The Well-posed-ness And The Asymptotic Stability Of Retrial M/G/1 Queue System With Bernoulli Feedback And Starting Failure

In this paper, we study a retrial queue with Bernoulli feedback, where the server is subjected to starting failure. We shall prove the system is well-posed and stabilizes asymptotically to its stead state in the sense of norm in a Banach space. We shall complete the proof by two steps. At first, we prove that the system operator generates a strongly continuous semigroup in a Banach space and hence the well-posed-ness follows from the semigroup theory of bounded linear operators. Futher, we show that the semigroup is a positive semigroup of contractions and has positive conservative property. In the second step, we study the stability of the system, our aim is maily to prove that the imaginary aixs except zero are in the resolvent set of the operator and the zero is a simple eigenvalue of theoperator and hence the asymptotic stability of the system follows.In the study, the main difficulty is that we can't get the resolvent of the operator explicit. To overcome this we use the approachof dual method. In the study of the well-posed-ness of the system, we prove that operator without Bernoulli feedback generates a C₀ semigroup of contraction by showing I is not an eigenvalue of adjoint operator of the system operator, and then get desired result by the perturbation theory of semigroup. When we study the asymptotic stability of the system, for ensuring the strong continuous property of the semigroup in the dual space, we use the sun-dual semigroup theory to restrict the dual space on the sun-dual space, then we get the desired result by studying the resolvent equations of the system without the coupling items and estimating the norm of coupling operator.