Exponential Stability and Availability Analysis of a Complex Standby System

Chunli Wang; Mingyan Teng; Fu Zheng

doi:10.1515/JSSI-2014-0279

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Exponential Stability and Availability Analysis of a Complex Standby System

Chunli Wang , Mingyan Teng and Fu Zheng

Published/Copyright: June 25, 2014

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From the journal Journal of Systems Science and Information Volume 2 Issue 3

Abstract

In this paper, we first investigate the solution of the two correlated units redundant system with two types of failure. By using the method of functional analysis, especially, the c₀ semigroup theory of bounded linear operators on Banach space, we prove the well-posedness and the existence of positive solution of the system. By analyzing the spectra distribution of the system operator, we prove that the dynamic solution of the system asymptotically converges to the nonnegative steady-state solution which is the eigenfunction corresponding to eigenvalue 0 of the system operator. Furthermore, we discuss the exponential stability of the system. Finally, we analyze the reliability the system with the help of our main results and present some reliability indices of the system at the end of the paper.

Keywords: redundant system; c0 semigroups; well-posedness; stability; availability

1 Introduction

Reliability is an important concept of various complex systems. The need of obtaining highly reliable system has became more and more important with the development of the modern technology. The high degree of reliability is usually achieved by introducing redundancy. The redundant system discussed here comes from [1]. It consists of two sub-systems A and B, connected in parallel having one regular and one expert repairman. This system has four states, viz. normal, degraded, failed and waiting. The sub-system B is in standby mode and consists of further two units, viz. B₁ and B₂, connected in parallel. It is assumed that when both the units of sub-system B start functioning simultaneously, then sub-system B works like the sub-system A. So, when some failure occurs in sub-system A, the two units of sub-system B will be in operable condition through perfect switching over device. If any one unit of subsystem B fails (after failing sub-system A), the system will work but in degraded stage. The system will be in down state only when both sub-system A and B fail or due to occurrence of environmental failure.

In [1], the well-posedness and asymptotic stability of the redundant system have been obtained. Because it is difficult to determine the convergence rate of the dynamic solution only by asymptotic stability, the study of the exponential stability is in demand, which is a more valuable property in application. We will continue to discuss the above system and obtain the exponential stability of the system. It should be pointed out that, in recent years, though there are several papers (e.g. see [3-6]) on the exponential stability of the repairable systems, the method of [2-5] is involved in quasi-compactness and is hard to be applied here. Moreover, as the applications of the main results of [1], we will calculate the stationary availability of the system with another method different from the traditional one [6-9].

The rest of this paper is organized as follows. In section 2, we introduce the mathematical model and some notations of the system. In the meantime, some results on the well-posedness and stability are introduced in this section. In section 3, the exponential of the system is presented. In the last section, we investigate the stationary availability of the system with another method different from the traditional one and give several numerical simulation examples.

2 Mathematical model formulation and some results

Most materials of this section come from [1] and for convenience, we do not restate and introduce them freely. Let a = λ₀ + λ₁ + λ₂, p₀(t) be the probability that the system is in operable state at time t, p₁(t) probability that the system is in waiting state at time t, p_i(t, x)dx probability that the system is in operable state at time and elapsed repair time lies between x and x + dx, where j = 8, 9, 10, 11, and p_i(t, x)dx probability that the system is in failed state at time t and elapsed repair time lies between x and x + dx, where i = 7, 12. With some additional assumptions, the differential equations of the system are obtained,

(1)p0(t)=∫0∞p2(t,x)μ1(x)dx+∫0∞p3(t,x)μ2(x)dx+∫0∞p4(t,x)μ(x)dx+∫0∞η(x)[p5(t,x)+p6(t,x)]dx+∫0∞p7(t,x)μ3(x)dx+∫0∞p12(t,x)β(x)dx

(2)[ddt+r1]p1(t)=λ0∫0∞p4(t,x)dx

(3)[∂∂t+∂∂x+μ1(x)+λ2]p2(t,x)=0

(4)[∂∂t+∂∂x+μ2(x)+λ1]p3(t,x)=0

(5)[∂∂t+∂∂x+μ(x)+λ]p4(t,x)=0

(6)[∂∂t+∂∂x+η(x)+λ1+λ2+r2]p5(t,x)=0

(7)[∂∂t+∂∂x+η(x)+λ1+λ2]p6(t,x)=0

(8)[∂∂t+∂∂x+μ3(x)]p7(t,x)=0

(9)[∂∂t+∂∂x+μ1(x)+r2]p8(t,x)=0

(10)[∂∂t+∂∂x+μ2(x)+r2]p9(t,x)=0

(11)[∂∂t+∂∂x+μ1(x)+λ2]p10(t,x)=0

(12)[∂∂t+∂∂x+μ2(x)+λ1]p11(t,x)=0

(13)[∂∂t+∂∂x+β(x)]p12(t,x)=0

and with the boundary conditions

(14)p2(t,0)=λ1p0(t),p3(t,0)=λ2p0(t)

(15)p4(t,0)=λ2∫0∞p2(t,x)dx+λ1∫0∞p3(t,x)dx

(16)p5(t,0)=λ0p0(t)+r1p1(t)+∫0∞μ1(x)p8(t,x)dx+∫0∞μ2(x)p9(t,x)dx

(17)p6(t,0)=r2∫0∞p5(t,x)dx+∫0∞μ1(x)p10(t,x)dx+∫0∞μ2(x)p11(t,x)dx

(18)p7(t,0)=λ3p0(t),p8(t,0)=λ1∫0∞p5(t,x)dx,p9(t,0)=λ1∫0∞p5(t,x)dx

(19)p10(t,0)=λ1∫0∞p6(t,x)dx+r2∫0∞p8(t,x)dx

(20)p11(t,0)=λ2∫0∞p6(t,x)dx+r2∫0∞p9(t,x)dx

(21)p12(t,0)=λ2∫0∞p10(t,x)dx+λ1∫0∞p11(t,x)dx

and the initial conditions p₀ = 1; p₁ = p₁(0, x) = · · · = p₂(0, x) = 0. You can refer to [1] for the concrete meanings of the symbols, such as λ₀ and r₁ etc.

The equations (1)~(21) can be rewritten as an the abstract Cauchy problem (see [11–12]) in Banach space X:

(ACP)dp(t)dt=Ap(t),t≥0,p(0)=(1,0,⋯,0)T.

in which 𝓐 is defined in [4] and X = ℝ²×(L¹(ℝ+))¹¹ is Banach space. For p = (p₀, p₁, p₂ (x), · · ·, p₁₂(x)) ∈ X, The norm of p is given by

||p||=|p0|+|p1|+∑i=212∫0∞|pi(x)|dx.

Let µ₄(x) = µ(x), µ₅(x) = β(x), µ₆(x)= η(x). The main results of this paper are obtained under the following assumptions.

General Assumptions

There exist positive constants H and c, such that for any t ≥ 0,

1t∫t∞e−∫0uμi(s)dsdu>H,

and

c=min{infx∈R+μi(x),i=1,2,⋯,6}.

The physical meaning of the general assumption is the mean repair rate is not too small. By the general assumptions, it is easily to obtain the following identify

∫0∞ui(x)e−∫0xui(τ)d(τ)dx=1.

The following theorems are main results of [4].

Theorem 1

𝓐 generates a positive C₀ contractive semigroup T(t) on X and the unique time dependent positive solution to the equations (1)~(21) exists.

Theorem 2

Zero is an algebraically simple eigenvalue of 𝓐 and the system is asymtotically stable. The steady-state solution is the positive unit eigenfunction corresponding eigenvalue 0 of the generator 𝓐.

3 The exponential stability of the system

Let T(t) be the semigroups generated by the operator 𝓐 and p(0) the initial value, the asymptotic stability shows that

T(t)p(0)=(p(0),Q)p~+T1(t)p(0)=p~+T1(t)p(0),limt→∞T1(t)p(0)=0,

in which p̃ is the positive unit eigenfunction corresponding to eigenvalue 0 of 𝓐 and Q = (1, · · · , 1) ∈ x*.

In general, the above stability does not implies the exponential stability, that is to say there exist positive ω and M such that

||T1(t)p(0)||≤Me−ωt,t≥0.

But in some special cases this implication holds. Now we will show the exponential stability of the system. First, we introduce some notations and results which come from references [3] and [14].

Let X be a Banach space and T(t) be a c₀ semigroups on X and 𝓐 its generator, for λ ∈ ρ(𝓐), then we have the symbols

ω1(T)=limt→+∞ln⁡||T(t)R(λ,A)||t,s0(A)=sup{ω>s(A):supReλ=ω||R(λ,A)||<+∞},

in which ω₁(T) is the growth bound of the semigroups T(t) and s₀(A) is abscissa of uniform boundedness of the resolvent of 𝓐. Their relationship is given by the following lemma which comes from [3].

Lemma 1

Let T(t) be a c₀ semigroup on Banach space X and 𝓐 its generator, then ω₁(T) < s₀(𝓐).

The following Lemma comes from [14].

Lemma 2

For each y ∈ X, b ∈ ℝ and ib ∈ ℂ, the unique solution to the equations (ib – A)p = y exists and p is given by

(22)(a+ib)p0=∫0∞p2(x)μ1(x)dx+∫0∞p3(x)μ2(x)dx+∫0∞p4(x)μ(x)dx+∫0∞η(x)[p5(x)+p6(x)]dx+∫0∞p7(x)μ3(x)dx+y0

(23)(r1+ib)p1=λ0∫0∞p4(x)dx+y1

(24)p2(x)=p2(0)e−∫0x[ib+λ2+μ1(s)]ds+∫0∞e−∫τx[ib+λ2+μ1(s)]dsy2(τ)d(τ)

(25)p3(x)=p3(0)e−∫0x[ib+λ1+μ2(s)]ds+∫0∞e−∫τx[ib+λ1+μ2(s)]dsy3(τ)d(τ)

(26)p4(x)=p4(0)e−∫0x[ib+λ0+μ(s)]ds+∫0∞e−∫τx[ib+λ0+μ(s)]dsy4(τ)d(τ)

(27)p5(x)=p5(0)e−∫0x[ib+λ1+λ2+r2+η(s)]ds+∫0∞e−∫τx[ib+λ1+λ2+r2+η(s)]dsy5(τ)d(τ)

(28)p6(x)=p6(0)e−∫0x[ib+λ1+λ2+η(s)]ds+∫0∞e−∫τx[ib+λ1+λ2+η(s)]dsy6(τ)d(τ)

(29)p7(x)=p7(0)e−∫0x[ib+μ3(s)]ds+∫0∞e−∫τx[ib+μ3(s)]dsy7(τ)d(τ)

(30)p8(x)=p8(0)e−∫0x[ib+r2+μ1(s)]ds+∫0∞e−∫τx[ib+r2+μ1(s)]dsy8(τ)d(τ)

(31)p9(x)=p9(0)e−∫0x[ib+r2+μ2(s)]ds+∫0∞e−∫τx[ib+r2+μ2(s)]dsy9(τ)d(τ)

(32)p10(x)=p10(0)e−∫0x[ib+λ2+μ1(s)]ds+∫0∞e−∫τx[ib+λ2+μ1(s)]dsy10(τ)d(τ)

(33)p11(x)=p11(0)e−∫0x[ib+λ1+μ2(s)]ds+∫0∞e−∫τx[ib+λ1+μ2(s)]dsy11(τ)d(τ)

(34)p12(x)=p12(0)e−∫0x[ib+β(s)]ds+∫0∞e−∫τx[ib+β(s)]dsy12(τ)d(τ)

(35)p2(0)=λ1p0,p3(0)=λ2p0

(36)p4(0)=λ2∫0∞p2(x)dx+λ1∫0∞p3(x)dx

(37)p5(0)=λ0p0+r1p1+∫0∞μ1(x)p8(x)dx+∫0∞μ2p9(x)dx

(38)p6(0)=r2∫0∞p5(x)dx+∫0∞μ1p10(x)dx+∫0∞μ2p11(x)dx

(39)p7(0)=λ3p0,p8(0)=λ1∫0∞p5(x)dx,p9(0)=λ2∫0∞p5(x)dx

(40)p10(0)=λ1∫0∞p6(x)dx+r2∫0∞p8(x)dx

(41)p11(0)=λ2∫0∞p6(x)dx+r2∫0∞p9(x)dx

(42)p12(0)=λ2∫0∞p10(x)dx+λ1∫0∞p11(x)dx

Theorem 3

The system is exponentially stable.

Proof

Let one rank projection P be the associated spectral projection of the eigenvalue 0 since 0 is an algebraically simple eigenvalue of 𝓐. Thus X has the invariant decomposition

X=X0+Y,X0=PX,Y=(I−P)X,

in which X₀ is the eigenfunction subspace whose dimension is one. Y is also Banach space equipped with the norm of X. Let T₁(t) and A₁ be the restrictions of T(t) and A on Y respectively. Obviously, we have σ(A₁) = σ(A) \ {0}. Now we will show that s₀(A₁) ≤ 0. However, 0 is a regular point of A₁, so it is sufficient to show that

supγ=ib,|b|>δ>0,b∈R||R(λ,A1)||<+∞.

In fact, for r = ib, b ∈ ℝ and each given y∈X, from Theorem 2 the equation (rI – A)p = y has solution p ∈ D(A) and p_i(x) is given by (17) and (18). Thus we have

(43)|b|∫0∞|pi(x)|dx≤∫0∞|pi(0)re−∫0x[r+γi+hiξ]dξ|dx+|b|∫0∞|∫0∞yi(τ)e−∫τx[r+γi+hi(ξ)d(ξ)]|dx

(44)|b|∫0∞|p2+i(x)|dx≤∫0∞|p2+i(0)re−∫0x[r+hiξ]dξ|dx+∫0∞|pi(0)r(e−∫0x[r+hiξ]dξ−re−∫0x[r+γi+hiξ]dξ)|dx+|b|c∫0∞y2+i(x)dx+|b|∫0∞|∫0xyi(τ)(e−∫τx[r+hi(ξ)]dξ−e−∫τx[r+γi+hiξ]dξ)dτ|dx

From (13), we have the following inequality

(45)(|b|+γ)p0≤y0+∑i=12|∫0∞pi(0)hi(x)e−∫0∞(r+γi+hi(ξ))dξdx|+∑i=12∫0∞|yi(x)|dx

Moreover, by the boundary conditions (14)~(16) we have

(46)|p1(0)|≤λ1|p0|+∫0∞|p2(0)(1−re−∫0x[r+h2(ξ)]dξ−h2(x)e−∫0x[r+h2(ξ)]dξ)|dx+∫0∞|p4(0)(1−re−∫0x[r+h2(ξ)]dξ)|dx+2∫0∞|y2(x)|dx+∫0∞|y4(x)|dx

(47)|p2(0)|≤λ2|p0|+∫0∞|p1(0)(1−re−∫0x[r+h1(ξ)]dξ−h1(x)e−∫0x[r+h1(ξ)]dξ)|dx+∫0∞|p3(0)(1−re−∫0x[r+h1(ξ)]dξ)|dx+2∫0∞|y1(x)|dx+∫0∞|y3(x)|dx

If adding (43)~(47) together, we would have the following estimate

(48)|b|(|p0|+∑i=14∫0∞|pi(x)|dx)≤y0+(1+|b|c)∑i=14∫0∞|yi(x)|dx+2∑i=12∫0∞|yi(x)|dx

By the definition of the norm on the space X and (48), we obtain that

||R(rI−A1)||≤1|b|(2+|b|c).

Therefore we know that,

limb→+∞||R(rI−A1)||=1c,

and

supr=ib,|b|>δ>0,b∈R||R(rI−A1)||<+∞.

From the definition of s₀(A) and Lemma 2, we obtain that s₀(A₁) ≤ 0 and ω₁(T₁) < 0 due to ω₁(T₁) < s₀ (A₁). However, by [14] we know semigroups T(t) is exponentially stable if and only if ω₁(T) < 0. Thus we complete the proof of Theorem 3.

In fact, the exponential stability of the repairable system was firstly given in [14]. You can refer to the reference [14] in the modified version for more information. To show the effectiveness of the reliability of the repair system, we include this part again in this note. The method is mainly employed from [3].

4 Some steady-state reliability indices

According to [2],

p0(t)+∑i=26∫0∞pi(x,t)dx+∑i=811∫0∞pi(x,t)dx

and

AV=limt→∞⁡(p0(t)+∑i=26∫0∞pi(x,t)dx+∑i=811∫0∞pi(x,t)dx)

are the instantaneous availability and the stationary availability of the repairable system, respectively. The availability of the system is one of the most important reliability indices and engineers are especially interested in the steady-state availability. Now we will obtain the stationary availability of the system with another method different from the traditional one. For convenience, set

f2(x)=e−∫0x[μ1(s)+λ2]ds,a2=∫0∞μ1(x)f2(x)dx,b2=∫0∞f2(x)dx,f3(x)=e−∫0x[μ2(s)+λ1]ds,a3=∫0∞μ2(x)f3(x)dx,b2=∫0∞f3(x)x,f4(x)=e−∫0x[μ(s)+λ0]ds,a4=∫0∞μ(x)f4(x)dx,b4=∫0∞f4(x)dx,f5(x)=e−∫0x[η(s)+λ1+λ2+r2]ds,a5=∫0∞η(x)f5(x)dx,b5=∫0∞f5(x)dx,f6(x)=e−∫0x[η(s)+λ1+λ2]ds,a6=∫0∞η(x)f6(x)dx,b6=∫0∞f6(x)dx,f7(x)=e−∫0xμ3(s)ds,b7=∫0∞f7(x)dx,f8(x)=e−∫0x[μ1(s)+r2]ds,a8=∫0∞μ1(x)f8(x)dx,b8=∫0∞f8(x)dx,f9(x)=e−∫0x[μ2(s)+r2]ds,a9=∫0∞μ2(x)f9(x)dx,b9=∫0∞f9(x)dx,f12(x)=e−∫0xβ(s)ds,b12=∫0∞f12(x)dx.

Theorem 4

The stationary availability of the system is

(49)AV=1+∑i=46bici+∑i=89bici+b2(c10+λ1)+b3(c11+λ2)1+c1+∑i=46bici+λ3b7+∑i=89bici+b2(c10+λ1)+b3(c11+λ2)+b12c12

in which

c1=r1−1λ0λ1λ2b4(b2+b3),c4=λ1λ2(b2+b3),c5=[1−b5(λ1a8+λ2a9)]−1[λ0+λ0λ1λ2b4(b2+b3)],c8=λ1b5[1−b5(λ1a8+λ2a9)]−1[λ0+λ0λ1λ2b4(b2+b3)],c9=λ2b5[1−b5(λ1a8+λ2a9)]−1[λ0+λ0λ1λ2b4(b2+b3)],c6=r2(b5c5+a2b8c8+a3b9c9)[1−b6(λ1a2+λ2a3)]−1,c10=λ1b6c6+r2b8c8,c11=λ2b6c6+r2b9c9,c12=λ2b2c10+λ1b3c11,

Proof

In order to compute the stationary availability of the discussed system, we need to obtain the formulation of the positive unit eigenfunction p corresponding to eigenvalue 0 of the operator 𝓐. Now we complete this job. In fact, 𝓐p = 0 is the following set of equations:

(50)p2′(x)=[μ1(x)+λ2]p2(x),p3′(x)=[μ2(x)+λ1]p3(x),p4′(x)=[μ(x)+λ0]p4(x),p5′(x)=[η(x)+λ1+λ2+r2]p5(x),p6′(x)=[η(x)+λ1+λ2]p6(x),p7′(x)=μ3(x)p7(x),p8′(x)=[μ1(x)+r2]p8(x),p9′(x)=[μ2(x)+r2]p9(x),p10′(x)=[μ1(x)+λ2]p10(x),p11′(x)=[μ2(x)+λ1]p11(x)

with the initial conditions:

(51)r1p1=λ0∫0∞p4(x)dx,p2(0)=λ1p0,p0(0)=λ2p0

(52)p4(0)=λ2∫0∞p2(x)dx+λ1∫0∞p3(x)dx

(53)p5(0)=λ0p0+r1p1+∫0∞μ1(x)p8(x)dx+∫0∞μ2(x)p9(x)dx

(54)p6(0)=r2∫0∞p5(x)dx+∫0∞μ1(x)p10(x)dx+∫0∞μ2(x)p11(x)dx

(55)p7(0)=λ3p0,p8(0)=λ1∫0∞p5(x)dx,p8(0)=λ2∫0∞p5(x)dx

(56)p10(0)=λ1∫0∞p6(x)dx+r2∫0∞p8(x)dx

(57)p11(0)=λ2∫0∞p6(x)dx+r2∫0∞p9(x)dx

(58)p12(0)=λ2∫0∞p10(x)dx+λ1∫0∞p11(x)dx

Solving equations (50), we have

(59)pi(x)=pi(0)fi(x),i=2,3,⋯,9,p10(x)=p10(0)f2(x)p11(x)=p11(0)f2(x),p12(x)=p12(0)f12(x)

By the Theorem 4.1 of [1], because 0 is a simple eigenvalue of 𝓐 and there exists a corresponding positive eigenvector, we can set p₀ = 1.With the help of

∫0∞ui(x)e−∫0xui(τ)dτdx=1

and the notations of c_i, it is easy to know that

p1=c1,p2(0)=λ1,p3(0)=λ2,pi(0)=ci(i=4,5,6),p7(0)=λ3,

and

pj(0)=cj(j=8,9,⋯,12).

Thus,

P=(1,c1,λ1f2(x),λ2f3(x),c4f4(x),c5f5(x),c6f6(x),λ3f7(x),c8f8(x),c9f9(x),c10f10(x),c11f11(x),c12f12(x))

is the positive eigenfunction corresponding to eigenvalue 0 of the operator 𝓐 and p̃ = ||P||^–1P. Therefore, we can get the expression AV = 1 – (c₁ + b₇c₇ + c₁₂c₁₂)||P||^–1 by the definition of the stationary availability, that is the formula (49) due to

||P||=1+c1+∑i=46bici+λ3b+∑i=89bici+b2(c10+λ1)+b3(c11+λ2)+b12c12.

We complete the proof of the Theorem 4. With the help of Theorem 4, we are able to calculate the mean up-time of the redundant system.

Theorem 5

The mean up-time (MUT) of the redundant system is equal to

MUT=1+∑i=46bici+λ3b+∑i=89bici+b2(c10+λ1)+b3(c11+λ2)λ0(1+b4c4)+λ2b2(c10+λ1)+λ1b3(c11+λ2)+(λ1+λ1)(b5c5+b6c6)+λ32b7.

Proof

Let M be the rate of failures of the system, then

M=||p||−1[λ0(1+p4)+λ2(p2+p10)+λ1(p3+p11)+(λ1+λ1)(p5+p6)+λ3b7].

From the proof of the Theorem 4, we know p₂ = λ₁b₂, p₃ = λ₂b₃, p₇ = λ₃b₂, p_i = c_ib_i, i = 4, 5, 6, p₁₀ = c₁₀b₂, and p₁₁ = c₁₁b₃. However, the mean up-time (MUT) of the redundant system is AVM. Computing directly leads to the formulation of MUT.

Corollary 1

The stationary availability and MUT of the system without standby are

AV′=11+λ3b7+λ0b12,MUT′=1λ0+λ32b7.

Supported by the National Natural Science Foundation of China (Grant No. 11201037, 11371070)

Acknowledgments

The authors would like to thank the referees for their helpful suggestions and comments.

References

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Received: 2013-10-24

Accepted: 2014-3-6

Published Online: 2014-6-25

Articles in the same Issue

https://doi.org/10.1515/JSSI-2014-0279

Keywords for this article

redundant system; c0 semigroups; well-posedness; stability; availability