Improving efficiency of service systems by performing a part of the service without the customer's presence

We propose a novel approach for service systems, according to which the service can be split into two phases: an opening service (OS), which can be provided only when the customer is present, and a complementary service (CS), which can be conducted without the customer. To improve the system's overall performance, after providing the OS, the server can postpone executing the CS to time intervals in which they are empty of customers. The prepared OSs are stored in a designated storage facility. Once the system becomes empty, the server executes the CS for the stored OSs, one at a time. We consider two unbounded dimensions system involving a multivariate probability generating function and derive closed-form expressions for the system's steady-state probabilities and various performance measures. An economic analysis compares the described model with a corresponding limited OS capacity model.