Analysis of a two-phase queue system with <p, F>-policy

Some kinds of arriving customers control policy are needed to maintain the service quality of working places. This paper studies the optimal management of a finite capacity M/M/1/K queueing system with < p, F>-policy, where all arriving customers demand the first essential service and some of them may further demand an additional optional service. In the presented system, no further arriving customers are allowed to enter the system as the number of customers reaches the system's capacity. Customers are allowed to enter the system or the customers are still unable to enter the system with probability when the queue length decreases to a certain threshold value. We use the matrix-form technique to derive some important performance measures, such as the expected number of customers in the system, the expected waiting time in the system, the expected number of customers when the server starts to allow customers entering the system and the expected number of customers when the system is blocked. We develop a cost model to determine the optimal control at a minimum cost. Finally, we present some managerial insights through an application example of the transport service system.