliability analysis of a high voltage system with dependent failures and imperfect coverage, Reliab. Eng. Syst. Saf. 37 (1992) 25–28. [5] S. Akhtar, Reliability of k-out-of-n:G system with imperfect fault coverage, IEEE Trans. Reliab. 43 (1994) 101–106. [6] M. Moustafa, Reliability analysis of k-out-of-n:G systems with dependent failures and imperfect coverage, Reliab. Eng. Syst. Saf. 58 (1997) 15–17. [7] K.S. Trivedi, Probability and Statistics With reliability. Queuing and computer Science Applications, John Wiley & sons, Inc., New York, 2002. [8] A. Myers, k-out-of-n:G system reliability with imperfect fault coverage, IEEE Trans. Reliab. 56 (2007) 464–473. [9] J.C. Ke, S.L. Lee, Y.L. Hsu, On a repairable system with detection, imperfect coverage and reboot: Bayesian approach, Simul. Model. Pract. Theory 16 (2008) 353–367. [10] J.C. Ke, Z.-L. Su, K.-H. Wang, Y.L. Hsu, Simulation inferences for an availability system with general repair distribution and imperfect fault coverage, Simul. Model. Pract. Theory 18 (3) (2010) 338–347. [11] K.-H. Wang, T.-C. Yen, Y.-C. Fang, Comparison of availability between two systems with warm standby units and different imperfect coverage, Qual. Technol. Quant. Manag. 9 (3) (2012) 265–282. [12] D. Powell, E. Martins, J. Arlat, Y. Crouzet, Estimators for fault tolerance coverage evaluation, IEEE Trans. Comput. 44 (2) (1995) 261–274.
Multistate Multifailures System Analysis
265
[13] A. Reibman, H. Zaretsky, in: Modeling fault coverage and reliability in a fault-tolerant network, Global Telecommunications Conference, vol. 2, 1990, pp. 689–692. [14] K.-Y. Cai, C.-Y. Wen, M.-L. Zhang, Fuzzy reliability modeling of gracefully degradable computing systems, Reliab. Eng. Syst. Saf. 33 (1991) 141–157. [15] S. Prabhudeva, A.K. Verma, Coverage modeling and reliability analysis using multistate function, Int. J. Autom. Comput. 4 (4) (2007) 380–387. [16] K. Kumar, J. Singh, P. Kumar, Fuzzy reliability and fuzzy availability of the serial process in butter- oil processing plant, J. Math. Stat. 5 (1) (2009) 65–71. [17] M. Ram, R.G. Varshney, S.B. Singh, Performance improvement of a parallel redundant system with coverage factor, J. Eng. Sci. Technol. 8 (3) (2013) 350–358. [18] P.P. Gupta, R.K. Sharma, Cost analysis of a three-state standby redundant electronic equipment, Microelectron. Reliab. 25 (6) (1985) 1029–1033. [19] M. Manglik, M. Ram, Behavioural analysis of a hydroelectric production power plant under reworking scheme, Int. J. Prod. Res. 53 (2) (2015) 648–664. [20] A. Kumar, M. Ram, System reliability measures in the presence of common cause failures, Int. J. Ind. Syst. Eng. 24 (1) (2016) 44–61. [21] M. Ram, M. Manglik, An analysis to multi-state manufacturing system with common cause failure and waiting repair strategy, Cogn. Eng. 3 (1) (2016) 1266185. [22] M. Ram, M. Manglik, Reliability measures analysis of an industrial system under standby modes and catastrophic failure, Int. J. Oper. Res. Inf. Syst. 7 (3) (2016) 36–56. [23] M. Ram, M. Manglik, Stochastic biometric system modelling with rework strategy, Int. J. Math. Eng. Manag. Sci. 1 (1) (2016) 1–17. [24] M. Ram, N. Goyal, Stochastic design exploration with rework of flexible manufacturing system under copula-coverage approach, Int. J. Reliab. Qual. Saf. Eng. 25 (2) (2017) 1850007. [25] G. Chopra, M. Ram, Stochastic analysis of two non-identical unit parallel system incorporating waiting time, Int. J. Qual. Reliab. Manag. 34 (6) (2017) 817–832. [26] N. Goyal, M. Ram, S. Amoli, A. Suyal, Sensitivity analysis of a three-unit series system under k-out-of-n redundancy, Int. J. Qual. Reliab. Manag. 34 (6) (2017) 770–784. [27] N. Goyal, M. Ram, Series-parallel system study under warranty and preventive maintenance, in: M. Ram, J.P. Davim (Eds.), Mathematics Applied to Engineering, Elsevier, 2017, pp. 97–113, ISBN: 978-0-12-810998-4. [28] B.S. Dhillon, Human errors: a review, Microelectron. Reliab. 29 (3) (1989) 299–304. [29] M. Ram, S.B. Singh, V.V. Singh, Stochastic analysis of a standby system with waiting repair strategy, IEEE Trans. Syst. Man Cybern. Syst. Hum. 43 (3) (2013) 698–707. [30] B.S. Dhillon, N. Yang, Probabilistic analysis of a maintainable system with human error, J. Qual. Maint. Eng. 1 (2) (1995) 50–59. [31] B.S. Dhillon, Y. Liu, Human error in maintenance: a review, J. Qual. Maint. Eng. 12 (1) (2006) 21–36. [32] N. Yang, B.S. Dhillon, Availability analysis of a repairable standby human-machine system, Microelectron. Reliab. 35 (11) (1995) 1401–1413. [33] N. Yang, B.S. Dhillon, Stochastic analysis of a general standby system with constant human error and arbitrary system repair rates, Microelectron. Reliab. 35 (7) (1995) 1037–1045.