Transient analysis of tree-Like processes and its application to random access systems

Abstract

A new methodology to assess transient performance measures of tree-like processes is proposed by introducing the concept of tree-like processes with marked time epochs. As opposed to the standard tree-like process, such a process marks part of the time epochs by following a set of Markovian rules. Our interest lies in obtaining the system state at the n -th marked time epoch as well as the mean time at which this n -th marking occurs. The methodology transforms the transient problem into a stationary one by applying a discrete Erlangization and constructing a reset Markov chain. A fast algorithm, with limited memory usage, that exploits the block structure of the reset Markov chain is developed and is based, among others, on Sylvester matrix equations and fast Fourier transforms. The theory of tree-like processes generalizes the well-known paradigm of Quasi-Birth-Death Markov chains and has various applications. We demonstrate our approach on the celebrated Capetanakis-Tsybakov-Mikhailov (CTM) random access protocol yielding new insights on its initial behavior both in normal and overload conditions.