Modeling and optimization of an M/M/1/K queue with single working vacation, feedback, and impatience timers under N-policy

This work presents an intensive study of a single server finite-capacity queueing model with impatience timers which depend on the server’s states, feedback, and a single working vacation policy operating under an \(N\)-policy discipline. We examine the scenario where the server must wait for the number of customers to reach \(N\) to start a regular busy period; otherwise, the server will initiate a working vacation or switch to the dormant state if the number of customers increases. By applying the Markov recursive method, the steady-state probabilities were derived. Various performance metrics were visually depicted to assess diverse system parameter configurations. After constructing the expected cost function of the model, Grey Wolf Optimization (GWO) algorithm is utilized to determine the optimum values of the service rates \(\mu^{*}\) and \(\mu_{v}^{*}\). Numerical examples are provided to validate the theoretical findings, offering insights into this intricate system.