Analyzing Feedback <i>M</i>/<i>G</i>/1 Double Retrial Orbit with Two-Phase Optional Service and Repair under Working Vacation Policy

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.