A queueing-inventory system in which customers can orbit during the service

We consider a service system in which the service can be performed without the customer's presence in the system. Thus, customers can place orders for the required services and leave the system to orbit, utilizing their time efficiently. If the service is completed before the customer's return from orbit, the ready service is stored in a designated storage facility until the customer returns and retrieves the ready service from the inventory. If, however, the service is not yet completed when the customer returns, the customer can go to orbit again. We call this customer policy "multiple orbit during the service". We formulate and analyze the queueing-inventory-multiple-orbit (QIMO) system using the matrix geometric method. Finally, based on the probabilistic study, we conduct an economic analysis using a practical.