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\(\mathrm{G}/\mathrm{GI}/N (+ \mathrm{GI})\) queues with service interruptions in the Halfin-Whitt regime. (English) Zbl 1344.60088

A standard \(\mathrm{G}/\mathrm{GI}/N\) multiserver queueing system under non-idling FCFS is subject to interruptions due to an external alternating renewal environment which generates up-down cycles for the complete system, i.e., servers break down concurrently and are repaired after the same time interval. The interruption of the service follows a preemptive-resume regime and customers are held in their service channel when interrupted. Customers waiting in line (not being assigned to a server) may abandon the queue due to expiring of their patience time (which is generally distributed), even during down times of the system. Whenever a customer enters a service channel he will no longer abandon even during down times. During down times, newly arriving customers are admitted to the system, and either send to a free server, if any, to wait there for service (without impatience behaviour) when the next up-period starts, or attend the queue possibly departing before their service starts due to impatience.
Several variants of the model are studied in the Halfin-Whitt regime, where the arrival rate and the number of servers grow appropriately, while service time distributions remain unchanged. The up-times and their distributions are of the same order as the service times while the down times become asymptotically negligible. The results for the queueing processes are: functional weak law of large numbers and functional central limit theorems, and for the virtual waiting time process: functional central limit theorem. For technical reasons and from the structure of the model, the proofs of the convergence utilize the Skorokhod \(M_1\)-topology.

MSC:

60K25 Queueing theory (aspects of probability theory)
60F17 Functional limit theorems; invariance principles
60F05 Central limit and other weak theorems
60J75 Jump processes (MSC2010)
90B22 Queues and service in operations research
68M20 Performance evaluation, queueing, and scheduling in the context of computer systems
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References:

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