Exploring non-Markovian dynamics: Stochastic analysis of a single-server retrial queue with recurrent clients and balking behavior under extended Bernoulli vacations.

The present study delves into the dynamics of a specific form of queueing system described as an retrial queue. Here, the queue comprises two distinct categories of clients: transit clients and recurrent clients. Transit clients are those who appear at the queue following a Poisson process, reflecting a random arrival pattern commonly seen in queueing scenarios. On the other hand, recurrent clients are predefined entities who immediately rejoin the queue once they've been served, demonstrating repetitive behavior in their interactions with the system. Once the server completes servicing a client, it initiates a vacation period. Moreover, in this approach, an optional extended vacation is also taken into account, i.e., the server may opt to indulge in an extended vacation following the initial essential Bernoulli vacation. Also, the consumers are allowed to balk. Further, the ergodicity requirement for the system's stability and then analytical findings for the stationary distribution are obtained. Additionally, various performance metrics for the system are also established. Furthermore, a comprehensive cost function is formulated and further optimized by incorporating a particle swarm optimization (PSO) approach. The convergence analysis is conducted as well, which is supported by illustrative figures. As a result, this work provides a beneficial understanding of enhancing the efficiency of such intricate queueing systems.