Stochastic interpretation for a single server retrial queue with Bernoulli feedback and negative customers

In this paper, we introduce a qualitative analysis in order to study the monotonicity and comparability properties of a single-server retrial queueing model with Bernoulli feedback and negative customers, relative to stochastic orderings. Performance measures of such a system are available explicitly, while their forms are cumbersome(these formulas include integrals of Laplace transform, solutions of functional equations, etc.). Therefore, they are not exploitable from the application point of view. To overcome these difficulties, we present stochastic comparison methods in order to get qualitative estimates of these measures. In particular, we prove the monotonicity of the transition operator of the embedded Markov chain. In addition, we establish conditions for which transition operators as well as stationary probabilities, associated with two embedded Markov chains, having the same structure but with different parameters, are comparable relative to the given stochastic orderings. Further, numerical examples are carried out to illustrate the theoretical results.