Modeling and cost optimization of controlling customer arrivals in queueing systems with two waiting areas

In current society, customers attach importance to good service quality when they arrive at a store. Owing to the limited space of the service waiting area inside a physical store, overcrowding may influence customer satisfaction. Thus, effectively controlling customers can avoid overcrowding in stores and help improve service quality. This study extends the classical $ F $-policy and introduces a novel queueing system with service waiting and queueing areas. When the service waiting area is occupied, newly arrived customers are blocked and must stay in the queueing area. Customers waiting outside the entrance are not allowed to enter the service waiting area until the number of customers in the service waiting area is reduced to the predetermined threshold. For the proposed queueing system, the joint steady-state probabilities of the number of customers in the two areas are obtained using a matrix rearrangement technique. We derive the stability condition for the proposed queueing model and compute the steady-state system performance measures. Subsequently, we construct an expected cost function per unit time and formulate a cost minimization problem. Finally, comparative numerical examples are presented to demonstrate the influence of the system parameters on the optimal threshold and optimal system capacity.