$G/{ GI}/N(+{ GI})$ queues with service interruptions in the Halfin–Whitt regime

Abstract

We study $G/GI/N(+GI)$ queues with alternating renewal service interruptions in the Halfin–Whitt regime. The systems experience up and down alternating periods. In the up periods, the systems operate normally as the usual $G/GI/N(+GI)$ queues with non-idling first-come–first-served service discipline. In the down periods, arrivals continue entering the systems, but all servers stop functioning while the amount of service that each customer has received will be conserved and services will resume when the next up period starts. For models with abandonment, interruptions do not affect customers’ patience times. We assume that the up periods are of the same order as the service times but the down periods are asymptotically negligible compared with the service times. We establish the functional central limit theorems for the queue-length processes and the virtual-waiting time processes in these models, where the limit processes are represented as stochastic integral convolution equations driven by jump processes. The convergence in these limit theorems is proved in the space ${\mathbb D}$ endowed with the Skorohod $M_1$ topology.