IN modern society that various systems have become more

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1 Developent of Reliability Function in -Coponent Standby Redundant Syste with Priority Based on Maxiu Entropy Principle Ryosuke Hirata, Ikuo Arizono, Ryosuke Toohiro, Satoshi Oigawa, and Yasuhiko Takeoto Abstract A -coponent standby redundant syste with priority is well-known as one of the ost fundaental redundant odels in the reliability theory. There are any studies about the -coponent standby redundant syste with priority. In late years, under the situation that only the average and variance of failure and repair tie of each coponent are provided, the ethod for evaluating ean tie to failure MTTF) in the entire syste has been proposed. Furtherore, by expanding the idea to evaluate MTTF under such a situation, an evaluation ethod for the variance of failure tie in the entire syste has been established. In this study, the procedure for obtaining the reliability function in the entire syste by utilizing the ean and variance of syste failure tie is developed based on the axiu entropy principle. Index Ters ethod of Lagrange ultiplier, axiu entropy principle, syste reliability, -coponent standby redundant syste with priority. I. INTRODUCTION IN odern society that various systes have becoe ore coplex and coplicated with progress of technology, high reliability in various systes in the society have been required. Hence, the research and developent in the reliability field is active. In the details, see textbooks in the reliability field such as Gertsbakh [] and Birolini []. A -coponent redundant syste with priority is wellknown as one of the ost fundaental redundant odels in the reliability theory. There are any studies about the -coponent standby redundant syste with priority. Zhang and Wang [3] and Yuan and Meng [4] have discussed a - coponent standby redundant syste with priority in use. Also, Leung et al. [5] have investigated a standby redundant syste with priority in repair. As their results, soe iportant reliability indices and optial operating policy of coponents in -coponent redundant syste have been suggested. Note that it has been assued in their studies that all distributions about failure and repair ties of each coponent are respectively specified and exactly provided as respective exponential distributions. Manuscript received Deceber, 7. R. Hirata is with Faculty of Engineering, Okayaa University, Okayaa, Japan e-ail: pklk9cdr@s.okayaa-u.ac.jp). I. Arizono is with Graduate School of Natural Science and Technology, Okayaa University, Okayaa, Japan e-ail: arizono@okayaa-u.ac.jp). R. Toohiro is with Graduate School of Natural Science and Technology, Okayaa University, Okayaa, Japan e-ail: pvkka7i@s.okayaau.ac.jp). S. Oigawa is with Graduate School of Natural Science and Technology, Okayaa University, Okayaa, Japan e-ail: p5pl9os@s.okayaau.ac.jp). Y. Takeoto is with Faculty of Science and Engineering, Kindai University, Osaka, Japan e-ail: takeoto@ech.kindai.ac.jp). On the other hand, Buzacott [6] and Osaki [7], [8] have considered the -coponent redundant syste consisting of priority and standby coponents, where the priority coponent is used whenever it is available. When the priority coponent falls in failure, it starts to be repaired iediately and the standby coponent starts to operate. If the standby coponent falls in failure during repairing the priority coponent, the entire syste becoes down. Then, under the assuption that the respective failure and repair ties of both coponents are explicitly provided as known probability distributions, Buzacott [6] and Osaki [7], [8] have derived individually their theoretical forulae of ean tie to failure MTTF) of the entire syste. Note that the evaluation forulae of MTTF derived by Buzacott and Osaki require any probability density functions PDFs) and/or cuulative distribution functions CDFs) about failure and repair ties in the respective coponents. That is, the theoretical evaluation ethods of MTTF of the -coponent standby redundant syste with priority suggested by Buzacott and Osaki have required the prerequisite that the probability distributions of failure and repair ties in each coponent are respectively specified as particular functions. However, it is considerably difficult to satisfy this prerequisite in practice. If we cannot identify explicitly the probability distributions about failure and repair ties in each coponent, the ethods entioned above would not be applicable. In contrast, the ean and variance are the required iniu inforation on characterizing the probability distribution. In ost practical cases, the ean and variance in a probability distribution will be at least known based on historical inforation. Hence, also for the probability distributions about failure and repair ties in respective coponents, we can assue that respective the ean and variance of failure and repair ties in respective coponents are respectively provided based on historical inforation. In recent years, Takeoto and Arizono [9] have proposed the evaluation ethod of MTTF in the -coponent standby redundant syste with priority under the liited inforation that only ean and variance about failure and repair ties in the respective coponents are provided. Then, Takeoto and Arizono have considered the situation that the probability distributions about failure and repair ties can be approxiated as a ixed Erlang distribution. Under this situation, Takeoto and Arizono develop a new approxiation procedure for evaluating the MTTF of the syste by cobining soe results of the Markov analysis based on Erlang distributions. As the result, the approxiate forula for coputing the MTTF in the the -coponent standby

2 redundant syste with priority has been constructed as the closed for under the liited inforation that only ean and variance about failure and repair ties are provided. Furtherore, through expanding the evaluation procedure considered by Takeoto and Arizono [9], Oigawa et al. [] have proposed the evaluation of the variance of failure tie in the -coponent redundant standby syste with priority in the case that only ean and variance about failure and repair ties of each coponent are provided. It goes without saying that MTTF is effective as an index for evaluating the reliability of the syste. Siilarly, the variance of failure tie in the -coponent standby syste with priority is also an iportant index. Through the above researches, we can evaluate MTTF and the variance of failure tie quantitatively. Then, we can evaluate qualitatively the reliability of a -coponent standby redundant syste with priority by MTTF and variance of failure tie. For exaple, even if MTTF is large and in the case that the variance of failure tie is relatively large, it is iagined to cause a proble in reliability. In this study, we consider the reliability function in order to evaluate quantitatively the syste reliability in the situation that only ean and variance about failure and repair ties are provided as the liited inforation. Since the ean and variance of failure tie in the entire syste have been evaluated based on the considerations by Takeoto and Arizono [9] and Oigawa et al. [], we consider to develop the probability density function about failure tie in the entire syste based on the the ean and variance of failure tie in the entire syste. However, there are any probability distributions with the sae values in the first and second order oents. Hence, we adopt the axiu entropy principle to develop the syste reliability function through identifying the the probability density function about failure tie in the entire syste under the constraints of the ean and variance of failure tie in the entire syste. The procedure based on the axiu entropy principle in the inforation theory is widely known as an elegant approach for deriving the probability density function under the constraints of the values of oents in the probability distribution. II. DETAILS OF -COMPONENT STANDBY REDUNDANT SYSTEM WITH PRIORITY We consider the -coponent standby redundant syste coposed of Coponent having priority and Coponent which is a standby coponent. In this case, the coponent with priority eans a coponent which is always used whenever it is available. In contrast, the standby coponent eans a coponent which stands by whenever the coponent with priority is available. When the priority coponent falls in failure, it starts to be repaired iediately and the standby coponent starts to operate. If the standby coponent falls in failure during repairing the priority coponent, the entire syste becoes down. The outline of behavior in this syste is illustrated in Figure. The solid line represents the operating status of each coponent. The dashed line represents the repair status of Coponent, the dasheddotted line represents the standby status of Coponent. In addition, the dotted arrowhead connecting Coponent and Coponent represents the switching of coponents. Fig.. Coponent priority Coponent) Coponent standby Coponent) in operation in standby in repair switching -coponent standby redundant syste with priority Syste Down Initially, at tie t =, Coponent is working, and Coponent is in the standby state. At this tie, the standby tie of the Coponent is not included in its lifetie. When Coponent fails, it starts to be repaired iediately and then Coponent starts to work. If the repair of Coponent has been copleted and Coponent is still working, then Coponent starts to work iediately, and Coponent is in a standby state. However, if Coponent fails during the repair of Coponent, the entire syste goes down. The switching of coponents is reliably executed, and it is assued that the switchover tie is instantaneous. III. OUTCOME OF PREVIOUS LITERATURE For the -coponent standby redundant syste of Figure, Buzacott [6] has derived MTTF under the situation that the failure and repair tie distributions of each coponent are explicitly prescribed as individual specific probability distributions On the other hand, under the situation that Coponent has an exponential failure tie distribution, Osaki [7], [8] has derived MTTF of the -coponent standby redundant syste of Figure. All the distribution functions of the failure and repair ties in each coponent have to be explicitly specified for evaluating MTTF by using the theoretical forulae by Osaki and Buzacott. However, in soe practical situations, it is difficult to know exactly the probability distributions of failure and repair ties. On the other hand, in ost practical cases, the ean and variance about failure and repair ties in respective coponents will be at least known based on historical inforation. In recent years, Takeoto and Arizono [9] and Oigawa et al. [] have respectively considered the evaluations of ean tie to failure and variance of failure tie in the - coponent standby redundant syste illustrated in Figure. In these evaluations, the followings have been assued: i) The failure tie distribution of Coponent has the cuulative distribution function CDF) F t) with ean E F = /λ and variance V F. ii) The repair tie of Coponent obeys the probability distribution G with ean E G and variance V G. iii) The failure tie of Coponent obeys the probability distribution F with ean E F and variance V F. They have considered the approxiation where the repair or failure) tie distribution in Coponent or ) is

3 transfored into a ixed Erlang distribution under the constraints of ean E G or E F ) and variance V G or V F ). Then, the syste has been reconstructed by four types of tentative systes based on the cobinations of respective Erlang distributions in the ixed Erlang distributions for the repair tie distribution of Coponent and the failure tie distribution of Coponent. The respective tentative systes have been analyzed using Markov process theory because the state transition in the respective tentative systes can be expressed by a state transition diagra based on an exponential distribution. MTTFs in the respective tentative systes are weightedly sued up using the ratios of ixture on the ixed Erlang distribution. As a consequence, Takeoto and Arizono [9] have defined MTTF of the entire syste as follows: where MT T F = p p MT T F n,n ) + p )p MT T F n+,n ) + p p )MT T F n,n +) + p ) p )MT T F n+,n +) µ, ) MT T F n,n ) = R n,n )) n /λ ã) = b d jn)) + b n j i + k k bi ) c k ) d jn)), i= n i + < φ i n i, 3) p i = n i + )φ i n i + ) n i φ i ) φ i +, 4) φ = V G /E G, 5) φ = V F /E F, 6) µ = n + p E G, 7) λ = n + p E F, 8) R n,n )s) = ũ n)s) i= n + ãs)ṽ n)s) i= d in)s) n j i + k b i s) c k s) d jn)s), 9) ãs) = s + µ + λ, ) bs) = µ s + µ + λ, ) k cs) = λ s + µ + λ, d n)s) =, 3) d jn)s) = w n)s) j n + c j k s) d kn)s), 4) ũ n)s) = ṽ n)s) = w n)s) = R s) s R s) b n s), 5) s R s) s R s) b n s), 6) s R s) b n s) s R s) b n s). 7) The notations of MT T F n,n ), MT T F n+,n ), MT T F n,n +) and MT T F n+,n +) in equation ) denote the ean tie to failure in the respective tentative systes. Further, the sybols of p i, φ, φ, µ and λ denote the paraeters in the case of approxiating the repair tie distribution of Coponent and the failure tie distribution of Coponent to the respective ixed Erlang distributions. In addition, the notations of R n,n )s), ãs), bs), cs), djn)s), ũ n)s), ṽ n)s) and ũ n)s) ean the functions derived by Takeoto and Arizono [9] for evaluating MTTF in the -coponent standby redundant syste. Further, R t) = F t) and then, R s) denotes the Laplace transforation for R t). Further, Oigawa et al. [] have addressed the derivation of the variance of the failure tie of the considered - coponent standby redundant syste with priority. Denote the probability density function PDF) of the failure tie distribution of the entire syste by ft). Then, the Laplace transforation for ft) is defined as: fs) = e st ft)dt. 8) On the other hand, since the reliability function Rt) of the entire syste is expressed as Rt) = t fτ)dτ, 9) the relationship between the Laplace transforations fs) and Rs) can be given as fs) = s Rs). Then, by applying the final-value theore and the L Hopital s rule to the differential calculus fors for this relationship, the following equations can be shown: d ds fs) s= f ) ) = E[t] = µ, d ds fs) s= f ) = E[t ], where superscript i) of the function differentiation of the function fs). f i) ) eans i-th

4 Therefore, the variance V [t] of the failure tie of the considered -coponent standby redundant syste with priority can be obtained as where V [t] = f ) f ) ) σ, ) f ) = p p f n,n ) ) + p )p f n +,n ) ) + p p ) f n,n ) ) = ũ) ũ ) + p ) p ) n ) ) n ũ n)) f n ),n +) + h n,n )) ã) f n +,n +) ), 3) d jn)) d ) jn ) ) n j i + k + ã)ṽ n)) k i= b i ) c k ) d jn))ã) + i + k ) d ) jn ) ), 4) V n ) = F + λ ) b λ bn ) n ã) + λ b, 5) d ) jn ) ) = w) n ) ) j n + c j k ) d kn)) j n + w n)) )ã) c j k ) d kn)) j n + + w n)) c j k ) w ) n ) = ṽ h n,n )) b h n,n )) = R n,n )) ) d kn ) ), 6) n ã) b ṽ, 7) bn ) + R n,n )) + n ã) b b, 8) i) where note that the notations of f n,n ) ), ũ) n ), ) d ) jn ) ) and w) n ) are siilar to the notation of f i) ). f n +,n ) ) f n,n +) ) and ) can be obtained. Reind that the concrete Through the siilar way, f n +,n +) atheatical expressions of f n +,n ) and f ) f n,n +) ) n +,n +) ) are oitted for convenience of the nuber of the paper. IV. DERIVATION OF RELIABILITY FUNCTION In this study, based on µ and σ derived as equations ) and ), we evaluate the probability density function ft) of the failure tie of the entire syste. As for derivation of µ and σ, we have considered that only the ean and variance of failure and repair ties in each coponent are given as the liited inforation. In other words, the basic condition in this study is that the distribution of failure and repair ties in each coponent is unknown and not specified. Hence, we do not suppose a specific distribution type to evaluate the probability density function ft) of the failure tie in the entire syste. Then, it is widely known that the axiu entropy principle brings an elegant solution for deriving the probability density function under the constraints of the values of oents. Therefore, we apply the axiu entropy principle to obtain the probability density function ft) of the failure tie in the entire syste. Based on the axiu entropy principle, the identification proble of ft) can be forulated as follows: Maxiize H f) = ft) f x) log f x) dx, 9) subject to f x) dx =, 3) xf x) dx = µ, 3) x µ) f x) dx = σ. 3) By using equations 9) 3), a Lagrange function is defined with introducing Lagrange ultipliers α, α and α 3 as V = H f) α f x) dx α xf x) dx µ α 3 x µ) f x) dx σ. 33) Through adopting the variational ethod to the Lagrange function V, we have δv = log fx) + δf α α x α 3 x µ) =. 34) Fro equation 34), f x) is expressed as fx) = e α e αx e α3x µ). 35) At the sae instant, by partially differentiating V with α, α and α 3, we have equations 3) 3). Therefore, based on equations 3) 3) and 35), fx) is rewritten as f x) = Ce x ) δ, 36)

5 where C = e α α3µ e α3 µα3 α α 3 ), 37) = µα 3 α α 3, 38) δ = α 3. 39) Accordingly, the derivation of f x) becoes a proble of finding C, and δ. Furtherore, by substituting equation 36) for equation 3), 3) and 3), the following equations can be shown: C C C e x ) δ dx =, 4) xe x ) δ dx = µ, 4) x µ) e x ) δ dx = σ. 4) Then, fro equations 4), we can derive as follows: C = e x ) δ dx. 43) Moreover, by ultiplying both sides of equation 43) by /, we obtain the following equation: C = + e x ) δ dx e x ) δ dx. 44) The right side eans the upper side probability function of the noral distribution with ean and variance δ. Through the variable transforation of t = x ) /δ, equation 44) is converted to C = π δ e t dt +. 45) Further, by replacing t by τ, equation 45) is expressed as C = + π = e τ δ dτ δ π e τ dτ = erf ). 46) δ Then, the function erf z) is known as the error function as follows: erf z) = z e t dt. 47) π The error function is frequently used in probability theory, statistics, aterial science, and partial differential equation. Further, the error function and confluent hypergeoetric function have the following relationship: erfx) = x F π ; 3 ) ; x. 48) ) Note that the confluent hypergeoetric function is classified as special functions. Then, the confluent hypergeoetric function is defined as F β; γ; z) = Γγ) Γβ) n= Γβ + n) z n Γγ + n) n!. 49) Furtherore, the following Kuer transforation can be applied to be confluent hypergeoetric function: F β; γ; z) = e z F γ β; γ; z). 5) Consequently, based on equation 48) and 5), equation 46) is transfored as follows C =. 5) πδ + e δ F ; 3 ; δ Further, as for equations 4) and 4), we have respectively the following equations: C = µ δ e δ, 5) C = µ + σ δ δ e δ. 53) Then, based on equations 5) and 53), the following equations can be shown: δ = µ + µ + σ. 54) Accordingly, by substituting equation 54) for equation 5) and 5) or 53)), we can obtain the following function for : π µ + µ + σ µ ) ) +e µ+µ +σ ) F, 3 ; µ + µ + σ ) ) µ + µ + σ )e µ+µ +σ ) =. 55) It is difficult to solve analytically equation 55) for because that equation 55) contains the confluent hypergeoetric function. However, we can derive the value of satisfying equation 55) nuerically by appropriate search algoriths such as linear search. It is clear that we can get the values of δ and C by calculated. Moreover, by substituting equation 36) for equation 9), we can derive as follows. πδ Rt) = C t ) ) e t ) δ F ; 3 t ) ; δ. 56) Thus, the reliability of the entire syste can be evaluated concretely by substituting calculated, δ and C for equation 56).

6 R t) V. NUMERICAL ANALYSIS In this section, we investigate the validity of our proposal on evaluating the reliability function with the following procedures: i) The ean and variance of failure and repair tie in Coponent are respectively given. Further, the ean and variance of failure tie in Coponent are also given. ii) The probability distributions for the failure and repair tie in Coponent and the probability distribution for the failure tie in Coponent are individually specified as a particular probability distribution. iii) Through a illion ties of trials iterated in the Monte Carlo siulation under the cobination of specified probability distributions, the reliability in the entire syste on tie t is estiated. iv) The proposed reliability function is copared with the reliability estiated through the Monte Carlo siulation. As an exaple, suppose that E F = /λ =., V F =., E G = 5., V G =.5, E F = 5., and V F =.5, respectively. Moreover, the failure and repair tie distributions in both coponents are respectively defined as the log-noral and/or Weibull distributions. The log-noral distribution and Weibull distribution are well known in the reliability field. Based on the above nuerical values and the supposition for the failure and repair tie distributions, the values of µ and σ for the failure ties in the entire syste are calculated as the following nuerical value: µ, σ ) = 6.89, ) Fig.. Coparison of reliability based on proposed reliability function and the siulation results VI. CONCLUSION In this study, we have addressed the evaluation ethod of the reliability function in the -coponent standby redundant syste with priority. Then, by utilizing the evaluation results for the ean tie to failure by Takeoto and Arizono [9] and the variance of the failure tie by Oigawa et al. [], we have succeeded in deriving the evaluation ethod of the reliability function in the -coponent standby redundant syste with priority. In concrete, by using only the ean and variance of the failure tie in the entire syste, the reliability function in the -coponent standby redundant syste with priority has been derived based on the axiu entropy principle. Then, it would like to be the future issue to establish the way of the optial preventive aintenance of the syste by using the reliability indices such as the reliability function. t Further,, δ and C are obtained as follows:, δ, C ) = 57.84, 34587, 44,.74). So, we can evaluate the reliability function in equation 56) by applying the above values of, δ, C ). One of the results of coparing the siulation and the proposed reliability is shown in Figure. In Figure, the failure tie distribution F t) and the repair tie distribution G t) in Coponent and the failure tie distribution F t) in Coponent are respectively supposed as the Weibull distribution with ean E F and variance V F, the log-noral distribution with ean E G and variance V G and the Weibull distribution with ean E F and variance V F. Fro Figure, it has been confired that the shape of the reliability function proposed in this study resebles that of siulation result. Moreover, reark that siilar results have been obtained under different conditions. Consequently, it has been confired that the proposed reliability function based on the axiu entropy principle is useful as an approxiation ethod for the reliability evaluation under the condition that only the ean and variance of failure tie in the entire syste can be used. REFERENCES [] I. Gersbakh, Reliability Theory with Applications to Preventive Maintenance, Springer-Verlag,. [] A. Birolini, Reliability Engineering: Theory and Practice 6th ed.), Springer-Verlag,. [3] Y. L. Zhang, G. J. Wang, A Deteriorating Cold Standby Repairable Syste with Priority in Use, European Journal of Operation Research, Vol. 83, No., pp , 7. [4] L. Yuan, X. X. Meng, Reliability Analysis of a War Standby Repairable Syste with Priority in Use, Applied Matheatical Modeling, Vol. 35, No. 9, , [5] K. N. F. Leung, Y. L. Zhang, K. K. Lai, Analysis for a Two-dissiilarcoponent Cold Standby Repairable Syste with Repair Priority, Reliability Engineering & Syste Safety, Vol. 96, No., pp ,. [6] J. A. Buzacott, Availability of Priority Standby Redundant Systes, IEEE Transactions on Reliability, Vol. R-, No., pp ). [7] S. Osaki, Reliability Analysis of a Two-unit Standby Redundant Syste with Priority, Canadian Journal of Operations Research, Vol. 8, No., pp ). [8] S. Osaki, Stochastic Models in Reliability and Maintenance, New York, Springer,. [9] Y. Takeoto, I. Arizono, A Study of MTTF in Two-unit Standby Redundant Syste with Priority under Liited Inforation about Failure and Repair Ties, Journal of Risk and Reliability, Vol. 3, No., pp ). [] S. Oigawa, I. Arizono, R. Toohiro, Y. Takeoto, Variance Evaluation of Tie to Failure in -Coponent Standby Redundancy Syste with Priority under Liited Inforation, Journal of Japan Industrial Manageent Association, Vol. 68, No., pp. 3 7) in Japanese).