Analysis of a deteriorating cold standby system with priority

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1 Lixia Ma, Genqi Xu, Nikos E. Mastoakis Analysis of a deteioating cold standby system with pioity Lixia Ma Tianjin Univesity Depatment of Mathematics Tianjin, 372 P. R. China lixiama@tju.edu.cn Genqi Xu Tianjin Univesity Depatment of Mathematics Tianjin, 372 P. R. China gqxu@tju.edu.cn Nikos E. Mastoakis Militay Institutes of Univesity Education Hellenic Naval Academy Tema Hatzikyiakou, Piaeus, Geece masto@hna.g Abstact: A deteioating cold standby epaiable system consisting of two dissimila components and one epaiman is studied in this pape. Suppose that the life of each component satisfies the exponentially distibution and epai time of the component satisfies the geneal distibution, the component 1 has pioity in use and epai. Fistly, a mathematical model is built via the diffeential and patial diffeential equations. And then using the C -semigoup theoy of bounded linea opeatos, the existence and uniqueness of the solution, the non-negative steady-state solution and the exponential stability of the system ae deived. Based on the stability esult, some eliability indices of the system and an optimization poblem ae pesented at the end of the pape. Key Wods: C Semigoup, Well-Posedness, Asymptotic Stability, Exponential Stability, Availability. 1 Intoduction With the development of the moden technology and extensive use of electonic poducts, the eliability poblem of the epaable systems has become a hot topic. It is well known that the edundancy can enhance the eliability of the system. In the ealie study, the standby component doesn t fail in its standby peiod since they ae machine system. Howeve, the electonic poduct doesn t satisfy this popety. In fact, fo an electonic poduct, it has less failue ate if it is in use, othewise, it has geate failue ate afte cetain time, which means that it deteioates. Theefoe, in this pape we will discuss the eliability of a system consisting of the electonic poduct with epai. Fo the eliability poblem of system, it has been a hot topic in engineeing and Mathematics. Using mathematical model and Makov enewal theoy, one studied the indices of eliability of the epaable system, fo example, see [1], [2], [3]. Notice that, in the ealie eseach, the component afte epai is as good as new, only ecent a deteioating case is consideed in ([8]). Although some nice esults have been obtained, thee exists cetain difficulty to obtain some index of the system, fo instance, how long time one can see the stable state of the system. The mainly difficulty comes fom the mathematical model; this is because one cannot give an exact solution to the model. To ovecome this difficulty, many authos have woked on the analysis of the epaable systems using functional analysis method (see [4]-[6], etc), moe pecisely, semigoup theoy of bounded linea opeatos ([7]) to pove the well-posedness and the stability of the system. In this pape, we will give a new model about a cold standby epaiable system, and then discuss the system by functional analysis method. The est is oganized as follows. In section 2, a mathematical model fo the system unde consideation is established. And then in section 3, the existence and uniqueness of nonnegative time-dependent solution of the system ae pesented via C semigoup theoy. In section 4, by analyzing spectal distibution of the system opeato, the main esults on stability of the system ae obtained. In section 5 some indices of stationay state of the system and the estimation of instantaneous availability of the system ae given. Finally, an optimal epai ate poblem is studied. 2 Modeling fo a system with pioity Fistly, we descibe the system unde consideation. Suppose that a system consists of two dissimila components and one epaiman, the component 1 is the main woking unit and the component 2 is cold standby unit. The component 1 has pioity in use and epai. The system satisfies the following assumptions: Assumption 1. Initially, the two components ae both new, and component 1 is in a woking state while component 2 is in a standby state. Assumption 2. Assume that both components afte epai ae as good as new. ISSN: Issue 2, Volume 1, Febuay 211

2 Lixia Ma, Genqi Xu, Nikos E. Mastoakis Assumption 3. The component 1 has pioity in woking and epai. When both components ae good, the component 1 has the highe use pioity than the component 2, even if component 2 is woking, it must be switched immediately into the standby state as soon as component 1 afte epaied so that the component 1 becomes the woking state; When both components fail (i.e. the system is down), component 1 has the highe epai pioity than component 2, even if the epaiman is epaiing component 2, he must switch to epai the component 1. He will wok on the epai of component 2 afte completing the epai on component 1. Assumption 4. The standby component will pehaps fail in the standby time fo some eason. The failue and epai times of both components follow exponential distibution and geneal distibution espectively. λ j and ε denote the failue ate of component j(j 1, 2) and the standby component; µ j (x) denote the epai ate of component j(j 1, 2). Assumption 5. All failues ae independent of each othe. Unde these assumptions, we can divide the system into the following states: : The components 1 and 2 ae in good condition; 1: The component 1 is failue unde epai and the component 2 is woking; 2: The component 1 is woking and component 2 is failue unde epaiing; 3: The components 1 and 2 ae failue unde epai. P (t) denotes the pobability that component 1 is in woking state and component 2 is in cold standby state at time t; p 1 (t, x) epesents the pobability that the epaiman is dealing with component 1 with the elapsed time lying in [x, x + ) and component 2 is in wok at time t; p 2 (t, x) epesents the pobability that the epaiman is dealing with component 2 with the elapsed time lying in [x, x + ) and component 1 is in wok at time t; p 3 (t, x) epesents the pobability that the epaiman is dealing with component 1 with the elapsed time lying in [x, x + ) and component 2 is waiting fo epai at time t. By the supplementay vaiables technique, the model of the system can be fomulated as dp (t) dt (λ 1 + ε)p (t) + µ 1 (x)p 1 (x, t) + µ 2 (x)p 2 (x, t), p 1 (x,t) t p 1(x,t) x (µ 1 (x) + λ 2 )p 1 (x, t), p 2 (x,t) t p 2(x,t) x (µ 2 (x) + λ 1 )p 2 (x, t), p 3 (x,t) t p 3(x,t) x µ 1 (x)p 3 (x, t). (1) The bounday conditions ae p 1 (, t) λ 1 P (t), p 2 (, t) εp (t) + µ 1 (x)p 3 (x, t), p 3 (, t) λ 1 p 2 (x, t) + λ 2 p 1 (x, t), (2) whee (x, t) R + R +. 3 The well-possedness of the system solution In this section, we will discuss the well-posed-ness of the system. To this end, we fomulate the system (1) into a suitable Banach space. Based on the pactical backgound of ou model, we can assume that µ j (x)(j 1, 2) satisfy M sup x R + µ j (x) < ; µ j (s)ds. (3) In the sequel, we denote by R + [, ). Fom the physical meaning of the poblem, we take the s- tate space X R [L 1 (R + )] 3, equipped with the nom P P + 3 p j L 1, fo each P (P, p 1, p 2, p 3 ) X. Obviously, X is a Banach space. Define an opeato A : X X by A P p 1 (x) p 2 (x) p 3 (x) with domain D(A) P X (λ 1 + ε)p + 2 µ j(x)p j (x) ( d µ 1(x) λ 2 )p 1 (x), ( d µ 2(x) λ 1 )p 2 (x), ( d µ 1(x))p 3 (x) (4) p j (x) is absolutely continuous, p j (x), p j (x) L1 (R + ), p 1 () λ 1 P, p 2 () εp + µ 1(x)p 3 (x), p 3 () λ 1 p 2(x) + λ 2 p 1(x), j 1, 2, 3, (5) Obviously, A is a linea opeato in X. By the definition of A, the system (1) can be ewitten as an abstact Cauchy poblem in Banach s- pace X: { dp (t) dt AP (t), t, (6) P () (1,,, ). ISSN: Issue 2, Volume 1, Febuay 211

3 Lixia Ma, Genqi Xu, Nikos E. Mastoakis whee P (t) (P (t), p 1 (x, t), p 2 (x, t), p 3 (x, t)). By Theoem II 6.7 and Definition II 6.8 of [9], we know that the well-possedness of the system (1) is equivalent to the opeato A being the geneato of some C semigoup T (t). Theefoe, the main task of this section is to veify that the opeato A geneates a C semigoup. In paticula, the semigoup T (t) geneated by the opeato A is also positive and contactive. Theoem 1. Let A be defined by (4) and (5). Then the system opeato A is a closed and densely linea opeato in X. The poof is a diect veification (fo example, see [?]), so we omit. Theoem 2. A is a dissipative opeato in X, and γ ρ(a) fo Rγ >. Poof: We know that the dual space of X is X R (L (R + )) 3 by the diect veification, and the nom fo Q X is given by Q max{ q 1, q j, j 2, 3, 4}. Step I. We show A is dissipative. Fo any P D(A), we choose Q P ( P sign(p ), P sign(p 1 (x)), P sign(p 2 (x)), P sign(p 3 (x)) X Obviously, Q P F(P ) {Q X (P, Q P ) Q P 2 P 2 }. Moeove, (AP, Q P ) P { (λ 1 + ε) P + sign(p ) 2 3 µ j (x)p j (x) p j (x)sign(p j(x)) (µ 1(x)+λ 2 ) p 1 (x) (µ 2(x) + λ 1 ) p 2 (x) µ 1 (x) p 3 (x) }, while sign(p ) µ j (x)p j (x) µ j (x) p j (x), p j (x)sign(p j(x)) p j (), (j 1, 2) we have (AP,Q P ) P (λ 1 + ε) P + 3 p j () λ 2 p 1 (x) λ 1 p 2 (x) µ 1 (x) p 3 (x) (λ 1 + ε) P + λ 1 P + εp + µ 1 (x)p 3 (x) + λ 1 p 2 (x) +λ 2 p 1 (x) λ 2 p 1 (x) λ 1 p 2 (x) µ 1 (x) p 3 (x), i.e., A is dissipative. Step II. We pove that {γ C Rγ > } ρ(a). Fo any F (f, f 1, f 2, f 3 ) X, we conside the esolvent equation (γi A)P F, that is (γ + λ 1 + ε)p µ 1 (x)p 1 (x) µ 2 (x)p 2 (x) f, dp 1 (x) + (γ + µ 1 (x) + λ 2 )p 1 (x) f 1 (x), dp 2 (x) + (γ + µ 2 (x) + λ 1 )p 2 (x) f 2 (x), dp 3 (x) + (γ + µ 1 (x))p 3 (x) f 3 (x), p 1 () λ 1 P, p 2 () εp + µ 1 (x)p 3 (x), p 3 () λ 1 p 2 (x)+λ 2 p 1 (x). Solving the diffeential equations in (7) yields p 1 (x) p 1 () (γ+µ 1(s)+λ 2 )ds + f 1() p 2 (x) p 2 () (γ+µ 2(s)+λ 1 )ds + f 2() p 3 (x) p 3 () (γ+µ 1(s))ds + f 3() (γ+µ 1(s))ds d. By assumption to µ j (x), we have f 1 () (γ+µ 1(s)+λ 2 )ds d, (γ+µ 2(s)+λ 1 )ds d, (γ+µ 1(s)+λ 2 )ds d f 1 () (γ+µ 1(s)+λ 2 )ds d f 1 () d (γ+µ 1(s)+λ 2 )ds 1 Rγ + λ 2 f 1 L 1 1 Rγ f 1 L 1, (7) (8) so, p 1 L 1 (R + ) and µ 1 (x)p 1 (x) f 1 (x) + p 1 () (γ + λ 2 ) p 1 (x) is finite. Similaly, we can pove that p 2, p 3 L 1 (R + ) and µ 2 (x)p 2 (x) < +. Now we substitute (8) into (7) into bounday conditions (2) and get algebaic equations about (P, p 1 (), p 2 (), p 3 ()): (γ + λ 1 + ε)p p 1 ()G 1 (γ) p 2 ()G 2 (γ) f + H 1 (γ) + H 2 (γ), λ 1 P + p 1 (), εp + p 2 () p 3 ()G 3 (γ) H 3 (γ) λ 2 G 4 (γ)p 1 () λ 1 G 5 (γ)p 2 () + p 3 () λ 2 H 4 (γ) λ 1 H 5 (γ), (9) ISSN: Issue 2, Volume 1, Febuay 211

4 Lixia Ma, Genqi Xu, Nikos E. Mastoakis whee G 1 (γ) µ 1 (x) (γ+µ 1(s)+λ 2 )ds, H 1 (γ) µ 1 (x) f 1() G 2 (γ) µ 2 (x) (γ+µ 1(s)+λ 2 )ds d, (γ+µ 2(s)+λ 1 )ds, H 2 (γ) µ 2 (x) f 2() G 3 (γ) µ 1 (x) (γ+µ 1(s))ds, H 3 (γ) µ 1 (x) f 3() G 4 (γ) H 4 (γ) G 5 (γ) H 5 (γ) (γ+µ 2(s)+λ 1 )ds d, (γ+µ 1(x))ds d, (γ+µ 1(s)+λ 2 )ds, f 1() (γ+µ 1(s)+λ 2 )ds d, (γ+µ 2(s)+λ 1 )ds, (γ+µ 2(s)+λ 1 )ds d, f 2() (1) Denote the coefficient matix of (9) by (γ) and the deteminant by D(γ), then (γ) γ+λ 1 +ε G 1 (γ) G 2 (γ) λ 1 1 ε 1 G 3 (γ) λ 2 G 4 (γ) λ 1 G 5 (γ) 1 When Rγ >, it holds that G 1 (γ) + λ 2 G 4 (γ) (µ 1(x) + λ 2 ) (γ+µ 1(s)+λ 2 )ds < 1, G 2 (γ) + λ 1 G 5 (γ) (µ 2(x) + λ 1 ) (γ+µ 2(s)+λ 1 )ds < 1. (11) This shows that (γ) is column stictly diagonal dominant, so D(γ) and linea equations (9) has a unique solution (P, p 1 (), p 2 (), p 3 ()). Thus, it can be deived fom (8) that the esolvent equations (7) has a unique solution (P, p 1 (x), p 2 (x), p 3 (x)) D(A), which means γi A is sujection. Because γi A is closed and D(A) is dense in X, then (γi A) 1 exists and is bounded by Invese Opeato Theoem. So γ ρ(a). As a diect esult of Lume-Phillips Theoem (see, [7]), we have the following esult. Theoem 3. Let A and X be defined as befoe. Then A geneates a C -semigoup of contactions on X. Hence the equation (6) has a unique solution. Next, we will pove the existence of positive solutions to (1) since it is a pactical poblem. We complete the poof by showing A geneates a positive semigoup on X. Theoem 4. A geneates a positive C -semigoup of contactions on X. Poof: Accoding to the positive semigoup theoy (see [11]), A geneates a positive C -semigoup of contactions if and only if A is a dispesive and R(I A) X. Since Theoem 2 has asseted that R(I A) X, we only need to pove A is a dispesive opeato. Take Φ (sign + (P ), sign + (p 1 (x)), sign + (p 2 (x)), sign + (p 3 (x))) { 1, p > ; instead of Q P, whee sign + (P ), p. The othes is simila to the pocess of poving the dissipativity of the opeato A, so we omit. Theoem 5. The semigoup T (t) geneated by A is andom one. Poof: It is sufficient to pove that T (t) is positive consevative accoding to definition of andom semigoup. Fo any positive vecto P D(A), P (P,, p 1 (x), p 2 (x), p 3 (x)), we have T (t)p > and T (t)p D(A), since T (t) is positive. Set P (t, ) (P (t), p 1 (x, t), p 2 (x, t), p 3 (x, t)), obviously, T (t)p dt (t)p satisfies the equation dt AP, t >. Integating the diffeential equations in (1) fom to with espect to x, we have dp (t) dt + 3 d dt which shows that P (t) + 3 p j (x, t), p j (x, t) is a constant. Since T (t)p is continuous in t, we obtain T (t)p P (t) + 3 p j (x, t) P. So T (t) is a andom semigoup, which coincides with the physical meaning. 4 Stability of the system 4.1 Asymptotical stability In this section, we will discuss the stability of the system by analyzing the specta of the system opeato A. Lemma 6. Set µ γ µ(x) (iγ+µ(ξ))dξ, then{ µγ 1, if γ, and µ µ γ 1, if γ, γ < 1, γ. Theoem 7. γ is a simple eigenvalue of A. Moeove if fo any ξ R +, sup ξ ξ ξ µ 1(s)ds <. (12) Thee is no othe spectum of A besides zeo in the imaginay axis. ISSN: Issue 2, Volume 1, Febuay 211

5 Lixia Ma, Genqi Xu, Nikos E. Mastoakis Poof: Let us conside the equation AP, i.e., (λ 1 + ε)p µ 1 (x)p 1 (x) µ 2 (x)p 2 (x), dp 1 (x) + (µ 1 (x) + λ 2 )p 1 (x), dp 2 (x) + (µ 2 (x) + λ 1 )p 2 (x), dp 3 (x) + µ 1 (x)p 3 (x), p 1 () λ 1 P, p 2 () εp + µ 1 (x)p 3 (x), p 3 () λ 1 p 2 (x)+λ 2 p 1 (x). (13) Since 1 λ 1 G 5 () >, diectly solving the equations we get and p 1 () λ 1 P, p 2 () ε+λ 1λ 2 G 4 () 1 λ 1 G 5 () P, p 3 () ελ 1G 5 ()+λ 1 λ 2 G 4 () 1 λ 1 G 5 () P, p 1 (x) λ 1 P (µ 1(s)+λ 2 )ds, ε+λ p 2 (x) P 1 λ 2 G 4 () 1 λ 1 G 5 () e p 3 (x) P ελ 1 G 5 ()+λ 1 λ 2 G 4 () 1 λ 1 G 5 () (µ 2(s)+λ 1 )ds, (14) µ 1(s)ds. (15) So is an eigenvalue of A with geometic multiplicity one and P (P, p 1 (x), p 2 (x), p 3 (x)) given by (15) is the eigenvecto of A coesponding to. Next, we will pove that the algebaic multiplicity of is one. Obviously, the dual space of X is X R [L (R + )] 3. Let A be the dual opeato of A. By the definition of the dual opeato, we can obtain A Q (λ 1 + ε)q + λ 1 q + εq 2 () µ 1 (x)q + q 1 (x) (µ 1(x) + λ 2 )q 1 (x) + λ 2 q 3 () µ 2 (x)q + q 2 (x) (µ 2(x) + λ 1 )q 2 (x) + λ 1 q 3 () q 3 (x) µ 1(x)q 3 (x) + µ 1 (x)q 2 () whee Q (q, q 1 (x), q 2 (x), q 3 (x)) and D(A ) { Q X q j (x), q j (x) L (R + ) j 1, 2, 3 }. Clealy, Q (1, 1,, 1) D(A ) and A Q, so is an eigenvalue of A and Q (1, 1, 1, 1, 1, 1) is a coesponding eigenvecto. Let P be an eigenfunction of A given by (15), then(p, Q) 1. Theefoe, is a simple eigenvalue of A. Now, we show that {ib b } ρ(a). Fo b R, b, F X, let us conside the equation (ibi A)P F, (i.e., eplace γ with ib in (7)) and solving the diffeential equation in (7) yields p 1 (x) p 1 () (ib+µ 1(s)+λ 2 )ds + f 1() (ib+µ 1(s)+λ 2 )ds d, p 2 (x) p 2 () (ib+µ 2(s)+λ 1 )ds + f 2() (ib+µ 2(s)+λ 1 )ds (16) d, p 3 (x) p 3 ()e (ib+µ 1(s))ds + f 3() (ib+µ 1(s)ds d. It is easy to see that p 1, p 2 L 1 (R + ). Fo p 3, p 3 (x) p 3 () + f 3 () d By the assumption (12), we have f 3 () d µ 1(s)ds µ 1(s)ds µ 1(s)ds < (17) fo any f 3 L 1 (R + ), so p 3 L 1 (R + ). Accoding to the poof of Theoem 2, the algebaic equation about (P, p 1 (), p 2 (), p 3 ()) and the coefficient matix is (ib) having the same fom with (26). When b, fom Lemma 6 we know that G 1 (ib) + λ 2 G 4 (ib) (µ 1 (x) + λ 2 ) (ib+µ 1(s)+λ 2 )ds < 1 and G 2 (ib) + λ 1 G 5 (ib) < 1, G 3 (ib) < 1. Thus, (ib) is also column stictly diagonal dominant, so D(ib) and by the same method in Theoem 2, ib ρ(a). That is ir\{} ρ(a). Remak 8. The condition (12) is necessay fo ir\{} ρ(a), othewise the estimate (17) doesn t hold evidently. Let us conside a counteexample. Fom this example we also see that the existence of expectation of the epai time does not ensue the condition (12). If (12) doesn t hold, maybe we have ir σ(a). α 2+x Example 4.1. Let us conside function µ(x) with α > 1. We have e x µ(s)ds (2 + ) α (2 + x) α 2 + α 1. So sup µ(s)ds. Fo f() 1 e ib L 1 (R + ), we have (2+) 4 3 (ib+µ(s))ds f()d d (2 + ) (2 + ) d. α 1 α 2+s ds ISSN: Issue 2, Volume 1, Febuay 211

6 Lixia Ma, Genqi Xu, Nikos E. Mastoakis Summay discussion above, we have the following assetion. Theoem 9. Let space X and opeato A be defined as befoe, the condition (12) hold. Then the following statements ae tue 1)A geneates a positive C -semigoup of contactions T (t); 2) Thee exists a unique dynamic solution P (t) fo any initial value P (). In paticula, when the initial value is nonnegative, the dynamic solution P (t) is positive. 3) The system (1) has a positive steady-state ˆP (P, p 1 (x), p 2 (x), p 3 (x)) which is given by (15) with P 1 Z, whee Z 1 + λ 1 + ε+λ 1λ 2 G 4 () 1 λ 1 G 5 () + ελ 1G 5 ()+λ 1 λ 2 G 4 () 1 λ 1 G 5 () (µ 1(s)+λ 2 )ds (µ 2(s)+λ 1 )ds µ 1(s)ds (18) Futhemoe, the dynamic solution P (t) conveges to the nonnegative steady-state ˆP in the sense of nom, i.e. lim P (t) lim T (t)p () (P (), Q) ˆP. t t Poof: Recalling [1], the eigenfunction coesponding to eigenvalue of the system opeato defined by (15) is the stable solution the system (1), we denote it by ˆP. Note that ˆP 1, then the esult is deived immediately. 4.2 The exponential convegence In the pevious subsection we see that the dynamic solution of the system conveges to the steady state when condition (12) holds. In this subsection, we conside the ate of convegence unde some conditions about µ j (x)(j 1, 2) stonge than (12). Since sup µ 1(s)ds sup µ 1(s+)ds, we define non-negative eal numbe µ 1 by µ 1 sup{η sup e ηx µ 1(s+)ds < }. (19) In what follows, we always assume that µ 1 >, which implies that condition (12) holds. Obviously, when η < µ 1, the integal fo, sup (µ 1(s+) η)ds <, (2) while fo η > µ 1, it must be Set (µ 1(s) η)ds. µ min{ µ 1, λ 1 }. (21) Theoem 1. Let X, A and µ be defined as befoe. Then we have (I) The half-plane {γ C Rγ + µ < } ae in the spectum of A; (II) The set {γ C Rγ + µ >, D(γ) } is in the esolvent set of A and the set {γ C Rγ + µ >, D(γ) } consists of all eigenvalues of A; (III) δ >, thee ae at most finitely many eigenvalues of A in the egion {γ C Rγ + µ δ}; (IV) Thee exists a constant ω 1 > such that the egion {γ C Rγ > ω1 } has only one eigenvalue γ, thus it is stictly dominant. Poof Fo any γ and F (f, f 1 (x), f 2 (x), f 3 (x)) X, we conside the esolvent equation (γi A)P F, that is (γ + λ 1 + ε)p µ 1 (x)p 1 (x) µ 2 (x)p 2 (x) f, dp 1 (x) + (γ + µ 1 (x) + λ 2 )p 1 (x) f 1 (x), dp 2 (x) + (γ + µ 2 (x) + λ 1 )p 2 (x) f 2 (x), dp 3 (x) + (γ + µ 1 (x))p 3 (x) f 3 (x), p 1 () λ 1 P, p 2 () εp + µ 1 (x)p 3 (x), p 3 () λ 1 p 2 (x) + λ 2 p 1 (x). (22) Solving the diffeential equations in (22) yields p 1 (x) p 1 () (γ+µ 1(s)+λ 2 )ds + f 1() (γ+µ 1(s)+λ 2 )ds d, p 2 (x) p 2 () (γ+µ 2(s)+λ 1 )ds + f 2() (γ+µ 2(s)+λ 1 )ds d, p 3 (x) p 3 ()e (γ+µ 1(s))ds + f 3() (γ+µ 1(s))ds d, (23) When Rγ + µ >, using (2) the fist pat in the expession p 1 (x), p 2 (x), p 3 (x) ae bounded. Fo the second tems in the expession p 1 (x), p 2 (x) ae also bounded. Fo the second tem of p 3 (x), f 3 () (γ+µ 1(s))ds d f 3 () d M 2 (Rγ) f 3 L 1, (Rγ+µ 1(s))ds ISSN: Issue 2, Volume 1, Febuay 211

7 Lixia Ma, Genqi Xu, Nikos E. Mastoakis whee M 2 (Rγ) sup (Rγ+µ 1(s+))ds <. Theefoe, p j L 1 (R + ), j 1, 2, 3. Obviously, when Rγ + µ <, at least p 3 (x) in (23) is not in L 1 (R + ). Thus (I) follows. (II) Now we substitute (23) into bounday conditions in (22) and get the algebaic equations about (P, p 1 (), p 2 (), p 3 ()): (γ + λ 1 + ε)p p 1 ()G 1 (γ) p 2 ()G 2 (γ) f + H 1 (γ) + H 2 (γ), λ 1 P + p 1 (), εp + p 2 () p 3 ()G 3 (γ) H 3 (γ) λ 2 G 4 (γ)p 1 () λ 1 G 5 (γ)p 2 () + p 3 () λ 2 H 4 (γ) λ 1 H 5 (γ), (24) The coefficient deteminant of the equations (24) is D(γ) γ+λ 1 +ε G 1 (γ) G 2 (γ) λ 1 1 ε 1 G 3 (γ) λ 2 G 4 (γ) λ 1 G 5 (γ) 1 Then the algebaic equations have solution if and only if D(γ). When D(γ), the equations (24) has unique solution (P, p 1 (), p 2 (), p 3 ()), futhemoe, the vecto detemined by (23) (P, p 1 (x), p 2 (x), p 3 (x)) D(A), and (γi A)P F, so γ ρ(a). When D(γ), the equations (24) has nonzeo solution (P, p 1 (), p 2 (), p 3 ()), futhemoe, the vecto (P, p 1 (x), p 2 (x), p 3 (x)) D(A), and AP γp, so γ is an eigenvalues of A. (III) Note that G 1 (γ) + λ 2 G 4 (γ) (µ 1(x) + λ 2 ) (γ+µ 1(s)+λ 2 )ds, 1 γ (γ+µ 1(s)+λ 2 )ds 1 γg 4 (γ) G 2 (γ) + λ 1 G 5 (γ) 1 γ (γ+µ 2(s)+λ 1 )ds 1 γg 5 (γ), G 3 (γ) 1 γ (γ+µ 1(s))ds. (25) and whee D(γ) γ γg 4 (γ) γg 5 (γ) γπ(γ) λ 1 1 ε 1 G 3 (γ), λ 2 G 4 (γ) λ 1 G 5 (γ) 1 Π(γ) (γ+µ1(ξ))dξ. (26) So, fo all µ + δ Rγ, by condition (3) G 3 (γ) µ 1 (x)e (Rγ+ µ)x iiγx M e (Rγ+ µ)x (µ1(ξ) µ)dξ By Riemman Lemma, we have Similaly, G 4 (γ) e (Rγ+λ 2+ µ)x iiγx G 5 (γ) e (Rγ+λ 1+ µ)x iiγx Π(γ) e (Rγ+ µ)x iiγx The Riemman Lemma assets that lim G 4(γ), Iγ lim Iγ (µ 1(ξ) µ)dξ lim G 3(γ). Iγ (µ 1(ξ) µ)dξ (µ 2(ξ) µ)dξ (µ 1(ξ) µ)dξ lim Iγ lim G 5(γ). Hence, Iγ D(γ) γ 1 λ 1 1 ε (27) (28) Π(γ), The limit is unifomly in the egion µ+δ Rγ <. Moeove D(γ) is analytic function. So D(γ) has at most finite numbe of zeos in µ + δ Rγ <. Thus A has at most finite numbe eigenvalue in the egion µ + δ Rγ <. Moeove, thee is no othe spectum of A besides zeo in the imaginay axis accoding to Theoem 11. Thus (III) is tue. (IV) Set H 1 (γ) D(γ) γ. So γ is a zeo of D(γ) if and only if it is that of H 1 (γ). Since H 1 (γ) H 1 (γ), its zeos ae symmetically with espect to the eal axis. Note that µ > implies H 1 (ib), b R. Let the zeos of H 1 (γ) in the egion µ+δ Rγ be γ k, k 1, 2,, m. We can set ω 1 min Rγ k. 1 k m Thee is no zeo of H 1 (γ) as Rs > ω 1. Hence thee is only one eigenvalue γ of A in the egion {γ C Rγ > ω 1 }. Accoding to the finite expansion theoem of semigoups, we have the following esult. Theoem 11. Let X and A be defined as befoe, and let T (t) be the semigoup geneated by A. Suppose that µ > and < ω 1 < Rγ 1. Then fo any initial P (), we have P (t) P (), Q P 2e ω 1t P () (29) whee P (t) T (t)p (). ISSN: Issue 2, Volume 1, Febuay 211

8 Lixia Ma, Genqi Xu, Nikos E. Mastoakis Poof: Since the Riesz spectal poject coesponding to γ is given by E(γ, A)F 1 R(s, A)F ds F, Q 2πi P s ε fo F X. This leads to E(γ, A) 1. Since T (t) is a dissipative semigoup in (I E(γ, A))X, and the condition µ > ensues that the esolvent R(γ, A)F (P, p 1 (x), p 2 (x), p 3 (x) is bounded u- nifomly in the egion µ + δ Rγ when γ is lage enough. So we have The esult is deived. P (t) P (), Q P T (t)(i E(γ, A))P () e ω1t (I E(γ, A))P () 2e ω1t P (). 4.3 Special case In this subsection we discuss the special case that µ 1 (x), µ 2 (x) ae the constant functions. In this case, G 3 (s) µ 1 (γ+µ 1 ),G µ 4(γ) 1 (γ+µ 1 +λ 2 ) and G 5(γ) µ 2 (γ+µ 2 +λ 1 ). Hence and D(γ) whee γh 2 (γ) (γ + µ 1 )(γ + µ 1 + λ 2 )(γ + µ 2 + λ 1 ) H 2 (γ) γ 3 + a 2 γ 2 + a 1 γ + a a µ 1 µ 2 (λ 1 + λ 2 ) + λ 1 λ 2 (µ 1 µ 2 ) λ 2 1 λ 2 +µ 2 1 µ 2 + ε(µ λ 2µ 1 λ 1 µ 1 λ 1 λ 2 ), a 1 µ λ 1µ 1 + µ 2 λ 1 + λ 2 µ 1 + λ 2 µ 2 +2µ 1 µ 2 + 2εµ 1 + ελ 2 ελ 1, a 2 2λ 1 + 2λ 2 + 2µ 1 + µ 2 + ε, ae eal coefficients. Clealy, H 2 (γ) has thee zeos. 5 Reliability analysis of the system 5.1 Some indices deiving fom Steady-State Fom Theoem 9, we obtain the steady state of the system. Based on this, some indices of eliability can be deived. Theoem 12. The nomal steady-state availability of the system is A N v whee Z is given by (18). 1 Z (3) Poof: By the expession of steady state (18) and definition of the nomal availability A N v P, we know the esult holds. Accoding to the definition of abnomal availability we have A U v p 1 (x) + p 2 (x), Theoem 13. The abnomal steady-state availability of the system is A U v p 1 (x) + p 2 (x) λ 1 Z [ (µ 1(s)+λ 2 )ds + ε+λ 1 λ 2 G 4 () 1 λ 1 G 5 () e whee G 4 (), G 5 () is given by (1). (µ 2(s)+λ 1 )ds ] The steady-state availability of the system is A v A N v + A U v 1 Z + λ 1 Z [ + (µ 1(s)+λ 2 )ds ε+λ 1 λ 2 G 4 () 1 λ 1 G 5 () e (µ 2(s)+λ 1 )ds ] (31) (32) Remak : Fom the expession of Z, we can know if the system has stonge epai ate µ j (x), then µ j(s)ds has a smalle value, thus the nomal steady-state availability of the system will be lage, the failue pobability of the system p 3 (x) will be smalle, and system nomal eliability be enhanced. The aveage failue numbe of the system at time t is called the instantaneous failue fequency, denoted by W f (t), and its limit as t is called steady-state failue fequency of the system, denoted by W f. Fom [2] and the model of the system, we know We immediately have W f lim t p 3 (t, ) p 3 () (33) Theoem 14. The steady-state failue fequency of the system is W f ελ 1G 5 () + λ 1 λ 2 G 4 (). [1 λ 1 G 5 ()]Z 5.2 The estimation of instantaneous availability The instantaneous availability of the system is the pobability of the system being in wok, which is defined by A v (t) P (t) + p 1 (x, t) + p 2 (x, t). Fom Theoem 11, A v (t) A v P (t) P 2e ω t, ISSN: Issue 2, Volume 1, Febuay 211

9 Lixia Ma, Genqi Xu, Nikos E. Mastoakis whee A v is given by (32), we have A v (t) A v (t) A v + A v A v (t) A v + A v P (t) P + A v 2e ω t + A v. 3 ln 1 ln 5 Because when t > ω, P (t) P.1. So, we have A v (t).1 + A v. Obviously, the failue pobability of the system is p 3 (x, t) 1 A v (t). It has an estimate1 3 ln 1 ln 5 A v (t).99 A v, when t > ω. 6 Optimal epai ate fo steadystate eliability In this section, we shall conside a optimization poblem fo the epai ate µ(x) (µ 1 (x), µ 2 (x)). Suppose P is the expectable pobability of zeo state in steady-state and P (µ) is the fist component of solution of system coesponding to µ(x). Take the index functional S(µ) P (µ)) P 2. Ou aim is to find µ(x) (µ 1 (x), µ 2 (x)) such that it minimizes S(µ). Accoding to the assumptions (3), we let the admissible set be U { (µ1 (x), µ 2 (x)) [L (R + )] } 3 U ln(1+x) 1+x µ 1 (x), µ 2 (x) M Clealy U is a closed and convex set in [L (R + )] 3. Ou object is to find µ U such that S(µ ) inf µ U S(µ). (34) µ is said to be optimal epai ate of the system. To solve the poblem (34), let P R µ U such that (P, p 1 (x), p 2 (x), p 3 (x)) R (L 1 (R + )) 3 W is a nonnegative solution coesponding to µ(x) P +. p 1 (x) + p 2 (x) + p 3 (x) 1 (35) We fistly pove that thee exists a P W satisfying S(P ) inf S(P ) inf P P 2. (36) P W P W Theoem 15. W is a bounded and w -closed set in R. Poof: Fom the pevious section we know W is a bounded set. Now we pove that W is a closed set. Let P (n) W, and P (n) P. Then thee exist a sequence µ (n) (x) U such that P (n) (x) ae the nonnegative solution to (1) with bounday conditions (2). Since U [L (R + )] 3 is bounded and then w -sequence compact, thee exist a subsequence of µ (n) (x) without loss geneality itself such that µ (n) (x) w µ(x) ( µ 1 (x), µ 2 (x)). It is easy to see that µ 1 (x) and µ 2 (x) ae nonnegative functions. Fo any x R + fixed, the function χ [,x] (s) L 1 (R + ), the w convegence of µ (n) deduces that lim n lim n Since (µ (n) χ [,x] (s)µ (n) 1 (s)ds µ 1(s)ds, χ [,x] (s)µ (n) 2 (s)ds µ 2(s)ds. 1, µ(n) [ln(1 + x)]2 so it holds that 1 2 [ln(1 + x)]2 ) U implies that x µ (n) 1 (s)ds, x Mx, µ (n) 2 (s)ds µ 1 (s)ds, µ 2 (s)ds Mx. Theefoe, µ(x) U. On the othe hands, fom (15) we get that p (n) 1 (x) λ 1P (n) (µ(n) (n) ε+λ (x) P 1 λ 2 G 4 () p (n) 2 p (n) 3 (x) P (n) 1 (s)+λ2)ds, 1 λ 1 G 5 () e (µ(n) 2 (s)+λ1)ds, ελ 1 G (n) 5 ()+λ 1λ 2 G (n) 4 () e 1 λ 1 G (n) 5 () µ(n) 1 (s)ds. (37) The convegence of P (n) implies that fo each x R +, p (n) 1 (x), p(n) 2 (x) and p(n) 3 (x) ae convegent and the limit ae espectively p 1 (x) λ 1 ( µ 1(s)+λ 2 )ds P, p 2 (x) ε+λ 1λ 2 G 4 () 1 λ 1 G 5 () e ( µ 2(s)+λ 1 )ds P, p 3 (x) ελ 1G 5 ()+λ 1 λ 2 G 4 () 1 λ 1 G 5 () µ 1(s)ds P. Clealy, it holds that p 1 () λ 1 P p 2 () ε P + µ 1 (x) p 3 (x), p 3 () λ 1 p 2 (x, t) + λ 2 p 1 (x). Fom above expession we see that p 1 (x), p 2 (x) and p 3 (x) ae absolute continuous and P, p 1, p 2, p 3 (x), satisfy the diffeential equations in (1). ISSN: Issue 2, Volume 1, Febuay 211

10 Lixia Ma, Genqi Xu, Nikos E. Mastoakis Finally, we pove that the integal µ 1(s)ds is finite. Note that the elation 1 2 [ln(1 + x)]2 µ 1(s)ds Mx, we have µ(n) 1 (s)ds e 1 2 [ln(1+x)]2 <. The Fatou lemma assets that µ 1(s)ds e 1 2 [ln(1+x)]2. Thus, Z 1 + λ 1 + ε+λ 1λ 2 G 4 () 1 λ 1 G 5 () + ελ 1G 5 ()+λ 1 λ 2 G 4 () 1 λ 1 G 5 () ( µ 1(s)+λ 2 )ds ( µ 2(s)+λ 1 )ds µ 1(s)ds (38) has meaning. Theefoe, ( P, p 1 (x), p 2 (x), p 3 (x)) is a nonnegative solution of (1) coesponding to µ(x) ( µ 1 (x), µ 2 (x)), and satisfies condition P ( µ) + 3 p j (x) 1. So P W, and W is a closed set. Theoem 16. W is a convex set, and S(P ) is a stictly convex functional on W. Poof: Let P (1), P (2) W and P (1) P (2). By the definition of W thee exist µ (i) (µ (i) 1, µ(i) 2 ), i 1, 2 coesponding to P (i). Fo any < τ < 1, we set P (i) (P (i) (x), p(i) (x)) and, p(i) 1 (x), p(i) 2 P τ (x) τp (1) (x) + (1 τ)p (2) (x). A diect veification shows that P τ satisfy the diffeential equation (1) coesponding to µ τ τµ (1) +(1 τ)µ (2). So W is a convex set. Futhe, we have S(τP (1) + (1 τ)p (2) ) τp (1) + (1 τ)p (2) P 2 < τ P (1) P 2 + (1 τ) P (2) P 2 τs(p (1) ) + (1 τ)s(p (2) ). Theefoe, S(P ) is a stictly convex functional in W. Theoem 17. Thee exists a unique P that S(P ) inf S(P ). p W 3 W such Poof: Since W R + is a bounded and closed set, the existence and uniqueness of P W follows fom the theoy of the convex function on convex set. Theoem 18. If P W, then thee exists a unique µ U such that S(µ ) inf µ U S(µ). Poof: Note that S(P ) P P 2, S(µ) P (µ) P 2 and P (µ) 1 Z(µ) whee Z 1 + λ 1 + ε+λ 1λ 2 G 4 () 1 λ 1 G 5 () + ελ 1G 5 ()+λ 1 λ 2 G 4 () 1 λ 1 G 5 () (µ 1(s)+λ 2 )ds (µ 2(s)+λ 1 )ds µ 1(s)ds (39) whee G 4 () (µ 1(s)+λ 2 )ds, G 5 () (µ 2(s)+λ 1 )ds. Obviously, Z(µ) depends continuously on (µ 1, µ 2 ) U. In paticula, it holds that Z(µ) 1 + λ 1 M+λ 2 + ε(m+λ 2)+λ 1 λ 2 M(M+λ 2 ) + ελ 1(M+λ 2 )+λ 1 λ 2 (M+λ 1 ) M 2 ((M+λ 2 ) whee M is defined as in U. Theefoe, µ U, P (µ) 1 Z(µ) (4) M 2 (M+λ 2 ) M 2 ((M+λ 2 )+λ 1 M 2 +ε(m+λ 1 )(M+λ 2 )+λ 1 λ 2 M+λ 1 λ 2 (M+λ 1 ). Theefoe, when P W, it must have P > M 2 (M+λ 2 ) M 2 ((M+λ 2 )+λ 1 M 2 +ε(m+λ 1 )(M+λ 2 )+λ 1 λ 2 M+λ 1 λ 2 (M+λ 1 ). so S(µ) aives minimum at µ M(1, 1). Acknowledgements: The eseach is suppoted by the National Science Natual Foundation in China (NSFC ). Refeences: [1] K. C. Who, A Repaable Multistate Device With Abitaily Distibuted Repai Times, Micoelectonics and Reliability, 21(1981),2, pp [2] J. H. Cao and K. Cheng, Intoduction to Reliability Mathematics, Science Pess, Beijing, China, [3] R. B. Liu, Y. H. Tang and C. Y. Luo, A new kind of N-unit seies epaiable system and its eliability analysis. Applied Mathematical Modelling, 2 (27), 1,pp ISSN: Issue 2, Volume 1, Febuay 211

11 Lixia Ma, Genqi Xu, Nikos E. Mastoakis [4] W. L. Wang, and G. Q. Xu, Stability analysis of a complex standby system with constant waiting and diffeent epaiman citeia incopoating envionmental failue, Applied Mathematical Modelling, 33 (29), pp [5] L. N. Guo, H. B. Xu, C. Gao and G. T. Zhu, Stability analysis of a new kind seies system. IMA Jounal of Applied Mathematics, 75(21), pp [6] Z. F. Shen et al. Exponential asymptotic popety of a paallel epaiable system with wam standby unde common-cause failue. Mathematical Analysis and applications, 341(28), 1, pp [7] A. Pazy, Semigoup of linea opeatos and application to patial diffeential equations, Spinge-Velag, Belin, New Yok,1982. [8] Z. Y. Lin and G. J. Wang, A geometic pocess epai model fo a epaiable cold standby system with pioity in use and epai. Reliability Engineeing and System Safety, 94(29), pp [9] K. J. Engel and R. Nagel, One-paamete Semigoups fo Linea Evolution Equations, Gaduate Texts in Math, Vol.194, Spinge-Velag, 2. [1] G. Gupu, X. Z. Li and G. T. Zhu, Functional Analysis Method in Queuing Theoy, Hetfodshie, Reseach Infomation Ltd, United Kingdom, 21. [11] R. Nagel, One-Paamete Semigoup of Positive Opeatos, Lectue Notes in Mathematics, Spinge, New Yok, ISSN: Issue 2, Volume 1, Febuay 211