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.

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 multiplexer 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.

Constant bit rate (CBR) traffic is expected to be an important source of traffic in high-speed networks. Such sources may have stringent delay and loss requirements and in many cases, they should be delivered exactly as they were generated. A simple delay priority scheme will bound the cell delay and jitter for CBR flows, so that in the network switches, CBR traffic will only compete with other CBR traffic in the networks. In this paper, we will consider a slotted queue at a typical (intermediate) node in such an environment. The cell arrival process of each source is characterized by a dispersed periodic arrival process (DPAP), the cells of which are assumed to arrive periodically and distribute randomly within the period. We provide an exact analysis of buffer fill and delay distribution using Ballot Theorems. Another generalized source model, the dispersed batch periodic arrival process (DBPAP) is also investigated. A DBPAP source is assumed to have a number of batches of cell arrivals and the batches position randomly in the period. This work shows that the traditional Markovian approximation models may grossly overestimate the buffer fill and delay, unless the number of sources is very large.