Evaluation of the Queue Length Distribution for some Queues with Correlated Inputs

Abstract

THE need for further progress in the analysis of queues with correlated inputs has been referred to by Kendall¹ and Lindley²; while systems with randomly delayed regular inputs have been studied^3,4, explicit results do not appear to be available except when the disturbances from regularity are small. Consider the case of a single server queue, with exponential service times μe^−μtdt (μ⁻¹ = ρ, the traffic intensity) and an input generated by superposing independent random exponential delays λe^−λtdt on a grid of scheduled time-points …, − 1, 0, 1, 2, …. The characteristics of this system (“model A”, say) are so far unsolved, but Kendall has suggested that progress might be possible if the model were modified by redistributing the arrivals randomly within each scheduled interval (i, i + 1) (“model B”). Except when λ is large (the nearly regular case), model B approximates to model A. The purpose of this communication is to show, with examples, that explicit results can be obtained with this type of modified model.