Approximate controllability of stochastic bounds of stationary distribution of an M/G/1 queue with repeated attempts and two-phase service

This paper aims to study the monotonicity properties and the stochastic controllability of some performance measures of an M/G/1 queue with repeated attempts and two-phase service. First, we prove the monotonicity of the transition operator of the embedded Markov chain relative to convex ordering. Then, we obtain comparability conditions for the distribution of the number of customers in the system. Finally, we give insensitive bounds for the stationary distribution of the embedded Markov chain of the model under consideration. To do so, we use the partial information about the aging concepts of the first essential service time distribution and the second optional service time distribution. To highlight the different obtained theoretical results, numerical examples based on simulation are provided. More precisely, we discuss numerically the conditions under which the approximation of our considered model by an M/M/1 retrial queue with exponential two-phase service is valid.