An automated formulation of queues with multiple geometric batch processes

We present an automated formulation mechanism that facilitates the inclusion of batches of geometrically distributed size in Markov modulated, multiprocessor queues of finite or infinite capacity. This provides a practical approach for the analytical modelling of many present day communication and computer systems, e.g. the Internet and mobile networks. Geometric distributions can be scaled and superimposed to produce a range of convex probability mass functions. The geometric distribution produces an infinite range of batch sizes, which creates unbounded transitions in queue length, leading to Kolmogorov balance equations with an unbounded, possibly infinite, number of terms. Our method centres on producing equivalent transformed Kolmogorov balance equations of minimal, finite range. The key contribution is the mechanical derivation of these transformed equations. Previously such equations had to be derived from first principles for every variant of every applicable queueing system in a highly error-prone procedure.