Asymptotic Stability of the Dynamic Solution of an N-Unit Series System with Finite Number of Vacations

We investigate an N-unit series system with finite number of vacations. By analyzing the spectral distribution of the system operator and taking into account the irreducibility of the semigroup generated by the system operator we prove that the dynamic solution converges strongly to the steady state solution. Thus we obtain asymptotic stability of the dynamic solution of the system.


Introduction
The series repairable systems are the classical repairable systems in reliability theory.As a result of the strong practical background of series repairable systems, many researchers have studied them extensively under varying assumptions (see [1] [2] [3] [4]).In [4], the authors studied an N-unit series system with finite number of vacations and obtained some reliability expressions such as the Laplace transform of the reliability, the mean time to the first failure, the availability and the failure frequency of the system by using the supplementary variable method and the generalized Markov progress method as well as the Laplace transform.The authors used the dynamic solution and its asymptotic stability in calculating the availability and the reliability.But they did not prove the existence of the dynamic solution and the asymptotic stability of the dynamic solution.Motivated by this, A. Osman and A. Haji proved in [5] the existence of a unique positive dynamic solution of the system by using C 0 -semigroup theory of linear operators.In this paper, we further study this system and prove that the dynamic solution of the system converges strongly to its steady state solution by analyzing the spectral distribution of the system operator and taking into account the irreducibility of the semigroup generated by the system operator; thus we obtain the asymptotic stability of the dynamic solution of this system.
The rest of this paper is organized as follows: In Section 2, we present the mathematical model of the system and give some results obtained in [5].In Section 3, we obtain main result on stability of the system by analyzing the spectral distribution of the system operator and taking into account the irreducibility of the semigroup generated by the system operator.

Previous Results
According to [4], the N-unit series system with finite number of vacations can be described by the following integro-differential equations: With the boundary conditions and the initial conditions , , where t ω ∈ ∞ × ∞ and the symbols in the equations have the following meaning.
( ) 0 p t : The probability that n units at time t are in working state and the repairman is idle; ( ) , d k p t y y : The probability that at time t the repairman is repairing the failed unit ( ) In [5], the authors transformed the system (1), ( 2) and (3) into the following abstract Cauchy problem ( [6], Def.II.6.1) on the Banach space ( ) where , and proved the following results.

The Main Result
To prove our main result on the asymptotic stability of the dynamic solution of system, we first prove the some lemmas.In [7], A. ( ) ( ) and there exists 0 γ ∈  such that ( ) -matrix, as follows:      ( )            .Applying Lemma 3.1 1), we immediately obtain ( ) Using Lemma 3.1 2) we can show that 0 is the only spectral point of A on the imaginary axis.
Lemma 3.3: The spectrum ( ) Proof: By Lemma 3.1 it suffices to prove that ( ) Since the General Assumption 2.1 implies that there exists r and ( ) 0 , g y g y y g y y g y y g y y a where we used the strict positivity of ( )   ( From ( 67)-(72) we deduce 1 ai D Φ < , thus the spectral radius fulfills ( )

2 :
The system (1), (2) and (3) has a unique positive dynamic solution Haji and A. Radl gave the following result.Lemma 3.1: Let ( ) 0 A γ ρ ∈ , then Journal of Applied Mathematics and Physics 1) one can compute each column sum of the ( ) on the right hand side of the inequality (61) can be estimated as )-(66) we can estimate each column sum of absolute entries of ai D Φ as follows.