Recent years

discrete time queueing systems have been receiving attention due to their wide applications in design and control of manufacturing and teleconmunication systems

and in modeling and analyzing of computer communication networks.In this paper we consider the discrete time Geom/G/1 queue with multiple vacations and bulk arrival. By using the total probability decomposition technique and u-transform

we discuss the transient and equilibrium distributions of the queue length with the beginning of the any initial state

and obtain the recursion expressions of the u-transform for the transient queue length distribution and the steady queue length distribution.Furthermore

we also obtain the stochastic decomposition of the queue length at a random point in equilibrium.Especially we can obtain some corresponding results under some especial cases.