The well-posed-ness of retrial queue system with Bernoulli feedback and starting failure

We study the well-posed-ness of a retrial queue with Bernoulli feedback, where the server is subject to starting failure. We proved the operator determined by the system described by the model generates a strongly continuous semigroup in a Banach static space, and hence the well-posed-ness of the system follows from the semigroup theory of bounded linear operators. In the proof of semigroup generation of the system operator, we use the approach of adjoint operator. By showing 1 is not an eigenvalue of adjoint operator of the system operator, we prove that operator without Bernoulli feedback generates a C0 semigroup of contraction, and then get the desired result by the perturbation theory of semigroup.