Analysis of Single Server Unreliable and Removable Server Queueing model with  Partial Breakdown and Balking Behavior

In systems with an unreliable and removable server, this study investigates an M/M/1 queueing model that accounts for consumer behavior and setup. During maintenance, working vacations, and setup, the server experiences partial failures. Customers arrive at a Poisson distribution with an arrival rate of λ. The server goes into a working vacation mode at a rate of ψ while the system is empty. The service rate drops from from to due to unexpected partial server breakdowns during the working vacation. If there are still customers waiting at the end of a vacation, the server resumes normal operations at a rate of . At a rate of α, the server enters a shutdown state when there are no customers. If customers arrive when the system shuts down, the server has to be restarted. With chance q, the server will fail during this restart procedure; with probability p, it will return to its typical busy state. The study develops closed-form equations for the steady-state probabilities and assesses a number of performance measures, such as expected queue length, system waiting time, and probabilities of the system's state during various server operating phases. Numerous system parameter’s numerical effects on various performance metrics are displayed in tables and visualizations.