Cost analysis of predictive maintenance in a failure-prone two-stage service retrial queue with orbital search

This paper explores a single-server retrial queueing system with two-stage service and preventive maintenance. Customers arrive in batches according to a compound Poisson process. The server provides two-stages of essential service to all arriving customers. If the server is idle upon the arrival of a batch, one of the customers in the batch receives the first stage of service immediately, while the rest join the orbit. Otherwise, the entire incoming batch joins the orbit. Once the service is completed, the customer immediately moves to the second stage of the service facility to complete the remaining service. The maintenance policy is constructively utilised during the idle state, even if the system is non-empty. Breakdowns of the main server can occur at any point during the busy state, and the repairs of the server commence without delay. If the server is idle in a non-empty system, then the server may search for customers in the orbit. Using the supplementary variable method, the equilibrium condition of the system is derived, and the stochastic decomposition law is verified. The system's probabilities and performance metrics are obtained. The optimal maintenance rate is determined through cost analysis to minimise expenses for the proposed model.