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Multiphase fluctuation analysis in a queue with maintenance
In this article we introduce a single-vacation queue with enhanced maintenance work. During his vacation time, the server performs secondary jobs which are randomly generated in batches and counted. When the time of absence expires, the server is called off. However, he does not interrupt his service on any job and on any packet of jobs. Only when his work on the packet is finished and his maintenance time is longer than $T$ (an arbitrary random time) does the server returns to the system and waits for customers if none is present. We use game theoretical work in combination with fluctuation analysis to find an explicit Laplace-Carson inverse, all leading to explicit functionals (a Kendall-like formula) for the queueing system with compound Poisson input and general service time.