Availability prediction of imperfect fault coverage ...

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Int. J. Operational Research, Vol. 17, No. 3, 2013

Availability prediction of imperfect fault coverage system with reboot and common cause failure Madhu Jain Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Distt. Hardwar-247 667, India E-mail: [email protected] Abstract: This investigation is concerned with the availability characteristics of a multi-component repairable system with coverage factor and reboot delay. The time to repair the operating units is considered to be generally distributed whereas the life time, recovery time and reboot time of the units are assumed to be exponentially distributed. By introducing the supplementary variable corresponding to remaining repair time and using recursive approach, we evaluate steady state probabilities of the system states. The availability and failure frequency for different system configurations are obtained explicitly for some special distributions of repair time such as exponential, gamma and uniform distributions. The numerical experiments are performed to illustrate the computational tractability of the derived analytical results. The sensitivity analysis is carried out to examine the effect of system parameters on the reliability indices. Neuro-fuzzy results are also depicted graphically to explore the possibility of soft computing approach for the performance prediction of complex redundant systems. Keywords: availability; general repair; supplementary variable; imperfect fault coverage; reboot; common cause failure; neuro-fuzzy; failure frequency. Reference to this paper should be made as follows: Jain, M. (2013) ‘Availability prediction of imperfect fault coverage system with reboot and common cause failure’, Int. J. Operational Research, Vol. 17, No. 3, pp.374–397. Biographical notes: Madhu Jain received her MSc, MPhil, PhD and DSc in Mathematics from University of Agra. She has been a Gold Medalist of Agra University at MPhil level. There are more than 280 research publications in refereed international/national journals and more than 20 books to her credit in addition to two reference books. She was a recipient of the Young Scientific award and SERC Visiting Fellow of DST, and career award of UGC. She has successfully completed six sponsored major research projects of DST, UGC and CSIR. Her current research interest includes performance modelling, soft computing, bio-informatics, reliability engineering and queueing theory. Thirty candidates have received their PhD under her supervision. She has visited more than 30 reputed universities/institutes in the USA, Canada, UK, Germany, France, Holland and Belgium. She has participated and presented her research work in more than 35 international and 80 national conferences/seminars.

Copyright © 2013 Inderscience Enterprises Ltd.

Availability prediction of imperfect fault coverage system

1

375

Introduction

The assessment of availability which is defined as the percentage of time that a system is available to perform its required function plays a key role for the planning, design, development and operation of various multi-component complex systems. The evaluation of system descriptors with respect to redundancy and failure characteristics is of vital importance in many real time systems. The embedded computer-based system can be viewed as multi-component system. The redundancy and maintainability issues of such systems have drawn the attention of researchers working in the area of stochastic modelling of machine repair systems with the provisioning of maintainability and standby support. A typical computer system is an example of multi-component machining system with standby support wherein there may be provision of redundant processors, disk drives, printers, etc. The maintainability of computer system by recovery and corrective/preventive maintenance is quite popular. Fault coverage is one of the important methodologies to achieve high reliability and pre-specified availability of such system. By using this technique, the system can recover its operations from an erroneous condition. Some faults/failures may remain undetected or uncovered in many multi-component redundant systems. The uncovered faults have adverse effect on the availability of such systems which have applications in software, computer and communication systems, computing and distribution networks, electrical power distribution, dangerous fluid transportation, water supply pipelines, and many more. The fault coverage is the ability of a system to detect, isolate and correct the failures/faults of redundant elements. For highly reliable systems, such as avionic and space vehicles, nuclear missiles, etc. imperfect fault coverage has a hazardous impact on the system reliability. Most of the studies on redundant repairable systems have so far focused on the performance prediction of availability/reliability under the assumption of exponential repair time distribution. However in many scenarios, the repair time may not follow the Markovian property as such other distributions such as Erlangian, gamma, uniform, etc. should be taken into consideration. In this investigation, we study the availability analysis of multi-component repairable machining system with warm standbys under the assumption of general repair time distribution. The concepts of imperfect fault coverage and reboot have been incorporated. The rest of the paper is structured in different sections as follows. The review of the literature related to proposed investigation is presented in Section 2. In Section 3, we describe the model under consideration by stating the requisite assumptions and notations. The steady state probabilities of different states of the multi-component system are also obtained. Explicit expressions for the system availability and failure frequency for different configurations and specific repair time distributions are established in Section 4. Section 5 contains numerical results and sensitivity analysis for the illustration purpose. The effects of various system parameters and neuro-fuzzy results are examined graphically. The highlights of findings and future scope of the present investigation are summarised in the final Section 6.

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Literature review

The failure or fault occurrence cannot be avoided during the design, development and operation phases of any real time system. In order to achieve high reliability/availability of such systems, the use of redundancy is common practice. Due to wide applicability of multi-component redundant systems, several researchers have contributed in this area. Wang et al. (2006) provided a comparison of the reliability and availability between four systems with warm standby components and standby switching failures. The availability, reliability and downtime of a system with repairable components have been investigated by Kiureghian et al. (2007). A systematic approach for the reliability allocation by comprehending functional dependency and failure criticality for each component or subsystem was suggested by Yadav (2007). The availability of a general repairable system was studied by Gamiz and Roman (2008). They evaluated the measures for the effectiveness of repairable system with unknown failure and repair time distributions. Vasanthi and Arulmozhi (2009) proposed a genetic model to study the reliability optimisation in a linear consecutively connected system (LCCS) by allocating M statistically independent multi-state elements with different characteristics to the first N positions. By introducing the uncertainty of the operating environments as a random variable, Persona et al. (2009) presented a new mathematical function for predicting the reliability of the systems. Sarkar et al. (2010) determined optimal production lot size, safety stock and reliability of a machinery system under the realistic assumptions that the production facility is subject to random breakdown and also to change in the variable reliability parameter. Levitin and Amari (2010) proposed an approximation algorithm based on a universal generating function approach for evaluating the reliability indices of k-out-of-n systems with shared spare element and different arbitrary distributions of time-to-failure. Both time dependent and steady state availability under ideal as well as faulty preventive maintenance were examined by Garg et al. (2010). Ram and Singh (2010) proposed the reliability analysis of a complex system with common cause failure and two types of repair facilities. Cekyay and Ozekici (2010) studied the mean time to failure and availability of semi-Markov missions with maximal repair. Sarkar and Biswas (2010) obtained the availability of a one-unit system supported by n − 1 spares and r repair facilities. Recently, Brissaud et al. (2011) considered the reliability modelling of failure rates as a function of time and influencing factors. The quasi-birth-death process has been used to examine the shut-off rules for the availability of a general non-identical components k-out-of-n: G system by Moghaddass et al. (2011). Ozaki et al. (2011) determined the availability and mean time to first failure (MTTFF) of M-for-N shared protection systems (with M + N units) from the viewpoint of an arbitrary one of the N end users. Jain and Gupta (2011) presented a review article on the reliability concepts by discussing the failure consequences, methodologies of redundant systems along with software and hardware redundancy techniques. A Markovian model was used by Hajeeh (2011a) to derive a closed form expression for the steady state availability for a system that is allowed to undergo several stages of deteriorations subject to several failures at each state. Hajeeh (2011b) also considered the reliability and availability for four series configurations with both warm and cold standby with the existence of common cause failure of the system at all states. The mean time to failure (MTTF) and steady state availability are derived for all configurations. Azadeh et al. (2011) presented a fuzzy simulation algorithm for estimating availability functions in time-dependent complex

Availability prediction of imperfect fault coverage system

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systems especially when insufficient data are available. Hajeeh (2012) studied the availability and long run probabilities of the different states of the general system which upon failure could undergo one of several repairs; imperfect repair, minimal repair, or replacement (perfect repair). In real time engineering systems, it is impossible to eliminate the cause of failure/fault completely or recover the system perfectly using redundancy approaches. Due to imperfect fault-coverage, the system may fail even prior to the exhaustion of standbys due to uncovered component failures. In reliability literature, several researchers have examined the effects of imperfect fault-coverage on the performance of these systems. The effect of imperfect coverage on the reliability indices of K-out-of-M: G systems was discussed by Moustafa (1997), Amari et al. (2004), Myers (2007) and many more. The reliability of the systems subject to imperfect fault-coverage was discussed by Trivedi (2002), Xing and Dugan (2002), Vieira and Madeira (2004) and Wang and Chiu (2006). The reliability assessment of multi-state systems subject to imperfect fault coverage was done by Chang et al. (2005), Levitin (2007) and Levitin and Amari (2008). The mission reliability analysis of multiple-phased fault-tolerant system based on Markov regenerative process was suggested by Mo et al. (2008). Myers and Rauzy (2008) gave the assessment of the reliability of redundant systems with imperfect fault coverage by means of binary decision diagram. Chakravarthy and Corral (2009) considered the influence of delivery times on the performance measures of k-out-of-N reliability system by using a finite quasi-birth-and-death process. Fault tolerance is a critical issue for sensors deployed in places where they are not easily replaceable, repairable and rechargeable. Bein et al. (2009) considered the coverage problem for sensor networks from the fault tolerance and reliability point of view. Two imperfect multi-component repair models are examined by Hajeeh and Jabsheh (2009) by assuming that the performance of a system becomes inferior after each failure. Ke et al. (2010) studied simulation inferences for an availability system with general repair distribution and imperfect fault coverage. Chang et al. (2010) suggested a new efficient algorithm for the reliability evaluation of a distributed computing network with imperfect nodes. An optimisation model is developed by Shafiee and Asgharizadeh (2011) to determine the optimal burn-in time and optimal imperfect preventive maintenance strategy that minimises the total mean servicing cost of a warranted product with an age-dependent repair cost. An algorithm for evaluating the performance distribution of complex series-parallel multi-state systems with propagated failures and imperfect protections was proposed by Levitin et al. (2011). Uprety (2012) developed a stochastic model by including preventive maintenance times to determine the performance of a reheating-furnace system having different failure and repair modes under variable operational conditions. Various system characteristics such as mean time to system failures, availability and expected utilisation of servers, are obtained using regenerative point technique. In computer-based fault tolerant systems, during the operational phase, the corrective maintenance is performed to remove the faults that cause degradation or failure of the system. The reboot concept plays a significant role for the improvement of the availability of the fault tolerant systems. In many fault tolerant systems, the system does not successfully recover from a unit failure and suffers from a more severe impact. In such a case, the system needs to be rebooted with a longer averaged duration. A few papers have appeared in literature on redundant systems which incorporate the concept of

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reboot while predicting the availability/reliability of such systems in different frameworks. The notable work in this direction is presented by Trivedi (2002). The availability analysis and Bayesian estimation for a repairable system with detection, imperfect coverage and reboot have been provided by Ke et al. (2008). A comparative study of availability between three systems with general repair times, reboot delay and switching failures was made by Wang and Chen (2009). Hsu et al. (2009) proposed Bayesian and asymptotic estimation for a repairable system with imperfect coverage and reboot. Ke et al. (2011) evaluated the reliability measures of a repairable system with M operating units having warm standby units and multi-repairmen by incorporating the concepts of switching failures and reboot delay. Hsu et al. (2011) provided statistical analysis of standby system with general repair, reboot delay, switching failure under the assumption of unreliable repair facility. For many complex systems, it is not feasible to evaluate the performance indices by using analytical or numerical techniques. For such systems, the soft computing approaches based on neuro-fuzzy logic play an important role (cf., Brown and Harris, 1994). Recently, Zhao et al. (2010) studied the software reliability growth model based on fuzzy wavelet neural network (FWNN-SRGM). Based on the experimental results they showed that the presented method can be easily implemented by exploiting the failure data.

3

Model description and steady state probabilities

In order to improve the system availability, we have not only to include the redundancy and maintainability but also improve the coverage-factor. A redundant repairable system consisting of M active units is considered. Each of the active unit may fail independently of the state of the other. If any unit fails, the system will immediately take reconfiguration operation to restore its functionality. The reconfiguration operation will detect and remove the failed unit from the system whereas other operating units will continue to operate as it is. The probability of successful reconfiguration operation is defined as coverage factor. The transition diagram of multi-component redundant repairable system is shown in Figure 1. Initially all units are active. We assume that active unit fails according to Poisson distribution with parameter λ. As soon as an active component fails, a protection switch successfully restores service by switching on the other unit with probability C. With complementary probability (1 – C), the protection switch fails to cover the failure of the active unit. The uncovered failure of the active unit is then cleared by a reboot action. The reboot time and the recovery time are assumed to be exponentially distributed with rate σ and θ, respectively. The repair of the failed unit is done by a single server; the repair time is independent and identically distributed (i.i.d.) general random variable B. The cumulative distribution function (CDF), probability density function and mean repair time are denoted by B(x), b(x) and b1 = 1/µ, respectively. The first come and first served (FCFS) discipline is followed for the repair of failed unit. After repair, the failed unit is immediately put into active mode. The random variable X(t) denotes the number of failed units in the repairable system at time t. We denote the recovery and reboot states when the number for failed units in the system is n by (n, RC) and (n, RB), respectively. Then {X(t); t ≥ 0} is a continuous-time stochastic process with 3M – 1 states.

Availability prediction of imperfect fault coverage system Figure 1

379

State transition diagram for M unit repairable system (see online version for colours)

For our model, the service time is assumed to be general distributed as such the stochastic process X(t) is non-Markovian. We introduce supplementary variable U(t) corresponding to remaining repair time at time t for the analysis purpose. Now we define the system state probabilities as follows: Server state

Probabilities

Active

Pn(x, t)dx; X(t) = n, 0 ≤ n ≤ M; x < U(t) < x + dx

Recovery (RC)

Q n(t); X(t) = n, 1 ≤ n ≤ M – 1

Reboot (RB)

Rn(t); X(t) = n, 1 ≤ n ≤ M – 1

For steady state, we define

Pn ( x) = lim Pn ( x, t ), 0 ≤ n ≤ M t →∞

Qn = lim Qn ( x, t ), 0 ≤ n ≤ M − 1 t →∞

Rn = lim Rn ( x, t ), 0 ≤ n ≤ M − 1 t →∞

We construct Chapman Kolmogov equations governing the model by using appropriate transition rates as follows: 0 = − M λ P0 + P1 (0) 0=

0=

(1)

∂ Pn ( x) − ( M − n)λ Pn ( x) + θ b( x)Qn + σ b( x) + Rn + b( x) Pn +1 (0), ∂x 1 ≤ n ≤ M −1

(2)

∂PM ( x) λ PM −1 ( x) ∂x

(3)

0 = ( M − n) λ C Pn − θ Qn +1 , 0 ≤ n ≤ M − 2

(4)

0 = ( M − n)λ (1 − C ) Pn − σ Rn +1 , 0 ≤ n ≤ M − 2

(5)

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Now we define ∞

∞

∫

∫

B* ( s ) = e − su dB (u ) = e− su b(u )du ; 0

0

∞

∞

∫

Pn* ( s ) = e − su Pn (u )du ; Pn = Pn* (0) = 0

∫

Pn (u )du

0

Taking Laplace transforms of (2) and (3), we get [( M − n)λ − s ] Pn* ( s ) = θ B* ( s ) Qn + σ B* ( s ) Rn + B* ( s ) Pn +1 (0) − Pn (0), 1 ≤ n ≤ M − 1 − sPM* ( s ) = λ PM* −1 ( s ) − PM (0)

(6) (7)

From (1), (4) and (5), we have P1 (0) = M λ P0

(8)

Qn +1 = an Pn , n = 0,1, 2,..., M − 2

(9)

Rn +1 = bn Pn , n = 0,1, 2,..., M − 2

(10)

where an =

( M − n )λ C

θ

, bn =

( M − n)(1 − C )λ

σ

Also

(

)

θ Qn + σ Rn = M − n − 1 λ Pn −1 , n = 1, 2,..., M − 1

(11)

Substituting 0 = 0 and (M – n)λ in (6), we get

(

)

( M − n)λ Pn* (0) = M − n − 1 λ Pn −1 + Pn +1 (0) − Pn (0)

(12a)

and

(

)

(

)

(

)

0 = M − n − 1 λ Pn −1 B* M − n λ + B* M − n λ Pn +1 (0) − Pn (0)

(12b)

We note that ∞

∫

Pn = Pn* (0) = Pn ( x) dx ,

(13)

0

Solving (12a) and (12b) and (13) and using (1), we get

( M − n − 1){ 1 − B ( M − n λ ) } P = ( M − n) B ( M − n λ ) *

Pn

*

n −1

,

(14)

Availability prediction of imperfect fault coverage system

381

Differentiating (6) w.r.t. ‘s’ and then substituting s = 0, for n = 1, 2, …, M – 1, we obtain

(

)

− Pn* (0) + ( M − n) λ Pn*′ (0) = − M − n − 1 λβ1 Pn −1 − b1 Pn +1 (0)

(15)

Differentiating (7) w.r.t. ‘s’ and then putting s = 0 − PM* (0) = λ PM* −1′ (0)

which using (13) yields PM = −λ PM* −1′ (0)

(16)

Substituting n = M – 1 in (15) and using (16), we find PM = 2λβ1 PM − 2 − (1 − λ b1 ) PM −1

(17)

Solving (14) and (17), we get n

Pn = MP0

1 − B* ( M − i )

∏ B (M − i λ) ,

PM =

1 ≤ n ≤ M −1

*

i =1

(

MP0 λβ − 1 + B* (λ )

)

B (λ ) *

M −1

∏ i=2

1 − B* (iλ ) B* (iλ )

(18)

(19)

where P0 is determined by using normalising condition M −1

P0 +

∑(P + Q + R ) + P i

i

i

M

=1

(20)

i =1

4

Performance indices

In this section, we establish various performance measures which may be of interest of system designers and decision makers for exploring the transient behaviour of the fault tolerant multi-component system. An M units system that works if and only if at least R out of total M units are working, is called a R-out-of-M: G system; such configuration is very popular type redundancy in fault tolerant systems including industrial as well as military systems. System availability and failure frequency are the importance measures for the optimal design of any system working in machining environment. For R-out-of-M: G configuration of multi-component machining system having M active units, the steady state availability and failure frequency are given as follows: M −R

AR , M (∞) = P0 +

∑ ⎡⎣ P + Q ⎤⎦ j

j =1

j

(21)

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and F = M λ P0 + λ

M −R

∑ (M − j) ( P + Q ) j

(22)

j

j =1

Some other performance measures to characterise the systems are •

the probability that the system is in recovery state is given by M −1

E[ RC ] =

∑Q

(23a)

j

j =1

•

the probability that the system is in reboot state is given by M −1

E[ RB ] =

∑R

(23b)

j

j =1

•

expected number of active units in the system is given by M −1

E[ A] = M P0 +

∑ (M − j) ( P + Q j

j

j =1

)

+ Rj .

(23c)

Now we discuss special cases corresponding to three repair time distributions viz. exponential distribution, gamma distribution and uniform distribution. The system performance indices for each case are established by setting Laplace transform of CDF of repair time B*(λ) and first moment of repair time distribution β1 = E[B]. To compute the probabilities of the system states and the steady state system availability of different configuration, we use equations (1) to (20) and (21), respectively.

4.1 Exponential distribution for repair time μ 1 , β = . Now we obtain steady λ+μ 1 μ state probabilities for the systems with two, three and four units using equations (18) to (21) as follows: For exponential distribution, we have B * (λ ) =

4.1.1 Two unit system P1 =

P2 =

Q1 =

(

2 1 − B* (λ ) B (λ ) *

) P == 2 ρ P

(

0

2 λβ1 − 1 + B* (λ ) B (λ ) *

2 λC

θ

P0 = aP0

0

)P

0

= 2 ρ 2 P0

(24a)

(24b) (24c)

Availability prediction of imperfect fault coverage system R1 = P0 =

2 λ (1 − C )

σ

383

P0 = bP0

(24d)

B* (λ ) = 1 + a + b + 2ρ + 2ρ 2 ⎛ 2λ C 2λ (1 − C ) ⎞ * 2λβ + B (λ ) ⎜ 1 + + ⎟ θ σ ⎝ ⎠

(

)

−1

(25)

where a=

2λ C

θ

, b=

2λ (1 − C )

σ

, ρ=

λ μ

For the two units system with 1-out-of-2: G configuration, the availability is A12 (∞) = P0 + P1 + Q1 =

1 + a + 2ρ 1 + a + b + 2ρ + 2ρ 2

(26)

4.1.2 Three unit system Now using equations (18) to (20), we obtain P1 = 3ρ P0 ;

(27a)

P2 = 6 ρ 2 P0

(27b)

Q1 = a0 P0 ,

(27c)

Q2 = a1 P1 = 3ρ a1 P0

(27d)

R1 = b0 P0 ,

(27e)

R2 = b1 P1 = 3ρ b1 P0

(27f)

P0−1 = 1 + a0 + b0 + 3ρ [1 + a1 + b1 + 2 ρ (1 + ρ ) ]

(28)

The availability for 2-out-of 3: G system and 1-out-of 3: G system are A23 (∞) = P0 + P1 + Q1 =

1 + a0 + 3ρ 1 + a0 + b0 + 3ρ ⎡⎣1 + a1 + b1 + 2 ρ (1 + ρ ) ⎤⎦

(29a)

A13 (∞) = P0 + P1 + P2 + Q1 + Q2 =

1 + a0 + a1 + 3ρ + 6 ρ 2 1 + a0 + b0 + 3ρ [1 + a1 + b1 + 2 ρ (1 + ρ ) ]

(29b)

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4.1.3 Four unit system For four unit system, using equations (18) to (20) we obtain the steady state probabilities as P1 = 4 ρ P0 ,

(30a)

P2 = 12 ρ 2 P0 ,

(30b)

P3 = 24 ρ 3 P0 ,

(30c)

P4 = 24 ρ 4 P0

(30d)

Q2 = a1 P1 = 4 ρ a1 P0 ,

(30e)

Q3 = a2 P2 = 12 ρ 2 a2 P0

(30f)

R2 = 4 ρ b1 P0 ,

(30g)

R3 = 12 ρ 3b2 P0

(30h)

P0−1 = γ 0 + 4 ργ 1 + 12 ρ 2γ 2 + 24 ρ 3 (1 + ρ )

(31)

where γi = 1 + ai + bi; i = 0, 1, 2. Now using equation (21), we get the availability for 3-out-of-4: G, 2-out-of-4: G, 1-out-of-4: G systems respectively, as follows: A34 (∞) = [ P0 + P1 + Q1 ] =

1 + a0 + 4 ρ

(32a)

γ 0 + 4 ργ 1 + 12 ρ γ 2 + 24 ρ (1 + ρ ) 2

3

A24 (∞) = P0 + P1 + P2 + Q1 + Q2 =

1 + a0 + a1 + 4 ρ + 24 ρ 2

(32b)

γ 0 + 4 ργ 1 + 12 ρ γ 2 + 24 ρ (1 + ρ ) 2

3

A14 (∞) = P0 + P1 + P2 + P3 + Q1 + Q2 + Q3 =

1 + a0 + a1 + a2 + 4 ρ + 24 ρ 2 + 24 ρ 3

(32c)

γ 0 + 4 ργ 1 + 12 ρ γ 2 + 24 ρ (1 + ρ ) 2

3

4.2 Gamma distribution for repair time To obtain state probabilities and steady state availability from equations (18) to (20) and ⎛ rμ ⎞ 1 (21) respectively, we set B *(λ ) = ⎜ ⎟ and β1 = . λ μ μ r + ⎝ ⎠

4.2.1 For two unit system P1 = 2 [η1 − 1] P0

(33a)

Availability prediction of imperfect fault coverage system

385

P2 = 2 {( ρ − 1)η1 + 1} P0

(33b)

P0−1 = 1 + a0 + b0 + 3ρη1

(34)

A12 (∞) =

1 + a0 + 2 (η1 − 1) 1 + a0 + b0 + 2 ρη1

(35)

4.2.2 For three unit system P1 =

3 [η2 − 1] P0 2

(36a)

P2 = 3 ⎡⎣(η1 − 1)(η2 − 1) ⎤⎦ P0 ;

(36b)

P3 = 3 {( ρ − 1)η1 + 1}{η2 − 1} P0

(36c)

⎛1 ⎞ P0−1 = 1 + a1 + a2 + 3 (η2 − 1) ⎜ + 3ρη1 ⎟ ⎝2 ⎠

(37)

3 (η2 − 1) 2 A23 (∞) = ⎛1 ⎞ 1 + a1 + a2 + 3 (η2 − 1) ⎜ + 3ρη1 ⎟ 2 ⎝ ⎠

(38a)

1⎞ ⎛ 1 + a1 + a2 + 3 (η2 − 1) ⎜η1 − ⎟ 2⎠ ⎝ A13 (∞) = ⎛1 ⎞ 1 + a1 + a2 + 3 (η2 − 1) ⎜ + 3ρη1 ⎟ ⎝2 ⎠

(38b)

1 + a1 +

4.2.3 Four unit system Using (18) to (20), we obtain the steady state probabilities as P1 = (4 / 3) (η3 − 1) P0

(39a)

P2 = 2 (η3 − 1) (η 2 − 1) P0

(39b)

P3 = 4 (η3 − 1) (η2 − 1)(η1 − 1) P0

(39c)

P4 = 4 (η3 − 1) (η 2 − 1){1 + η1 ( ρ − 1)} P0

(39d)

⎡ 2 ⎤ P0−1 = 1 + a1 + a2 + a3 + (η3 − 1) ⎢ − + 4η2 − 1 + 4η1 ρ (η2 − 1) ⎥ ⎣ 3 ⎦

(40)

Using (21) and above probabilities, we can easily obtain steady state availabilities A14(∞), A24(∞) and A34(∞) in explicit form.

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4.3 Uniform distribution e − βλ − e −αλ α +β , b1 = . Using 2 β −α equation (21), we obtain the availability of the system in different configurations. For 1-out-of-2: G system, we get

For the uniform distribution U(α, β), we set B* ( λ ) =

A12 (∞) =

5

(

)

2λ ( β − α ) + e − βλ − e −αλ (a − 1)

{(

)

}

λ β 2 − α 2 λ + (1 + a + b)( β − α )

(41)

Numerical results and sensitivity analysis

The validity and utility of analytical results can be explored by numerical experiments. System availability and failure frequency are the importance measures for the optimal design of any system working in machining environment. Here we are interested in examining the effect of different parameters on the system reliability and mean time to failure by taking the illustration. In this section, numerical results are provided for three unit repairable system with standbys, imperfect coverage and provision of reboot. For the computation of availability and failure frequency, software ‘MATLAB’ is used to develop the computer programme. To predict the effects of various parameters, numerical results for availability are summarised in Tables 1 and 2 for different repair time distributions. Furthermore, graphs are drawn by varying failure rate (λ) for exponential, uniform and gamma repair time distributions, in order to explore the effect of imperfect fault coverage (C), reboot (σ) and recovery rate (θ) in Figures 2 to 4, respectively. For computational purpose we set the default parameters as M = 3, C = 0.3, θ = 0.6, σ = 1, µ = 3. Table 1 reveals the trends of availability which increases as C and µ increase in case of exponential and gamma distributions. In case of uniform distribution, for the smaller value of µ (i.e., µ = 1), the availability is higher but as µ increases, it attains constant value. It is observed that for all the distributions of the repair time, the availability increases as C increases. For uniform distribution, there is no change in the availability when we change the value of σ; this may be due to constant repair time. In Table 2, we examine the effects of θ, µ and σ on the availability and note that as σ increases, the availability increases in case of exponential and gamma distribution. However, for uniform distribution as seen in Table 1, there is no change with respect to σ. Similar to Table 1, the availability increases as µ increases in case of exponential and gamma distribution but it remains almost constant for the uniform distribution except lower value of µ = 1. On increasing the values of C, the availability increases for the uniform distribution but decreases for the exponential distribution. It is noticed that for the gamma distribution, the availability also changes with respect to parameter C.

σ=3

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0.409

0.409

0.409

07

09

11

13

15

0.409

15

05

0.409

13

0.409

0.409

11

03

0.409

09

0.434

0.409

01

0.409

0.409

15

07

0.409

13

05

0.409

11

0.409

0.409

09

0.434

0.409

07

03

0.409

05

01

0.409

03

0.444

0.444

0.444

0.444

0.444

0.444

0.444

0.479

0.444

0.444

0.444

0.444

0.444

0.444

0.444

0.479

0.444

0.444

0.444

0.444

0.444

0.444

0.444

0.479

0.491

0.491

0.491

0.491

0.491

0.491

0.491

0.519

0.491

0.491

0.491

0.491

0.491

0.491

0.491

0.519

0.491

0.491

0.491

0.491

0.491

0.491

0.491

0.519

C = 0.9

0.892

0.884

0.874

0.859

0.838

0.801

0.728

0.498

0.869

0.862

0.852

0.838

0.817

0.782

0.711

0.490

0.807

0.800

0.792

0.779

0.761

0.730

0.667

0.467

0.911

0.903

0.892

0.877

0.855

0.817

0.741

0.505

0.896

0.888

0.878

0.863

0.842

0.805

0.731

0.500

0.855

0.847

0.838

0.824

0.804

0.770

0.701

0.484

C = 0.8

0.927

0.919

0.908

0.893

0.870

0.831

0.753

0.512

0.920

0.912

0.901

0.886

0.863

0.825

0.748

0.509

0.900

0.892

0.881

0.867

0.845

0.808

0.733

0.501

C = 0.9

Exponential distribution C = 0.7

0.899

0.891

0.880

0.865

0.843

0.804

0.725

0.477

0.854

0.847

0.837

0.823

0.802

0.767

0.694

0.464

0.806

0.800

0.791

0.779

0.760

0.728

0.662

0.451

0.910

0.902

0.891

0.876

0.853

0.814

0.734

0.485

0.896

0.888

0.877

0.862

0.840

0.802

0.724

0.479

0.854

0.847

0.837

0.823

0.802

0.767

0.694

0.464

C = 0.8

0.927

0.918

0.907

0.892

0.868

0.828

0.745

0.487

0.920

0.911

0.900

0.885

0.861

0.822

0.740

0.485

0.899

0.891

0.880

0.865

0.843

0.804

0.725

0.477

C = 0.9

Gamma distribution C = 0.7

Table 1

σ=2

σ=1

0.434

C = 0.8

Uniform distribution

C = 0.7

01

µ

Availability prediction of imperfect fault coverage system 387

Effects of parameters C, µ and σ on the availability for different distributions of repair time

σ=3

0.429

0.429

0.429

0.429

0.429

0.429

0.429

0.429

0.429

0.429

0.429

0.429

0.429

0.429

0.429

0.429

0.429

0.429

0.429

0.429

0.429

0.429

0.429

0.429

01

03

05

07

09

11

13

15

01

03

05

07

09

11

13

15

01

03

05

07

09

11

13

15

θ = 0.7

0.502

.502

0.502

0.502

0.502

0.502

0.502

0.502

0.502

0.502

0.502

0.502

0.502

0.502

0.502

0.502

0.502

0.502

0.502

0.502

0.502

0.502

0.502

0.502

θ = 0.8

0.469

0.469

0.469

0.469

0.469

0.469

0.469

0.469

0.469

0.469

0.469

0.469

0.469

0.469

0.469

0.469

0.469

0.469

0.469

0.469

0.469

0.469

0.469

0.469

θ = 0.9

Uniform distribution

0.805

0.798

0.790

0.778

0.760

0.730

0.668

0.469

0.750

0.745

0.738

0.727

0.711

0.684

0.629

0.449

0.624

0.620

0.615

0.608

0.596

0.577

0.537

0.399

θ = 0.7

0.779

0.774

0.766

0.755

0.738

0.710

0.652

0.463

0.718

0.713

0.707

0.697

0.683

0.659

0.610

0.442

0.581

0.577

0.573

0.567

0.558

0.543

0.509

0.388

θ = 0.8

0.763

0.757

0.750

0.740

0.724

0.698

0.643

0.459

0.697

0.693

0.687

0.678

0.666

0.644

0.598

0.437

0.554

0.551

0.548

0.543

0.535

0.522

0.493

0.381

θ = 0.9

Exponential distribution

0.805

0.798

0.790

0.778

0.760

0.729

0.666

0.464

0.750

0.745

0.737

0.727

0.711

0.684

0.628

0.445

0.624

0.620

0.615

0.607

0.596

0.576

0.535

0.395

θ = 0.7

0.779

0.774

0.766

0.755

0.739

0.711

0.654

0.467

0.718

0.713

0.707

0.698

0.684

0.660

0.611

0.446

0.718

0.713

0.707

0.698

0.684

0.660

0.611

0.446

θ = 0.8

0.763

0.758

0.751

0.741

0.725

0.700

0.646

0.469

0.697

0.693

0.687

0.679

0.667

0.645

0.601

0.446

0.554

0.552

0.548

0.544

0.536

0.523

0.496

0.389

θ = 0.9

Gamma distribution

Table 2

σ=2

σ=1

µ

388 M. Jain

Effects of parameters θ, µ and σ on the availability for different distributions of repair time

Availability prediction of imperfect fault coverage system Figure 2

Availability vs. λ by varying C for (a) exponential (b) uniform and (c) gamma distributed repair time (see online version for colours)

1

c=0.7 c=0.8 c=0.9

Availability

0.9 Availability

389

0.8

0.7

1

c=0.1

0.9

c=0.2 c=0.3

0.8 0.7 0.6 0.5 0.4

0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.3

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

λ

λ

(a)

(b) c=0.7

1

c=0.8

Availability

0.9

c=0.9

0.8 0.7 0.6 0.5 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

λ (c) Figure 3

Availability vs. λ by varying σ for (a) exponential (b) uniform and (c) gamma distributed repair time (see online version for colours)

1 0.9

σ =1 σ =2 σ =3

1

σ =1 σ =2 σ =3

0.9

Availability

Availability

0.8

0.8 0.7

0.7 0.6 0.5

0.6

0.4

0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.3

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

λ

λ

(a)

(b)

1

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M. Jain

Figure 3

Availability vs. λ by varying σ for (a) exponential (b) uniform and (c) gamma distributed repair time (continued) (see online version for colours) σ =1 σ =2 σ =3

1 0.9 Availability

0.8 0.7 0.6 0.5 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

λ (c) Figure 4

Availability vs. λ by varying θ for (a) exponential (b) uniform and (c) gamma distributed repair time (see online version for colours) θ=0.4

1

θ=0.9

0.8 Availability

0.8 0.7 0.6

0.7 0.6 0.5 0.4

0.5

0.3

0.4 0.1

0.2

0.3

0.4

0.5

λ

0.6

0.7

0.8

0.9

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

λ

(a)

(b) θ=0.4

1

θ=0.6 θ=0.8

0.9 Availability

Availability

θ=0.8

0.9

θ=0.8

0.9

θ=0.7

1

θ=0.6

0.8 0.7 0.6 0.5 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

λ

(c)

1

Availability prediction of imperfect fault coverage system

391

For exponential, uniform and gamma uniform distributed repair time, in Figures 2(a) to 4(c), we examine the availability by varying λ and different values of C, σ and θ, respectively. In all Figures 2(a) to 4(c), we see that the availability decreases as λ increases but decreasing pattern is more prevalent for lower values of λ. These figures also reveal the decreasing (increasing) trends of availability for the decrement in the values of c and σ (θ). In Figure 6, the numerical results obtained analytically for the availability are compared with the neuro-fuzzy results by constructing adaptive network-based fuzzy inference system (ANFIS) in MATLAB using neuro-fuzzy tool. Since fuzzy parameter is characterised by its membership; we use Gaussian function for describing the membership function. For the training purpose, ANFIS is trained for 100 epochs. In Figure 5, fuzzy membership functions by considering five linguistic values (say very low, low, average, high, very high) for the input parameters λ is shown for the variation in availability depicted in Figures 6(a) to 6(b). Figures 7(a) to 7(d) display the failure frequency with respect to parameter C and different values of µ, θ, β and λ for exponential distributed repair time. In Figure 5, we depict the shape function of input parameter λ. For different values of parameter r of the gamma distribution, in Figures 6(a) to 6(d), we plot the availability with the variation in the service rate (µ), arrival rate (λ), reboot rate (θ) and recovery rate (σ), respectively. The analytical results as well as nero-fuzzy results are displayed. As expected, the availability slightly decreases when r increases. In Figures 6(a) and 6(d), the availability increases with respect to µ and σ, respectively; the increasing trend is more significant for the lower values of the parameters. On the contrary, in Figures 6(b) and 6(c), we notice that the availability diminishes as the arrival rate (λ) and θ decrease. The numerical results obtained by using the soft computing approach shown by tick marks are quite close to the analytical results which are shown by continuous curves.

Degree of membership

Figure 5 Membership functions for input parameter (a) μ (b) λ (c) λs (d) σ (see online version for colours) in1mf1

1

in1mf2

in1mf3

in1 mf4

in1mf5

0.8

0.6

0.4

0.2

0 1

1.2

1.4

1.6

1.8

2 2.2 input1

2.4

2.6

2.8

3

392

M. Jain For gamma distributed repair time, the availability vs. (a) μ (b) λ (c) λs (d) σ by varying r (see online version for colours)

Figure 6

r=0.5(Analytical Set 1) r=0.5(Afnis Set 1) r=2(Analytical Set 2) r=2(Afnis Set 2) r=3.5(Analytical Set 3) r=3.5(Afnis Set 3)

1

0.8

0.8

Availability

Availability

0.9

r=0.5(Analytical Set 1) r=0.5(Afnis Set 1) r=2(Analytical Set 2) r=2(Afnis Set 2) r=3.5(Analytical Set 3) r=3.5(Afnis Set 3)

1

0.7 0.6

0.6 0.4 0.2

0.5

0 0.4

0.5 1

1.5

2

2.5

μ

1

1.5

2

0.7

0.7

0.68

0.68

Availability

0.66

0.62

0.3 0.4 0.5 0.6 0.7 0.8 0.9

λs

1

0.6 0.5

1

1.5

2

2.5

3

5

1

0.5 0.4 0.3

Failure Frequency

0.7 0.6

0.8 0.6 0.4

3 2

1 0.1

0.2

(a)

0.3

0.4

c

0.5

0.6

0.2

0.3

0.2

μ

4.5

Failure Frequency by varying c and (a) μ (b) θ (c) σ (d) λ

0.9

4

4

(d)

0.8

5

3.5

σ

1

6

5

r=0.5(Analytical Set 1) r=0.5(Afnis Set 1) r=2(Analytical Set 2) r=2(Afnis Set 2) r=3.5(Analytical Set 3) r=3.5(Afnis Set 3)

(c) Figure 7

4.5

0.66 0.64

r=0.5(Analytical Set 1) r=0.5(Afnis Set 1) r=2(Analytical Set 2) r=2(Afnis Set 2) r=3.5(Analytical Set 3) r=3.5(Afnis Set 3) 0.1 0.2

4

1.1

c

0 1.1 1.3 0.9 0.7 0.3 0.5

θ

(b)

Failure Frequency

Availability

0.72

0.6

3.5

(b)

0.72

0.62

3

λ

(a)

0.64

2.5

3

Availability prediction of imperfect fault coverage system

393

Failure Frequency by varying c and (a) μ (b) θ (c) σ (d) λ (continued) 1

0.8 0.7 0.6 0.5 1.3

F ailu re F re q u en cy

0.9

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

0.4

1.1 0.9

0.7

c

0.1

1

0.9

F a ilu re F re q u e n c y

Figure 7

0.8

0

0.7

0.5

0.6 0.3

0.5 0.4

(c)

σ

0.6

0.5

0.4

λ

0.3

0.2

0.3 0.1

0.5

0.7

0.9

1.1

1.3

c

(d)

In Figures 7(a) to 7(d), by varying the coverage parameter C, we exhibit the failure frequency for different values of parameters µ, θ, λ and σ, respectively. It is observed that the failure frequency decreases as we increase µ and σ whereas the effect of λ, C and θ are reverse, i.e., the failure frequency increases as we increase these parameters. Overall we conclude that for all the distributions of repair time, the availability (failure frequency) decreases (increases) as λ, C and θ increase and µ and σ decrease. Based on numerical experiment we infer that the availability can be improved significantly by controlling the parameters C and θ because both descriptors prevent the entire system failure by replacing the available standbys with the failed units.

6

Concluding remarks

The concept of redundancy is prevalent in many day-to-day as well as industry where multi-component machining systems is equipped with standby units to replace the faulty unit. The present investigation on the assessment of system reliability indices for different configurations by applying the redundancy and maintainability approaches can play an important role to improve the availability and efficiency of the concerned real time systems. Further, the role of fault tolerant system in hardware and software computer systems is well established. With the advancement of technology, the fault tolerant software and hardware systems are likely to be more complex in future and the availability of the same will depend upon recovery procedure which includes the redundancy, reboot, repair/replacement measures. In many real time applications, fault-tolerant systems are expensive and time consuming to develop and deploy. In our study, we have established the explicit results for the availability and failure frequency of multi-component redundant systems. Many noble features such as

394

M. Jain

a

general repair time

b

imperfect fault coverage

c

reboot, have been incorporated.

The investigation done deals with the availability prediction of more versatile real time systems which has wide applicability as multi units system with the provision of redundancy as well as maintainability are quite popular in machining environment. The stochastic model developed may be helpful in the evaluation of the performance indices of real time fault-tolerant systems embedded in manufacturing/production systems, computer and communication systems, transportation and distribution networks, etc. The performance indices can be easily computed by using the soft computing approach as demonstrated. This shows the scope of the development of numeric controlled machining systems and may be helpful to the production engineers and system analysts for the design of ANFIS-based embedded engineering systems wherein desired level of reliability/availability of the system can be achieved. Our study can be further employed for the determination of the optimal number of standbys and repairmen under specified techno-economic constraints.

Acknowledgements The author would like to thank the anonymous referees and editor-in-chief of the journal for their valuable suggestions and critical comments which helped a lot in improving the quality and clarity of the paper. The grant from MHRD, India for research project is gratefully acknowledged.

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