Existence and Uniqueness of Solution of an N-unit Series Repairable System with a Repairman Doing Ot

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Journal of Modern Mathematics Frontier Volume 3 Issue 3, September 2014 doi: 10.14355/jmmf.2014.0303.03

Existence and Uniqueness of Solution of an N-unit Series Repairable System with a Repairman Doing Other Work Abdukerim Haji*1, BilikizYunus2, Abdulla Hoxur1 College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

1

College of Mechanical Engineering, Xinjiang University, Urumqi 830047, China

2

abdukerimhaji@sina.com.cn; 2bilikiz62@sina.com.cn; 31565446975@qq.com

*1

Abstract Weinvestigate an N-unit series repairable system with a repairman doing other work. By using the method of functional analysis, especially, C0 -semigroup theory and spectral theory of positive operators, we prove the wellposedness and the existence of a unique positive solution of the system.

Keywords Series Repairable System; C0 -semigroup; Dynamic Solution; Well-posedness

Introduction Repairable system is not only a kind of important system discussed in reliability theory but also one of the main objects studied in reliability mathematics. In reliability analysis of repairable systems, it is usually assumed that the repairman has two states, either repairing the failed unit or idle. However, in actual practice, in order to increase system income, the administrator of system often will consider that the repairman should service for customer of outside system without affecting the repairman's own work at the same time. In (Liu and Tang 2005), Liu and Tang studied an N-unit series repairable system with a repairman doing other work, and obtained the expression of Laplace transforms of some primary reliability indices of the system by using the supplementary variable method, the generalized Markov progress method and the Laplace-transform technique. In (Liu and Tang 2005), the authors used the dynamic solution in calculating the availability and the reliability. But they did not discuss the existence of the positive dynamic solution. In this paper, we prove the well-posedness and the existence of a unique positive dynamic solution of the system, by using C0 -semigroup theory of linear operators

52

from (Engel and Nagel 2000, Nagel 1986). First, we reformulate the system as an abstract Cauchy problem. Next, we show that the system operator generates a positive contraction C0 -semigroup.

Furthermore, we prove the well-posedness of the system and the existence of a unique positive dynamic solution of the system for given initial condition. The system is described by the following equations, see (Liu and Tang 2005). n  dp0 ( t ) = ( c + Λ ) p0 ( t ) + ∫0∞ µ ( x ) p2 ( t , x ) dx + ∑ ∫0∞ µi ( y ) p1i ( t , y ) dy,  i =1  dt  ∂p1i ( t , y ) ∂p1i ( t , y ) 1, 2, , n, + = − µi ( x ) p1i ( t , y ) , i =  ∂y  ∂t  ∂p ( t , x ) ∂p ( t , x ) 2 + 2 = − ( c + µ ( x ) + Λ ) p2 ( t , x ) ,  t ∂ ∂x   ∂p3i ( t , x ) ∂p3i ( t , x ) + = − µ ( x ) p3i ( t , x ) + λi p4 ( t , x ) , i = 1, 2, , n,  ∂x  ∂t  ∂p4 ( t , x ) ∂p4 ( t , x ) + = − ( Λ + µ ( x ) ) p4 ( t , x ) + cp2 ( t , x ) ,  ∂x  ∂t

where Λ =∑ in=1 λi , it’s the boundary condition  p1i ( t , 0 ) = 1, 2, , n, λi p0 ( t ) + ∫0∞ µ ( x ) p3i ( t , x ) dx, i =  ∞ p = ( t , 0 ) cp0 ( t ) + ∫0 µ ( x ) p4 ( t , x ) dx, ( BC )  2 ) 0,=i 1, 2, , n,  p3i ( t , 0=   p4 ( t , 0 ) = 0,

and its initial condition

 p0 ( 0 ) = 1,  ) 0,=i 1, 2, , n,  p1i ( 0, y=  ( IC )  p2 ( 0, x ) = 0,  ) 0,=i 1, 2, , n,  p3i ( 0, x=  p ( 0, x ) = 0,  4

Journal of Modern Mathematics Frontier Volume 3 Issue 3, September 2014

where (t , x) ∈ [0, +∞) × [0, +∞) , (t , y ) ∈ [0, +∞) × [0, +∞) ,

p0 (t ) gives the probability that at time t all the units are in working state and the repairman is idle; Analogously, p1i (t , y )dy represents the probability that at time t the repairman is repairing the failed unit i , and the hours that the failed unit has been repaired lies in ( y, y + dy ] ; p2 (t , x)dx represents the probability that at time t all the units are in working state and repairman is servicing for customer, p3i (t , x)dx represents the probability that at time t the failed unit i is waiting to be repaired and the repairman is servicing for customer, and the hours that the customer has spent on the service lies in ( x, x + dx] ,

p4 (t , x)dx represents the probability that at time t all the units are in working state and repairman is servicing for one customer, the other customers is waiting for service.

Throughout the paper we require the following assumption for the functions µ ( x ) and µi ( x ) .

Generalassumption1.1.

The

x →∞

)

µ∞ := min µ , µ∞(1) , µ∞( 2) ,..., µ∞( n ) .

Cauchy problem ( Engel and Nagel 2000)

on the

Banach space ( X , . ) , where

) ( n

)

(

n

+ p2

+ p4

n+2

.

ψ i : L1y [ 0, ∞ ) → C , f  ψ i ( f ) = ∫0∞ µi ( y ) f ( y ) dy ,

ψ : L1 [ 0, ∞ ) → C ,

f  ψ ( f ) = ∫0∞ µ ( x ) f ( x ) dx .

Moreover, the operators

are defined respectively as

d d − − µi ( y ) , i = D1i = 1, 2,..., n , D2 = − − ( c + µ ( x ) + Λ ) dy dx

abstract approach as in e.g. (Casarino, Engel, Nagel and Nickel 2003). To this purpose we introduce the boundary space ∂X = C 2 n + 2 and then define the following ``boundary operators'' L and Φ by

 p0         p11 ( y )   p11 ( 0 )       p12 ( y )   p12 ( 0 )           p1n ( y )   p1n ( 0 )      L : D ( Am ) → ∂X ， p2 ( x )  = p2 ( 0 )  p x  p 0   31 ( )   31 ( )   p32 ( x )   p32 ( 0 )               p3n ( x )   p3n ( 0 )   p ( x )   p (0)   4   4 

n

+ ∑ p3i

L1x [ 0, ∞ )

L1x [ 0, ∞ )

 P0 , P11 ( y ) , P12 ( y ) ,..., P1n ( y ) , P2 ( x ) ,    ∈ X  P31 ( x ) , p32 ( x ) ,..., P3n ( x ) , P4 ( x ) 

The Space ( X , .

)

denote the linear functionals

t

P

n

0   0  ...   0  0   λ1  λ2  ...   λn  D4 

Here and in the following ψ i , i = 1, 2,..., n and ψ

n+2

L1x [ 0, ∞ ) i 1 =i 1 = L [ 0, ∞ ) 1 y

) (

0 0 ...

... 0 ... 0 ... 0 ... 0 ... ... ... D3n ... 0

D ( Am ) = C × Wy1,1 [ 0, ∞ ) × Wx1,1 [ 0, ∞ )

And

p = p0 + ∑ p1i

0 0 0 ...

0 0

... ... ...

To model the boundary conditions ( BC ) we use an

To apply semigroup theory we transform in this section the system ( R ) , ( BC ) , ( IC ) into an abstract

(

... ... ... ... ... ...

ψ 0 0 0 0 0 ... ... ...

0 0 0 D2 0 0 λ1 D31 0 λ2 0 D32 ... ... ... λn 0 0 c 0 0

d d D3i = − − µ ( x),i = 1, 2,..., n , D4 = − − ( Λ + µ ( x ) ) . dx dx

The Problem as an Abstract Cauchy Problem

X =C × L1y [ 0, ∞ ) × L1x [ 0, ∞ )

... ψ n ... 0 ... ... ... D1n

= D1i , i 1,= 2,..., n, D2 , D3i , i 1, 2,..., n and D4 on W 1,1 [ 0, ∞ )

µ : lim µ ( x ) > 0 ,= = µ∞( i ) : lim µi ( x ) = > 0, i 1, 2,..., n ,

(

 − (c + Λ ) ψ1  D11 0   ... ...  0 0   0 0 Am =  0 0   0 0   ... ...  0 0   0 0 

µ ( x) and

functions

µi ( x) : R+ → R+ is measurable and bounded such that x →∞

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) will also be called state space.

In a first step we introduce a ``maximal operator''

( Am , D ( Am ) ) describing only ( R ) . It is given by

and

Φ : X → ∂X ,

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Journal of Modern Mathematics Frontier Volume 3 Issue 3, September 2014

 p0     λ 0 ...  p11 ( y )   1    λ2 0 ...  p12 ( y )      ... ... ...    λn 0 ...  p1n ( y )      c 0 ... Φ  p2 ( x )  = 0 0 ...  p x   ( ) 0 0 ...  31   p32 ( x )      ... ... ...    0 0 ...     p3n ( x )   0 0 ...  p ( x)   4 

0 0 ...

0 0 0 0 ...

0 0

By a direct calculation we obtain the explicit form of  p0    the elements in ker ( γ − Am ) as follows.

0 ψ 0 ... 0 0  p y   11 ( )  0 0 ψ ... 0 0    p12 ( y )  Lemma 3.1: For γ ∈ ρ ( A0 ) , we have  ... ... ... ... ... ...      p ∈ ker ( γ − Am )  0 0 0 ... ψ 0   p1n ( y )   0 0 0 ... 0 ψ   ⇔  0 p ( x)  t 0 0 0 ... 0 0   2  p0 , p11 ( y ) , p12 ( y ) ,..., p1n ( y ) , p2 ( x ) ,  p x   ( ) p   ∈ D ( Am ) , 0= 0 0 ... 0 0  31   p ( x ) , p ( x ) ,..., p ( x ) , p ( x )  31 32 3n 4    p x ( ) ... ... ... ... ... ...   32  y  n 1 −γ y − ∫0 µi (ξ ) d ξ  ∞ 0 0 0 ... 0 0   p c µ y e dy = ∑ ( ) [ ∫ 1 i i 0   γ + c + Λ i =1 0 0 0 ... 0 0   p3n ( x )  x − ( γ + cΛ ) x − ∫ µ (ξ ) d ξ  p ( x)  0 +c2 ∫0∞ µ ( x ) e dx ] ,  4 

Now the system operator ( A, D ( A ) ) on X given by

−γ y −

Φp} Ap = Am p, D ( A ) = { p ∈ D ( Am ) | Lp =

p3i ( x ) = c3i e

Cauchy problem in the Banach space X as follows.

+ ( c2 + c4 )

In this section we characterise σ ( A ) by the spectrum of a scalar (2n + 2) × (2n + 2) -matrix, i.e. an operator on the boundary space ∂X . Clearly, it is in general much easier to determine the spectrum of an operator on the boundary space then to determine σ ( A ) directly. The abstract background can be found in (Greiner 1987). It has been widely used since then, see (Greiner 1987, Casarino, Engel, Nagel and Nickel 2003 or Kramar and Sikolya 2005) to mention a few applications. We start from the operator ( A0 , D ( A0 ) ) defined by

         Dγ =          

54

0 0 0

0 0 0

0  

0 

0 0

0 0

− ( γ + c +Λ ) x − ∫ µ (ξ ) d ξ x

0

，

− γ x − ∫ µ (ξ ) d ξ x

0

λi

(1 − e ) e γ Λ

− x − ∫ µ (ξ ) d ξ

−Λx

(

)

x

0

− ( γ +Λ ) x − ∫0 µ (ξ ) d ξ x

p4 ( x ) = c2 1 − e − cx e

,i = 1, 2,..., n，

+ c4 e

− ( γ +Λ ) x − ∫0 µ (ξ ) d ξ x

.

Using Lemma 1.2(Greiner 1987), the domain D ( Am ) of

          

D ( Am ) = D ( A0 ) ⊕ ker ( γ − Am ) .

Moreover, since L is surjective,

L |ker (γ − A ) : ker ( γ − Am ) → ∂X is m

invertible

for

each

γ ∈ ρ ( A0 ) , see Lemma 1.2 (Greiner 1987).

We denote its inverse by Dγ : =

(L |

ker ( γ − Am )

)

−1

: ∂X → ker ( γ − Am )

and call it ``Dirichlet operator''.

We can give the explicit form of Dγ as follows.

Lemma 3.2: For each γ ∈ ρ ( A0 ) , the operator Dγ has

D ( A0 ) := 0} , A0 p := Am p { p ∈ D ( Am ) | Lp = d1,2 0 d3,2 

µi ( ξ ) d ξ

the maximal operator Am decomposes as

Characteristic Equation

d1,1 d 2,1 0 

0

p2 ( x ) = c2 e

Describes the system completely. The above equations ( R ) , ( BC ) and ( IC ) are equivalent to the abstract

 dp ( t )  = Ap ( t ) , t ∈ [ 0, ∞ ) ( ACP )  dt t  =  p ( 0 ) (1, 0,..., 0 ) ∈ X

y

∫ = p1i c= , i 1, 2,..., n， 1i e

the form

d1, n 0 0 

d1, n +1 0 0 

0 0 0 

0 0 0 

d n +1, n 0 0

0 d n + 2, n +1 d n + 3, n +1

0 0 d n + 3, n + 2

0 0 0

0 

d n + 4, n +1 

0 

d n + 4, n + 3 

0 0

d 2 n + 2, n +1 d 2 n + 3, n +1

0 0

0 0

        

0 0 0 



d 2 n + 2,2 n +1 0



0 0 0 0 

       0  0  d n + 3,2 n + 2   d n + 4,2 n + 2       d 2 n + 2,2 n + 2  d 2 n + 3,2 n + 2  0 0 0 

Journal of Modern Mathematics Frontier Volume 3 Issue 3, September 2014

y 1 − γ y − ∫ µ i (ξ ) d ξ ∞ 0 = dy, i 1, 2 , n, ∫0 µi ( y )e γ +c+Λ

d1,i

d1, n +1 =

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a1, n +1 =

x 1 − (γ + c +Λ ) x − ∫ µ (ξ ) d ξ ∞ 0 dx, ∫0 µ ( x)e γ +c+Λ

λ1

γ +c+Λ

a1,2 n + 2 =

y

−γ y − ∫0 µi (ξ ) d ξ

, i 1, 2 , n, di +1,i e= =

∞

∫0

λ1

− ( γ + c +Λ ) x − ∫0 µ (ξ ) d ξ x

µ ( x) e +

λ1 Λ

(

)

∞ −Λx e ∫0 µ ( x ) 1 − e

(

)

∞ −Λx e ∫0 µ ( x ) 1 − e

Λ

λ

dx

y

−γ x − ∫0 µ (ξ ) d ξ

,

x

−γ x − ∫0 µ (ξ ) d ξ

dx

x

dx,

µ (ξ ) d ξ

∞ ∫0 i 2 a2,i = dy, i 1, 2,  , n, = ∫0 µi ( y ) e λ1 − (γ + c +Λ ) x − ∫ µ (ξ ) d ξ −Λx γ +c+Λ (1 − e ), , d n + 3,= =e n +1 Λ −γ y −

x

d n + 2, n +1

0

 d n + 3, n + 2 = e

x

−γ x − ∫0 µ (ξ ) d ξ

λ1

, d n + 3,2= n+2

Λ

a2, n +1 =

(1 − e

−Λx

x

)e

− γ x − ∫ µ (ξ ) d ξ 0

λ2

γ +c+Λ

,

λ1 Λ

(1 − e −Λx )e

x

−γ x − ∫0 µ (ξ ) d ξ

, d n + 4, n + 3 = e

λ2

0

− x−

x

d

0

n ,i

Λ

d 2 n + 2, n + 2 = e

(1 − e ) −Λx

− γ x − ∫ µ (ξ ) d ξ 0

,

d 2 n + 2,2= n+2

Λ

= d 2 n + 3,2 n +1 e

−γ x − ∫0 µ (ξ ) d ξ

−γ x − ∫0 µ (ξ ) d ξ

,

x

dx

x

dx,

Λ

(

)

∞ −Λx e ∫0 µ ( x ) 1 − e

λ

y

−γ x − ∫0 µ (ξ ) d ξ x

dx,

µ (ξ ) d ξ

∞ ∫o i n = dy, i 1, 2, , n, ∫ µi ( y ) e γ +c+Λ 0

an, n +1 =



λn

λ2

−γ y −

,

x

)

 a2,2 n + 2 =

−Λx

λn

Λ

(

dx

∞ −Λx e ∫0 µ ( x ) 1 − e

,

λ ∫ µ (ξ ) ξ 1 − e ) e= , ( a Λ

d 2 n + 2,= n +1

λ2

x

x

− γ x − ∫ µ (ξ ) d ξ

 d n + 4,2= n+2

+

− ( γ + c +Λ ) x − ∫0 µ (ξ ) d ξ

a2, n + 3 = ∫0∞ µ ( x ) e

 d n + 4,= n +1

∞

∫0

µ ( x) e

λn

∞

γ +c+Λ

∫0

µ ( x) e +

λn Λ

− ( γ + c +Λ ) x − ∫ µ (ξ ) d ξ x

0

(

dx

)

∞ −Λx e ∫0 µ ( x ) 1 − e

− γ x − ∫ µ (ξ ) d ξ x

0

dx,



(1 − e ) e −Λx

−γ x − ∫0 µ (ξ ) d ξ x

(

− ( γ +Λ ) x − ∫ µ (ξ ) d ξ x

an,2 n +1 = ∫0∞ µ ( x )e

,

)

1 − e − cx , an +1,i =

0



−γ x − ∫0 µ (ξ ) d ξ x

dx,

y c −γ y − ∫0 µi (ξ ) d ξ ∞ dy, i 1, 2, , n, = ∫0 µi ( y ) e γ +c+Λ

− ( γ + c +Λ ) x − 0 µ (ξ ) d ξ x

d 2 n + 3,2 n + 2 = e

− ( γ +Λ ) x − ∫0 µ (ξ ) d ξ x

.

an +1, n +1

For γ ∈ ρ ( A0 ) , the operator ΦDγ can be represented by the (2n + 2) × (2n + 2) -matrix

 a1,1 a1,2   a2,1 a2,2      an,1 an,2 a ΦDγ = a  n +1,1 n +1,2 0 0   0 0      0  0

a1,n +1

0



0

 a2,n

a2,n +1

0 a2,n +3 

0











an,n +1

0

0

 an,2n +1

 an +1,n an +1,n +1 0

0



0



0

0

0

0



0



0

0

0

0



0

















0

0

0

0



0



 an,n

0

a1,2n + 2   a2,2n + 2     an,2n + 2  an +1,2n + 2   0  0      0 

where

a1,i

(

)

x

0



 a1,n 

∫ dx µ ( x) e c ∞ = ∫0 γ + c + Λ + ∞ µ x 1 − e− cx e−(γ +Λ ) x − ∫ µ (ξ ) dξ dx, ∫0 ( )

an +1,2 n + 2 = ∫0∞ µ ( x )e

− ( γ +Λ ) x − ∫ µ (ξ ) d ξ x

0

dx.

The Following result, which can be found in (Haji and Radl 2007), plays important role for us to obtain our main result in Section Ⅳ.

Lemma 3.3(The characteristic equation): If γ ∈ ρ ( A0 ) and there exists γ 0 ∈ C such that 1 ∈ σ (ΦDγ 0 ) , then

γ ∈ σ ( A ) ⇔ 1 ∈ σ ( ΦDγ ) .

Well-Posedness of the System

λ

y

µ i (ξ ) d ξ

∞ ∫0 1 = dy, i 1, 2, , n, ∫ µi ( y )e γ +c+Λ 0 −γ y −

Our aim in this section is to prove the well-posedness

55

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Journal of Modern Mathematics Frontier Volume 3 Issue 3, September 2014

and the existence of a unique positive solution of the system. We will first prove that generates

( T ( t ) )t ≥ 0

a

positive

( A, D ( A) )

contraction

in fact

C0 -semigroup

using Phillips' theorem (Nagel 1986). To this

purpose let us check that

( A, D ( A) )

( A, D ( A) )

= q1i ( y )

and 1 if sgn + ( p0 ) =  0 if

surjective for some γ > 0.

Lemma 4.1:For γ ∈ R, γ > 0 , we have γ ∈ ρ ( A) .

Proof: Let γ ∈ R, γ > 0 . Then all the entries of ΦDγ are positive and using only elementary calculations one can show that both column sums are strictly less than

ΦDγ < 1, and thus 1 ∉ σ ( ΦDγ ) . Using

1. Hence,

Lemma 3.3 we conclude that γ ∈ ρ ( A) .

Lemma 4.2: A : D ( A ) → R ( A ) ⊂ X is a closed linear

If X ' denotes the dual space of X , then

X

=C × ( L∞y [0, ∞)) n

×

n

i =1

n

+ ∑ q3i i =1

L∞x [0, ∞ )

L∞y [0, ∞ )

+ q4

.

L∞x [0, ∞ )

p2 ( x) > 0, p2 ( x) ≤ 0,

p3i ( x) > 0, p3i ( x) ≤ 0,

1 if sgn + ( p4 ( x)) =  0 if

i = 1, 2 n,

i = 1, 2 n,

p4 ( x) > 0, p4 ( x) ≤ 0,

.

From Lemma 4.1- 4.3 we see that all the conditions in Phillips' theorem (Nagel 1986) are fulfilled and thus we obtain the following result. positive contraction C0 -semigroup (T (t ))t ≥ 0 .

,

From Theorem 4.4 and Cor.II.6.9 (Engel and Nagel 2000) we can characterize the well-posedness of ( ACP ) as follows.

L1x [0, ∞ )

q = (q0 , q11 ( y ), q12 ( y ), , q1n ( y ), q2 ( x), q31 ( x), q32 ( x), , q3n ( x), q4 ( x)) ∈ X t

'

Theorem 4.5: The associated abstract Cauchy problem ( ACP ) is well-posed.

.

Lemma 4.3: The operator ( A, D ( A ) ) is dispersive.

Proof: For

p = ( p0 , p11 ( y ), p12 ( y ), , p1n ( y ), p2 ( x), p31 ( x), p32 ( x), , p3n ( x), p4 ( x))t ∈ D( A)' we define

q = (q0 , q11 ( y ), q12 ( y ), , q1n ( y ), q2 ( x), q31 ( x), q32 ( x), , q3n ( x), q4 ( x))t ∈ X '

56

p1i ( y ) ≤ 0,

,

Theorem 4.4: The operator ( A, D( A)) generates a

+ q2

where

where

p1i ( y ) > 0,

1 if sgn + ( p2 ( x)) =  0 if

1 if sgn + ( p3i ( x)) =  0 if

p0 ≤ 0,

( A, D( A)) is a dispersive operator.

X ( L∞x [0, ∞)) n + 2

It is obvious that X ' is a Banach space endowed with the norm

q = q0 + ∑ q1i

1 if sgn + ( p1i ( y )) =  0 if

p0 > 0,

Noting the boundary condition, it is not difficult to see that Ap, q ≤ 0 . By (Nagel 1986, p.49) we obtain that

operator and D( A) is dense in X .

'

= p sgn + ( p1i ( y )), i 1, 2, n,

q4 ( x) = p sgn + ( p4 ( x))

is dispersive and γ − A is

The following lemma in particular shows the surjectivity of γ − A for any γ > 0.

= p sgn + ( p1i ( y )), i 1, 2, n,

q2 ( x) = p sgn + ( p2 ( x)),

fulfills the

conditions in Phillips' theorem, namely that D ( A ) is dense in X ,

q0 = p = sgn + ( p0 ), q1i ( y )

,

From Theorem 4.5 and Prop.II.6.2 (Engel and Nagel 2000) we can state our main result.

Theorem 4.6: The system ( R ) , ( BC ) and ( IC ) has a

,

unique positive dynamic solution

P ( x, t ) which

satisfies p (., t ) = 1, t ∈ [0, ∞) .

Proof: From Theorem 4.4, Theorem 4.5 and Prop.II.6.2 (Engel and Nagel 2000) we obtain that the associated abstract Cauchy problem ( ACP ) has a unique positive time-dependent expressed as

solution

P ( x, t )

which

can

be

Journal of Modern Mathematics Frontier Volume 3 Issue 3, September 2014

= P( x, t ) T= (t ) p (0) T (t )(1, , 0) .

Since

Greiner G, Perturbing the boundary conditions of a

p = ( p0 , p11 ( y ), p12 ( y ), , p1n ( y ), p2 ( x), p31 ( x), p32 ( x), , p3n ( x), p4 ( x)) ∈ D( A) t

'

satisfies

the system ( R ) , ( BC ) and ( IC ) , it is not difficult to

generator, Houston J. Math. 1987, 13, 213--229.

Haji A and Radl A, A semigroup approach to queueing systems, Semigroup Forum, 2007, 75, 610--624.

Kramar Mand Sikolya E, Spectral properties and asymptotic

know that

periodicity of flows in networks, Math. Z. 2005, 249, 139--

d p (t , x) d T (t ) P(0) = = 0. dt dt

t , x) Therefore, p (=

T (t )= P(0)

162.

= P(0) 1 . This shows

p (t ,.) = 1, ∀t ∈ [0, ∞) .

Luo, C.Y, An N-unit series

repairable system with a repairman doing other work, J. Nagel R, One-parameter Semigroups of Positive Operators,

This work was supported by the Natural Science Foundation of Xinjiang Uighur Autonomous Region (No. 2014211A002).

REFERENCES

Casarino V, Engel K-J, Nagel Rand Nickel G, A semigroup approach to boundary feedback systems, Integr. Equ. Oper. Theory, 2003, 47, 289--306.

Engel K-J and Nagel R, One-Parameter Semigroups for Evolution

Liu, R.B, Tang, Y.H. and

Natu. Science .Heilongjiang Uni., 2005, 22(4), 493--496.

ACKNOWLEDGMENT

Linear

www.sjmmf.org

Equations,

Graduate

Mathematics, vol. 194, Springer-Verlag, 2000.

Texts

in

Springer-Verlag, 1986.

Abdukerim Haji was born in Kashkar, Xinjiang Uighur Autonomous Region, China, on July 14, 1963. He earned his master's degreein 1990 from XinjiangUniversity, Urumqi, China and his Ph.D. degree in Applied Mathematics from Tuebingen University, Tuebingen, Germany, in 2007. He was a Lecturer of Mathematics in Xinjiang University, Urumqi, China from 1990 to 1996. Since 1996, he has been working as an associate Professor of Mathematics in Xinjiang University Urumqi, China. His current research interests are semigroup theory of linear operators and reliability analysis of systems.

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