Exponential asymptotic property of a parallel repairable system with warm standby under common-cause failure

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1 J. Math. Anal. Appl. 341 (28) Exponential asymptotic property of a parallel repairable system with warm stanby uner common-cause failure Zifei Shen, Xiaoxiao Hu, Weifeng Fan Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 3214, China Receive 26 January 27 Available online 22 October 27 Submitte by L. Guo Abstract We iscuss exponential asymptotic property of the solution of a parallel repairable system with warm stanby uner commoncause failure. This system can be escribe by a group of partial ifferential equations with integral bounary. First we show that the positive contraction C -semigroup T(t)[Weiwei Hu, Asymptotic stability analysis of a parallel repairable system with warm stanby uner common-cause failure, Acta Anal. Funct. Appl. 8 (1) (26) 5 2] which is generate by the operator corresponing to these equations is a quasi-compact operator. Then by using [Weiwei Hu, Asymptotic stability analysis of a parallel repairable system with warm stanby uner common-cause failure, Acta Anal. Funct. Appl. 8 (1) (26) 5 2] that is an eigenvalue of the operator with algebraic inex one an the C -semigroup T(t)is contraction, we conclue that the spectral boun of the operator is zero. By using the above results the exponential asymptotical stability of the time-epenent solution of the system follows easily. 27 Elsevier Inc. All rights reserve. Keywors: C -semigroup; Quasi-compact operator; Asymptotic stability 1. Introuction As the evelopment of science an technology, electron prouctions an network are use everywhere. So the stability analysis of the system becomes more an more important. In [1], the author evelope the mathematical moel, which is a parallel repairable system with warm stanby uner common-cause failure. In [2] the author prove the asymptotic stability of the system an the steay-state solution is shown to be the eigenvector of the system operator corresponing to the eigenvalue. Whereas, the velocity of the time-epenent solution converging to the steay one has not mentione. In this paper, we will iscuss the converging velocity. We first convert the moel into an abstract Cauchy problem in a Banach space, then show that the operator corresponing to this moel generates a positive contraction C -semigroup. We then prove that C -semigroup T(t) is a quasi-compact operator, an that spectral boun of this operator is zero. Thus by Theorem 2.1 in [3, p. 343] we obtain our esire result. Supporte by the National Science Founation of China ( ), Natural Science Founation of Zhejiang Province (Y66292). * Corresponing author. aress: szf@zjnu.cn (Z. Shen) X/$ see front matter 27 Elsevier Inc. All rights reserve. oi:1.116/j.jmaa

2 458 Z. Shen et al. / J. Math. Anal. Appl. 341 (28) Fig. 1. State space iagram of the system. 2. System moel The following assumptions are associate with this system (see Fig. 1): (i) The system has three ientical units (two active an one on stanby). (ii) The harware an common-cause failures are statistically inepenent. (iii) A common-cause failure can occur when both active parallel units an the stanby are in perfect working conition as well as when the system is operating in a egrae state. (iv) The system is in an up-state as long as one unit is working. (v) All failure rates are constant. (vi) A unit s repair rate is constant. (vii) The faile system repair times are assume to be arbitrarily istribute. (viii) A repaire unit or system is assume to be as goo as new. (ix) The switching mechanism for the stanby is consiere automatic an instantaneous. (x) The stanby may fail in its stanby moe, in aition, to the switching mechanism. The following symbols are use in this article: t: time; λ 1 : constant failure rate of a unit; λ 2 : constant failure rate of switching mechanism an/or stanby itself; λ C : constant critical common-cause failure rate; λ C1 : constant common-cause failure rate of the system when one of the parallel units has faile; λ C2 : constant common-cause failure rate when the switching mechanism an/or stanby itself is isable; λ C3 : constant common-cause failure rate when two units have faile; p i (t): the probability that the system is in state i at time t;fori =, 1, 2, 3; μ 1 : constant repair rate when one of the parallel units is isable; μ 2 : constant repair rate for the switching mechanism an/or the stanby itself; μ 3 : constant repair rate when two units have been isable;

3 Z. Shen et al. / J. Math. Anal. Appl. 341 (28) p i (x, t): probability ensity (with respect to repair time) that the faile system is in state i an has an elapse repair time of x, fori = 4, 5; μ i (x): repair rate when the faile system is in state i an has an elapse repair time of x,fori = 4, 5. Accoring to [1], the moel for a two ientical units parallel repairable system with warm stanby subject to common-cause failures can be expresse by a group of integro-ifferential equations: where t p (t) (2λ 1 λ C λ 2 )p (t) = μ 1 p 1 (t) μ 2 p 2 (t) p i (x, t)μ i (x) x, (1) t p 1(t) (2λ 1 λ C1 μ 1 )p 1 (t) = 2λ 1 p 1 (t) μ 3 p 3 (t), (2) t p 2(t) (2λ 1 λ C2 μ 2 )p 2 (t) = λ 2 p (t) μ 3 p 3 (t), (3) t p 3(t) (λ 1 λ C3 2μ 3 )p 3 (t) = 2λ 1 p 1 (t) 2λ 1 p 2 (t), (4) ( t ) x μ 4(x) p 4 (x, t) =, (5) ( t ) x μ 5(x) p 5 (x, t) =, (6) p 4 (,t)= λ 1 p 3 (t), (7) p 5 (,t)= λ C p (t) λ C1 p 1 (t) λ C2 p 2 (t) λ C3 p 3 (t), (8) p () = 1, p i () =, i = 1, 2, 3, p j (x, ) =, j = 4, 5, (9) μ i (x), G i = sup μ i (x) <, x< For simplicity, let x μ i (s) s <, μ i (x) x =, i = 4, 5. h = 2λ 1 λ C λ 2, h 1 = 2λ 1 λ C1 μ 1, h 2 = 2λ 1 λ C2 μ 2, h 3 = λ 1 λ C3 2μ 3. Take state space as follows: { X = y R R R R L 1 [, ) L 1 [, ) y = 3 y i i= } y i L 1 [, ). It is obvious that X is a Banach space. In the following we efine several operators an their omains: p 1 p 1 Ap = 2 p 2 3 p, 3 x μ 4(x) p 4 (x) x μ 5(x) p 5 (x) { } p i (x) x L 1 [, ), p i (i = 4, 5) are absolutely continuous functions, D(A) = p X p 4 () = λ 1 p 3,p 5 () =, 3 i= λ Cj p j

4 46 Z. Shen et al. / J. Math. Anal. Appl. 341 (28) μ 1 μ 2 μ 4 (x) x μ 5 (x) x p 2λ 1 μ 3 p 1 Ep = λ 2 μ 3 p 2 2λ 1 2λ 1 p, 3 p 4 (x) p (x) D(E) = X. Then the above equations (1) (9) can be written as an abstract Cauchy problem in the Banach space X, p(t) = (A E)p(t), t [, ), t p() = (1,,,,, ). (1) (11) In [2], we prove the following result. Theorem 1. A E generates a positive contraction C -semigroup T(t). Theorem 2. A generates a positive contraction C -semigroup S(t). The proof is similar to the [2,6]. In this paper, we first prove that S(t) is a quasi-compact operator by stuying two operators V(t)an W(t), then we obtain that T(t)is a quasi-compact operator by using the compactness of E, an last by using [2], is an eigenvalue of A E an (A E) with geometric multiplicity one. Thus by Theorem 2.1 in [3, p. 343] we euce our esire result. 3. Main results Proposition 1. For φ X, p(x,t) = (S(t)φ)(x) is a solution of the following system: p(t) = Ap(t), t p() = φ, φ X, then φ e t φ 1 e 1t φ 2 e 2t φ 3 e 3t, x <t, p 4 (,t x)e x μ 4(τ) τ p(x,t) = ( S(t)φ ) p 5 (,t x)e x μ 5(τ) τ (x) = φ e t φ 1 e 1t φ 2 e 2t φ 3 e 3t, x t, φ 4 (t x)e x x t μ 4(τ) τ φ 5 (t x)e x x t μ 5(τ) τ where p 4 (,t x), p 5 (,t x) are given by (7) an (8). ( )

5 Z. Shen et al. / J. Math. Anal. Appl. 341 (28) Proof. Since p is a solution of the following system: t p (t) = p (t), (12) t p 1(t) = 1 p 1 (t), (13) t p 2(t) = 2 p 2 (t), (14) t p 3(t) = 3 p 3 (t), (15) ( t ) x μ 4(x) p 4 (x, t) =, (16) ( t ) x μ 5(x) p 5 (x, t) =, (17) p 4 (,t)= λ 1 p 3 (t), (18) p 5 (,t)= λ C p (t) λ C1 p 1 (t) λ C2 p 2 (t) λ C3 p 3 (t), (19) p () = φ, p i () = φ i, i = 1, 2, 3, p j (x, ) = φ j, j = 4, 5. (2) If we set ξ = x t an efine Q 4 (t) = p 4 (t ξ,t) an Q 5 (t) = p 5 (t ξ,t), then from (16), (17) we know that Q 4 (t) = μ 4 (ξ t)q 4 (t), (21) t Q 5 (t) = μ 5 (ξ t)q 5 (t). (22) t If ξ<, then integrating (21) (22) from ξ to t, an using Q 4 ( ξ) = p 4 (, ξ) = p 4 (,t x), Q 5 ( ξ) = p 5 (, ξ)= p 5 (,t x) we have p 4 (x, t) = Q 4 (t) = Q 4 ( ξ)e ξ μ4(ξτ)τ, (23) p 4 (x, t) = p 4 (,t x)e x μ 4(τ) τ, (24) p 5 (x, t) = p 5 (,t x)e x μ 5(τ) τ. (25) From (12) (15) we obtain p (t) = φ e t, p 1 (t) = φ 1 e 1t, p 2 (t) = φ 2 e 2t, p 3 (t) = φ 3 e 3t. (26) If ξ, then integrating (21) (22) from to t, an then using relations Q 4 () = p 4 (ξ, ) = Q 4 (x t) an Q 5 () = p 5 (ξ, ) = Q 5 (x t), an by similar argument to (23) (25) we obtain φ e t φ 1 e 1t φ 2 e 2t p(x,t) = φ 3 e 3t. φ 4 (t x)e x x t μ 4(τ) τ φ 5 (t x)e x x t μ 5(τ) τ The proof of Proposition 1 is complete. Remark. By Theorem 2 an similar argument in [2] or [6], we know that the system ( ) has a unique nonnegative solution p(x,t) = (S(t)φ)(x),byusingtheC -semigroup theory in [5], we can know that p(x,t) is not only the weak solution of the system ( ) but also the strong solution.

6 462 Z. Shen et al. / J. Math. Anal. Appl. 341 (28) If we efine two operators as follows, for p X, { ( ), x [,t), V(t)p (x) = (S(t)p)(x), x [t, ), { ( ) (S(t)p)(x), x [,t), W(t)p (x) =, x [t, ), then S(t)p = V(t)p W(t)p. (27) (28) Lemma 1. (See [4, p. 29].) A close an boune subset Y of L 1 [, ] is compact if an only if the following two conitions hol: (i) (ii) lim h lim h h φ(x h) φ(x) x =, uniformly for φ Y, φ(x h) x =, uniformly for φ Y. From Lemma 1 it is easy to prove the following result. Lemma 2. A close an boune subset Y X is compact if an only if the following two conitions hol: (i) (ii) lim h j=4 lim h j=4 h φ j (x h) φ j (x) x =, uniformly for φ = (φ,φ 1,φ 2,φ 3,φ 4,φ 5 ) Y, φ j (x) x =, uniformly for φ = (φ,φ 1,φ 2,φ 3,φ 4,φ 5 ) Y. Theorem 3. W(t) is a compact operator on X. Proof. Accoring to the efinition of W(t), it suffices to prove conition (ii) in Lemma 2. For boune φ X we set p(x,t) = (S(t)φ)(x), x [,t), then p(x,t) is a generalize solution of the system ( ). So by Proposition 1, we have, for x,h [,t), x h [,t), p i (x h, t) p i (x, t) x = = p 4 (x h, t) p 4 (x, t) x p 5 (x h, t) p 5 (x, t) x p4 (,t x h)e xh μ 4 (τ) τ p 4 (,t x)e x μ 4(τ) τ x p 5 (,t x h)e xh μ 5 (τ) τ p 5 (,t x)e x μ 5(τ) τ x p4 (,t x h) e xh μ 4 (τ) τ e x μ 4(τ) τ x p 4 (,t x h) p 4 (,t x) e x μ 4(τ) τ x

7 Z. Shen et al. / J. Math. Anal. Appl. 341 (28) p5 (,t x h) e xh μ 5 (τ) τ e x μ 5(τ) τ x p 5 (,t x h) p 5 (,t x) e x μ 5(τ) τ x. (29) We will estimate each term in (29). By using (18), (19) an Theorem 2 we have p 4 (,t x h) = λ 1 p 3 (t x h) λ 1 p(,t x h) X = λ 1 S(t x h)φ( ) X λ 1 φ X, (3) p5 (,t x h) = λ C p (t x h) λ C1 p 1 (t x h) λ C2 p 2 (t x h) λ C3 p 3 (t x h) max{λ C,λ C1,λ C2,λ C3 } p (t x h) p 1 (t x h) p 2 (t x h) p 3 (t x h) max{λ C,λ C1,λ C2,λ C3 } p(,t x h) X = max{λ C,λ C1,λ C2,λ C3 } S(t x h)φ( ) X max{λ C,λ C1,λ C2,λ C3 } φ X. By using (3), (31) we estimate the first an thir term in (29) as follows: p4 (,t x h) e xh μ 4 (τ) τ e x μ 4(τ) τ x (31) λ 1 φ X e xh μ 4 (τ) τ e x μ 4(τ) τ x as h, uniformly for φ, (32) p5 (,t x h) e xh μ 5 (τ) τ e x μ 5(τ) τ x max{λ C,λ C1,λ C2,λ C3 } φ X e xh μ 5 (τ) τ e x μ 5(τ) τ x as h, uniformly for φ. (33) By using (18), (26) we have p4 (,t x h) p 4 (,t x) = λ1p3 (t x h) p 3 (t x) ( = λ φ3 1 e 3 (t x) e 3(t x) ) as h, uniformly for φ. (34) By using (19), (26) we have p5 (,t x h) p 5 (,t x) λ C p (t x h) λ C p (t x) λ C1 p 1 (t x h) λ C1 p 1 (t x) λc2 p 2 (t x h) λ C2 p 2 (t x) λc3 p 3 (t x h) λ C3 p 3 (t x). By using (26) an o as (34) we have λ C p (t x h) λ C p (t x) as h, uniformly for φ, λ C1 p 1 (t x h) λ C1 p 1 (t x) as h, uniformly for φ, λ C2 p 2 (t x h) λ C2 p 2 (t x) as h, uniformly for φ.

8 464 Z. Shen et al. / J. Math. Anal. Appl. 341 (28) So we obtain p 5 (,t x h) p 5 (,t x) ash, uniformly for φ. (35) (34) an (35) imply that the secon term an fourth term in (29) satisfy p 4 (,t x h) p 4 (,t x) e x μ 4(τ) τ x ash, uniformly for φ, p5 (,t x h) p 5 (,t x) e x μ 5(τ) τ x ash, uniformly for φ. (36) Combining (32), (33), (36) with (29), for x h [,t), we euce p i (x h, t) p i (x, t) x as h, uniformly for φ. (37) If h [ t,), x (,t), then from the relation p 4 (x h, t) =, p 5 (x h, t) =, for x h<, we have p i (x h, t) p i (x, t) x = = = p 4 (x h, t) p 4 (x, t) x p 5 (x h, t) p 5 (x, t) x p4 (x h, t) p 4 (x, t) x p4 (x h, t) p 4 (x, t) x p 5 (x h, t) p 5 (x, t) x p 5 (x h, t) p 5 (x, t) x p4 (x h, t) p 4 (x, t) x p4 (x, t) x p 5 (x h, t) p 5 (x, t) x p 5 (x, t) x. (38) Since x h [,t),forx [,t), h [ t,), for the first term an the thir term in (38), similar way to (37) we have p4 (x h, t) p 4 (x, t) x ash, uniformly for φ, (39) p 5 (x h, t) p 5 (x, t) x ash, uniformly for φ. (4) By using Proposition 1 an (3), we estimate the secon term in (38) as follows: p 4 (x, t) x = p 4 (,t x)e x μ 4(τ) τ x λ 1 φ X e x μ 4(τ) τ x as h, uniformly for φ. (41)

9 Z. Shen et al. / J. Math. Anal. Appl. 341 (28) It is the same as (41), by using Proposition 1 an (31) we euce p 5 (x, t) x as h, uniformly for φ. (42) Combining (39) (42) with (38), for h [ t,), we obtain p i (x h, t) p i (x, t) x ash, uniformly for φ. (43) From (37) an (43) we know that the result of this theorem is right. Theorem 4. Assume there exist four positive constants μ j, μ j, j = 4, 5 such that <μ j μ j (x) μ j <, j = 4, 5, then V(t)satisfies V(t)φ X e min{h,h 1,h 2,h 3,μ 4,μ 5 }t φ X, φ X. (44) Proof. For any φ X from the efinition of V(t)an (26), we have V(t)φ X = p (t) p1 (t) p2 (t) p3 (t) t φ4 (t x)e x x t μ 4(τ) τ x φ5 (t x)e x x t μ 5(τ) τ x φ e t φ 1 e 1t φ 2 e 2t φ 3 e 3t sup e x x t μ 5(τ) τ x [t, ) t φ 5 (t x)x t sup e x x t μ 4(τ) τ x [t, ) t φ 4 (t x)x e min{h,h 1,h 2,h 3 }t ( φ φ 1 φ 2 φ 3 ) e μ 4t φ 4 L 1 [, ) e μ 5t φ 5 L 1 [, ) e min{h,h 1,h 2,h 3,μ 4,μ 5 }t φ X, φ X. (45) (45) shows that the result of this theorem hols. From Theorems 3 an 4 we euce S(t) W(t) = V(t) e min{h,h 1,h 2,h 3,μ 4,μ 5 }t, as t. From which together with Definition 2.7 in [3, p. 214], we erive the following result. Theorem 5. S(t) is a quasi-compact operator on X. Since E is a compact operator on X by Theorem 5 an Proposition 2.9 in [3, p. 215], we conclue Corollary 1. T(t)is a quasi-compact operator on X. Lemma 3. (See [2].) σ p (A E) an its algebraic inex is one, σ p (A E) an its geometric multiplicity is one. From Lemma 3 we euce that σ p (A E) an its algebraic multiplicity is one. Lemma 4. (See [2].) {γ C Re γ> or γ = ia, a, a R} belongs to the resolvent set of (A E).

10 466 Z. Shen et al. / J. Math. Anal. Appl. 341 (28) Combining Lemmas 3 an 4 we conclue that spectral boun of (A E) is zero. Thus by using Lemma 3, Theorem 1 an Corollary 1 an Theorem 2.1 [3, p. 343], we conclue the following result. Theorem 6. If there exist four positive constants μ j, μ j, j = 4, 5, such that <μ j μ j (x) μ j <, j = 4, 5, then exist a positive projection P of rank one, an suitable constants δ>, M such that T(t) P Me δt, where P = 2πi 1 Γ (zi A B E) 1 z, Γ is a circle with center an sufficiently small raius. Combining Lemmas 3 an 4, [5, Theorem 14], Theorem 6, Theorem 2.1 of [3, p. 216], we obtain the following result. Theorem 7. If there exist four positive constants μ j, μ j, j = 4, 5 such that <μ j μ j (x) μ j <, j = 4, 5, then the time-epenent solution of the system (1) (9) strongly converges to its steay-state solution that is an lim p(x,t) =ˆp, t p(x,t) ˆp Ce εt, ε>, C 1, where ˆp is the eigenvector corresponing to. Proof. By using Theorem 2.1 of [3, p. 216], Theorem 6, we obtain T(t)= T 1 (t) R(t), where T 1 (t) = P, P is the positive projection of, R(t) Ce εt, ε>, C 1. Then p(x,t) = T(t)p() = Pp() R(t)p() = p,q ˆp R(t)p() =ˆp R(t)p(), where p() = (1,,,,, ), Q = (1, 1, 1, 1, 1, 1) is the eigenvector corresponing eigenvalue of the ajoint matrix (A E). As a result, the exponential asymptotical stability of the solution of the system is obtaine. Acknowlegments The authors are grateful to the anonymous referees of the paper for their helpful comments. References [1] B.S. Dhillon, O.C. Anue, Common-cause failure analysis of a parallel system with warm stanby, Microelectron. Reliab. 33 (9) (1993) [2] Weiwei Hu, Asymptotic stability analysis of a parallel repairable system with warm stanby uner common-cause failure, Acta Anal. Funct. Appl. 8 (1) (26) 5 2. [3] Nager Rainer, One-Parameter Semigroups of Positive Operators, Lecture Notes in Math., vol. 1184, Springer, Berlin, [4] G.F. Webb, Theory of Nonlinear Age-Depenent Population Dynamics, Marcel Dekker, New York, [5] Gupur Geni, Xue-zhi Li, Guang-tain Zhu, Functional Analysis Metho in Queueing Theory, Research Information Lt, Hertforshire, 21. [6] Houbao Xu, Helong Liu, Jingyuan Yu, Guangtian Zhu, The asymptotic stability of a man-machine system with critical an non-critical human error, System Sci. Math. 1 (25(5)) (25)