Modeling a queueing–inventory system integrated with a chatbot service mechanism and hybrid replenishment policy

Online retail platforms increasingly integrate the automated customer-service interfaces, such as chatbots, to improve customer interactions and support. This paper analyzes a queueing–inventory system (QIS) that incorporates a rule-based chatbot mechanism and operates under a hybrid replenishment policy. The customer is engaged in the service process, as the interaction between the server and the customer continues until the customer’s needs are met. Additionally, in this type of system, a single server can serve multiple customers in parallel. The system is modeled as a stochastic process using a quasi-birth-and-death (QBD) process. By employing the Matrix-geometric method (MGM) developed by Neuts, we establish the model’s stability conditions and derive its steady-state probability vector. In order to evaluate system efficiency, we define several performance indicators, including customer waiting time, the number of customers waiting for server response, expected inventory level, emergency reorder rate, and total system cost. These measures directly address the service efficiency encompassing waiting time, utilization, and replenishment and cost minimization, which includes inventory holding costs, ordering costs, and customer loss. Furthermore, a detailed comparative analysis is conducted to benchmark the proposed QIS with the traditional system, including comparisons with conventional (s, Q) inventory policies. The results demonstrate that chatbot and hybrid replenishment strategies significantly improve both cost efficiency and service performance. Numerical results indicate that the implementation of chatbots can achieve a reduction of up to 30% in total cost and waiting time compared to conventional human-operated (physical server) systems under the consideration of hybrid reordering policy and customer eviction.