An Adaptable Single-Server Encouraged Arrival, Balking, and Symmetric Stochastic Markovian Queuing System with Threshold Policy

This study examines an adaptable single-server encouraged arrival, balking, and symmetric stochastic Markovian queuing system. We have first time introduced encouraged arrival and balking for this model. By employing a technique that allows the service to be a symmetric stochastic Markovian queuing system, based on a threshold level of “D,” either single or in batch, the server may adjust to the system size. The server serves one consumer at a time if there are fewer than ‘D’ consumers in the system. A group of ‘D’ consumers receives service from the server if the total amount of consumers in the process exceeds or is exactly equal to ‘D’. The service periods are exponential, and there are different service levels for both individual and batch service operations. Whether providing services to a single or a collection of consumers, the server will break down during the repair phase. The breakdowns occur after an encouraged arrival, and the breakdown collection for single and batch services is distinct. In addition, both single-service and batch-service instances have variable repair frequencies and exponential repair periods. The steady-state probabilities for system size are derived using the probability-generating function.