Finite Queues at the Limit of Saturation

A wide range of real life systems are modeled by queueing systems with finite capacity buffers. There are well established numerical procedures for the analysis of these queueing models when the load islower or higher than the system capacity, but these numerical methods become unstable as the loadgets close to the system capacity. We present simple modifications of the standard computational methodswhich remain numerically stable at saturation as well. We consider two specific Markov models: finite quasi birth death (QBD) processand finite Markov fluid queue (MFQ). The first one describes the behavior of queueing systems with discretebuffer content, while the second one describes the behavior of queueing systems with continuous buffer content. The stationary solution of a finite QBD process is a combination of two matrix geometric serieswhile the stationary fluid density of a finite MFQ is a combination of two matrix exponential functions. Apart of this there are several further similarities between the discrete and continuous buffer modelsat saturation. The proposed solution method exploits the similarities of the models.