Performance and economic analysis of an unreliable single-server queue with general retrial times and varied customer patience levels

This paper presents a comprehensive mathematical analysis of an unreliable single-server retrial queue with general retrial times, serving two types of customer arrivals: high-patience and low-patience customers. Customers arrive in the system following two Poisson processes with different service rates. In addition, the model incorporates essential features such as service times, reserved times, and repair times, all following general distributions. The proposed model has practical applications in diverse domains, including healthcare systems, web traffic management, and call centers. Using the supplementary variable technique, we carry out an extensive analysis of the model. This approach allows us to derive the ergodicity condition for this Markov chain and compute its stationary distribution. The main performance measures of the system are expressed through the stationary state probabilities. Numerical illustrations are presented. Finally, we conduct an economic study to assess the impact of various system parameters on performance measures and total cost, offering a visual overview of the system's effectiveness and profitability. A comparative analysis with existing models shows how our approach generalizes traditional retrial queue models, which typically consider a single type of customer arrival, by considering two distinct customer classes. This contributes to the advancement of queueing theory and provides insight into optimizing real-world systems.