A New Analysis Framework for Discrete Time Queueing Systems with General Stochastic Sources

This 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, be periodic, be Markovian or described by the moving average time-series model, etc. Our solution framework is novel and unifying. It uses a combination of multi-dimensional generating functions and combinatorial analysis using 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 finally spectral decomposition approaches. We provide a number of analytical and numerical examples including a simple multi-server model with i.i.d arrivals, an ATM multiplexor fed by a (random) number of periodic sources, and a new example considering the discrete moving average model for the arrival process where a simple closed-form expression for the stationary distribution of the system queue lengths is provided. Keywords— Combinatorial Analysis, Multi-dimensional Generating Function, Ballot Theorems, Batch Arrival and Departure Models