The paper considers a general class of discrete time systems with batch arrivals and departures. Such models appear frequently in the teletraffic analysis of computer and communications networks. Our arrival models are assumed to be quite general. They could be independent and identically distributed (i.i.d.) in successive slots, periodic, Markovian, or described by the moving average time-series model, etc. Our solution framework is novel and unifying and it uses a combination of multidimensional generating functions and combinatorial analysis utilizing extensions of classical ballot theorems. In general, we provide an explicit analytical expression as an infinite sum to obtain the system stationary probability distribution avoiding classical root-finding methods, matrix analytical methodologies, and spectral decomposition approaches. We provide a number of analytical and numerical examples including: i.i.d. models with Poisson and Binomial arrivals; multiserver queueing systems fed by Markovian sources; queues fed with a discrete moving average source of the first and second order; an i.i.d. discrete Pareto batch arrival model. Closed-form analytical expressions are obtained for the stationary distribution of the system queue lengths and numerical examples are also provided when appropriate.