The Israeli queue with a capacitated server: modeling and approximations

The Israeli Queue is a batch service polling system where a single server attends to multiple queues based on seniority. Each arriving customer belongs to one of several classes. Upon arrival, a customer either joins an existing queue for their class or initiates a new queue if they are the first of their class to arrive. Customers from the class with the most senior member are served together as a batch, with the service time remaining constant regardless of the batch size. This service model is found in applications like advanced elevator systems and on-demand shared mobility, where passengers heading to the same destination can share a ride. However, in many real-world scenarios, the vehicle capacities are small and constraining, which calls for a deeper exploration of the Israeli queue with a capacitated server (IQCS). In this paper, we formally define the IQCS and address the challenges of creating a mathematically tractable model to represent it. To approximate the IQCS, we develop a quasi-birth-death process and derive approximations for key performance measures. To validate our approach, we implement a simulation model and use it to compare the IQCS, the approximate model, and the original Israeli Queue. Our results across various scenarios demonstrate the accuracy of the approximate model. Nonetheless, the presence of a remaining gap underscores the ongoing challenge of precisely and efficiently modeling the IQCS, posing an open question for the research community.