We investigate the solution of a repairable parallel
system with primary as well as secondary failures. By using the method of functional
analysis, especially, the spectral theory of linear operators and the theory of

As science and technology develop, the theory of reliability has infiltrated into the basic sciences, technological sciences, applied sciences, and management sciences. It is well known that repairable parallel systems are the most essential and important systems in reliability theory. In practical applications, repairable parallel systems consisting of three units are often used. Since the strong practical background of such systems, many researchers have studied them extensively under varying assumptions on the failures and repairs; see [

The mathematical model of a repairable parallel system with primary as well as secondary failures was first put forward by Gupta; see [

The parallel repairable system with primary and secondary failures can be described by the following equations (see [

In [

The system has a unique positive time-dependent solution

The time-dependent solution

In this paper, we require the following assumption for the failure rate

The function

In this section, we rewrite the underlying problem as an abstract Cauchy problem on a suitable space

For simplicity, let

To model the boundary conditions (

The operator

In this section we investigate the boundary spectrum

Let

A combination of [

Clearly, knowing the operator matrix in (

The following consequence is useful to compute the boundary spectrum of

The imaginary axis belongs to the resolvent set of

The eigenvectors in

For

If for

The domain

Moreover, since

We can give the explicit form of

For each

The operator

For

The operators

(i)

(i) Note that

The part

Hence,

The spectrum of

Let

If, in addition, there exists

Let us first show the equivalence

To prove (i) observe first that

Conversely, if we assume that

Using the Characteristic Equation

For the operator

By the Characteristic Equation

Indeed, 0 is even the only spectral value of

Under Assumption

For any

Substituting

The main gaol in this section is to prove the well-posedness of the system. In order to prove this, we will need some lemmas.

We will prove the assertion in two steps.

We first prove that

We now prove that

For

If

Let

From Lemmas

The operator

We now characterize the well-posedness of (

For a closed operator

From Theorem

The system (

From Theorems

In this section, we prove the asymptotic stability of the system by using

Let

Under our assumption, we see from the Characteristic Equation

The above representation for the resolvent of

The semigroup

We know from [

We now use the information obtained on

To study the asymptotic behaviour of the semigroup

The set

From

We can now show the convergence of the semigroup to a one-dimensional equilibrium point.

The space

Since by Lemma

We now consider the generator

Combining Lemmas

There exists

Since the semigroup gives the solutions of the original system, we obtain our final result.

The time-dependent solution of the system (

In this paper, we considered a repairable system involving primary as well as secondary failures. By using the

The author expresses his gratitude to Professor Rainer Nagel and Dr. Agnes Radl as well as the editor and referee for the constructive comments and valuable suggestions. The author also wishes to thank DAAD for the financial support. This work was supported by Doctor Foundation of Xinjiang University (no. BS080108) and the Natural Science Foundation of China (no. 10861011).