On the analysis of the discrete-time system with batch service and Early Arrival discipline

This paper investigates a discrete-time queueing system with batch arrivals and batch service under the Early Arrival discipline. Using the method of generating functions, we derive the functional equation for the system's state. To determine the unknown constants in the numerator of the resulting expression, we apply Rouché’s Theorem, proving the existence of a sufficient number of roots within the unit disk. This approach allows us to obtain a closed-form solution for the probability generating function (PGF) of the stationary distribution of the number of requests. The results are applicable for optimizing performance in synchronized data transmission networks like DECT-2020, IAB 5G/6G NR.