By means of ancestor line, we first derive that the stable system size of GeomX /G/1 queue with exhaustive service discipline and multiple vacations(denoted by Geomx /G(E,MV)/1 ) has decomposition properties and by compairing GeomX /G(E,MV)/1 with Geom/G(E,MV)/1 queue, we show that the stablewaiting time of GeomX /G(E,MV)/1 is distributed as the sum of two independent random variables, which help to illustrate the system size and waiting time of GeomX /G/1 queue system with exhaustive service discipline and adaptive multistage vacations(denoted by GeomX /G(E,AMV)/1 ) have decomposition properties.Secondly, GeomX1GeomX2 /G/1 queue with exhaustive service discipline is considered , the PGF of the stable system size is given as well. The stable system size of the GeomX1GeomX2 /G/1 queue with single vacation and multiple vacationsare discussed. Then GeomX /G(E,AMV) /G/1 queue is discussed and in the steady state, the PGF of the system size and waiting time are obtained.In the end , adaptive multistage vacations are first introduced into GeomX /G/1repairable queue system. We discussed the system under two different service disciplines, and some indexes such as system availability, failure frequency and reliability function are offered.
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