Existence and Uniqueness of the Dynamic Solution of a Series-Parallel Repairable System Consisting of Three-Unit with Multiple Vacations of a Repairman

We investigate a series-parallel repairable system consisting of three-unit with multiple vacations of a repairman. By using CC00-semigroup theory of linear operators in the functional analysis, we prove that the system is well-posed and has a unique positive dynamic solution.


Introduction
The series-parallel repairable systems consisting of three-unit are frequently used in practice.Since the strong practical background of the series-parallel repairable systems consisting of three-unitrepairable system, many researchers have studied them extensively under varying assumptions on the failures and repairs, see [1] [2].Repairman is one of the essential parts of repairable system, and can affect the economic benefit of the system.The repairman leaves for a vacation when there are no failed units for repair in system, which can have important influence to performance of system.In [3], the authors studied series-parallel repairable system consisting of three-unit with multiple vacations of a repairman 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.In [3], 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.Motivated by this, we study in this paper the wellposedness and the existence of a unique positive dynamic solution of the system, by using ∁ 0 -semigroup theory of linear operators.
The series-parallel repairable system consisting of three-unit with multiple vacations of a repairman can be described by the following equations (see [3]).
where () = � 1,  = 0, 0,  ≠ 0 Here (, ) ∈ [0, +∞) × [0, +∞), (, ) ∈ [0, +∞) × [0, +∞);  0 (, ) gives the probability that at time tall the three units are operating, the repairman is in vacation, the system is good and the elapsed vacation time lies in [,  + );  1 (, ) represents the probability that at time tunit 1 and one of unit 2 and unit 3 are operating, another one is waiting for repair, the repairman is in vacation, the system is good and the elapsed vacation time lies in [,  + );  2 (, ) represents the probability that at time t unit 2 and unit 3 are temporarily halted, unit 1 is waiting for repair, the repairman is in vacation, the system is down and the elapsed repair time lies in [,  + );  3 (, ) represents the probability that at time tone of unit 2 and unit 3 is temporarily halted, another one is waiting for repair, unit 1 is also waiting for repair ,the repairman is in vacation, the system is down and the elapsed vacation time lies in [,  + ); 4 (, ) represents the probability that at time tunit 1 is temporarily halted, unit 2 and unit 3 are waiting for repair, is also waiting for repair, the re-pairman is in vacation, the system is down and the elapsed vacation time lies in [,  + ); p 5 (, ) represents the probability that at time t unit 1 and one of unit 2 and unit 3 are operating, another one being repaired by the repairman, the system is good and the elapsed repair time of unit 2 or unit 3 lies in [,  + );  6 (, ) represents the probability that at time tunit 2 and unit 3 are temporarily halted, unit 1 being repaired, the system is down and the elapsed repair time of unit 1 lies in [,  + );  7 (, ) represents the probability that at time tone of unit 2 and unit 3 are temporarily halted, another one is waiting for repair, unit 1 being repaired, the system is down and the elapsed repair time of unit 1 lies in [,  + );  8 (, ) represents the probability that at time t unit 1 is temporarily halted, one of unit 2 and unit 3 is waiting for repair, another one being repaired by the repairman, the system is down and the elapsed repair time of unit 2 or unit 3 lies in [,  + );  9 (, ) represents the probability that at time t unit 1 is waiting for repair, one of unit 2 and unit 3 is temporarily halted, another one being repaired by the repairman, the system is down and the elapsed repair time of unit 2 or unit 3 lies in [,  + ); λ, μ are positive constants; () is the vacation rate function;  1 () and  2 () are repair rate function of unit 1 and unit 2 (or unit3).
Throughout the paper we require the following assumption for the vacation rate function()and the repair rate functions   ()( = 1,2).

Characteristic Equation
In this section we characterize () by the spectrum of a scalar 7 × 7-matrix, i.e., or we obtain a characteristic equation which relates () to the spectrum of an operator on the boundary space ∂X.For this purpose, we apply techniques and results from [5].We start from the operator � 0 , ( 0 )� defined by ( 0 ) = { ∈ (  )|  = 0},  0  =   .
For γ ∈ ρ( 0 ), the operator F  can be represented by the 10 × 10-matrix The Following result, which can be found in [7], plays important role for us to prove the existence of unique dynamic solution of the system.

Existence of a Unique Dynamic Solution of the System
In this section, we prove the well-posedness and the existence of a unique positive dynamic solution of the system.For this purpose we first prove that the operator A generates a positive contraction ∁ 0 -semigroup (()) ≥0 using the Phillips' theorem, see ( [8], Thm.C-II 1.2).The following lemma shows the surjectivity of  −  for  > 0.
Proof: Let ∈ ℝ,  > 0. Then all the entries of   are positive and using only elementary calculations one can show that the column sums are strictly less than 1.Hence, ∥   ∥< 1and thus 1 ∉ �  �.Using Lemma 3.3 we conclude that ∈ ().
From Lemma 4.1-4.3we see that all the conditions in Phillips' theorem (see [8], Thm.C-II 1.2) are fulfilled and thus we obtain the following result.