Volume 35, Issue 1, 2026
DOI: 10.69980/03276716.2026.08
A Repairable M/M/1 Retrial Queue with Setup Times and Negative Arrivals: Conceptual Applications to Neural Information Processing and Cognitive Systems
Abstract
This study investigates a repairable M/M/1 retrial queue subjected to server setup times and the destabilizing effect of negative arrivals. The server is deactivated during idle periods to save energy. The server is reactivated once the primary customer arrives; however, it is not served immediately. The customer later joins a retrial orbit to attempt the service again. The analysis further integrates server breakdowns during active service periods, necessitating random repair intervals. A steady-state analysis is conducted using the probability-generating function method to derive the performance metrics, including the mean orbit length. Numerical experiments are conducted to assess the influence of various system parameters and to develop a cost-optimization model that identifies efficient operational policies, while considering energy consumption, retrial handling, and maintenance and repair costs. The developed framework is particularly applicable to systems where energy conservation and operational disruptions are critical, such as cloud computing infrastructures, telecommunication networks, and neural information-processing systems.
Keywords
M/M/1 Retrial Queue, Server Setup Times, Negative Arrivals, Server Breakdown and Repair, Idle–Time Server Shutdown, Probability-Generating Function, Cost Optimization, Cognitive Processing, and Neural Information Systems.