Study of an Early Arrival System on GeoX/G/1 Queue with Single Vacation

Abstract

This paper investigates the GeoX/G/1 queue with single vacation in an early arrival system. The time between arrival batches follows a geometric distribution, while the service and vacation durations are expressed as integral multiples of a slot duration and can follow arbitrary distributions. This study focuses on analyzing the system length distributions at different time epochs, the waiting time distribution for a random customer within a batch, and performing a cost analysis using the theory of difference equations and the supplementary variable technique. This method has an advantage over traditional queueing analysis techniques, as it eliminates the need to compute the transition probability matrix for the embedded system length process. We evaluate key performance metrics and demonstrate the computational process through numerical examples.