This paper investigates an unreliable $ M/G(P_{1}, P_{2})/1 $ retrial queueing system with a woking vacation. An arriving customer successfully starts the first phase service with the probability $ \alpha $ or the server fails with the probability $ \bar{\alpha} $. Once failure happens, the serving customer is taken to the orbit. The failed server is taken for repair with some delay. Once the repair is comleted, the server is ready to provide service once again. In this background, we implemented the working vacation scenario. During working vacation, the service will be provided at a slower rate, rather than entirely stopping the service. The supplementary variable method was adopted to find the orbit and system lengths. Additionally, some unique results and numerical evaluations have been presented.

The present study delves into the dynamics of a specific form of queueing system described as an M / G / 1 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.

This research presents a comprehensive analysis of the M/G/1 double retrial queue model with a two-phase optional service and repair, under a working vacation policy. The model addresses scenarios where patients are willing to pay more for enhanced comfort and care. To accommodate this, an optional service phase is introduced, allowing patients to choose between ordinary and premium service levels. The system is further refined by a two-phase service mechanism: in the first phase, patients receive an initial level of service, and depending on their satisfaction or specific needs, they may opt for an enhanced second phase of service. Additionally, the model incorporates feedback and repair processes, reflecting a more realistic representation of queue dynamics. Patients who are dissatisfied with the service or encounter issues can re-enter the system for further attention. The non-Markovian model equations are solved using the probability-generating function approach, which facilitates the analysis of complex queue behaviors and the calculation of key performance indicators, such as expected system length and queue length. The paper includes detailed numerical results with test data to demonstrate the accuracy of the model’s predictions and validate the proposed system design’s effectiveness.

Queueing systems (QS) play a critical role in modeling and optimizing various real-world processes by managing the flow of entities within systems involving queues. To address this, the proposed QS integrates features such as single arrivals, a retrial mechanism, optional re-service, working vacations, and delay repair scenarios. The system’s dynamics are comprehensively analyzed using the supplementary variable technique, which provides deeper insights into its behavior and performance metrics. Furthermore, to enhance operational efficiency, an advanced cost optimization technique is employed to identify the optimal cost structure for the system. This holistic approach not only sheds light on the operational intricacies of the QS but also offers practical strategies for cost reduction and validation of analytical findings, thereby advancing queueing theory and broadening its practical applications across multiple domains.