Analysis of a Queue Subject to Discouraged Arrivals, Customer Impatience, and Self-Switching Server Dynamics

This study examines a single-server queue approach that discourages arrivals, accounts for impatient customers, and includes a self-switching server is considered. A queue with a single self-switching server, subjected to discouraged arrivals and customer impatience, is analyzed. Customers arrive at a rate of λ/(n+1), where λ denotes a positive constant and n corresponds to the total number of customers in the system. The server provides service at a rate of µ1 until the size of the system is smaller than K. Once the system size reaches K, the server switches to a faster service rate of µ2. After joining the queue, customers become impatient and abandon the system at a rate of ξ until the server switches to the fast service rate. There is no impatience once the server starts serving at the faster rate. The steady-state probability distribution of this system is determined, and its performance metrics, such as expected wait time and the number of customers likely to join the queue or system, are provided. Numerical illustrations are also included.