Analysis of a GI/M/1/N Retrial Queue with Working Vacations, Bernoulli Feedback, and Impatient Customers: Steady-State Analysis, Asymptotics, and Cost Optimization for Next-Generation Networks

This paper analyzes a GI/M/1/N retrial queue with multiple working vacations, Bernoulli feedback, and impatient customers with retention. Customers who find the server busy do not wait in a physical queue; they join a retrial orbit in which the total retrial attempt rate is proportional to the orbit size (the classical linear retrial policy, rη). Using the supplementary variable technique combined with an embedded Markov chain at arrival epochs, we derive the steady-state distributions at pre-arrival and arbitrary epochs, and we obtain closed-form expressions for the matrix exponentials required for deterministic, Erlang, and hyperexponential inter-arrival times. A degenerate-limit result shows that, as the retrial rate tends to infinity, the model reduces in distribution to the classical GI/M/1/N waiting-line queue of Guendouzi and Saadaoui; the numerical validation reported in this paper is instead performed against an independent discrete-event simulation. We derive consistent performance measures (including a flow-balance identity usable as a numerical check), study the asymptotic behavior as the capacity N→∞, develop an economic model, and optimize it with particle swarm optimization (PSO) together with comparative GA and SA experiments and a robustness analysis of the cost parameters. Numerical illustrations highlight the impact of the system parameters on performance and profitability, with motivating applications to 6G networks, cloud computing, and smart healthcare.