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.

Triangular M/G/1-Type and Tree-Like Quasi-Birth-Death Markov Chains

“…We will focus on the specific case were these matrices are all triangular. Such Markov chains occur when applying the framework presented in Van Velthoven et al (2006b) to any scalar tree-like QBD Markov chain; various examples of such scalar Markov chains are given in He (2000). As with the G matrix for M/G/1-type Markov chains, we develop a simple recursive algorithm to compute the diagonals of the smallest nonnegative (triangular) solution V one at a time in O(m 3 d).…”

Time-bounded reachability in tree-structured QBDs by abstraction

“…The analysis of (tree-structured) QBDs mostly focuses on steady-state probabilities as these can be obtained using matrix-geometric techniques [5]. Transient analysis has received scant attention; the only existing approach is approxi-mate [30]. Recently, direct techniques based on uniformization or variants thereof [15], have been proposed for reachability properties for general infinite-state CTMCs [32] and for highly structured infinite-state CTMCs, such as normal QBDs [24] and Jackson queueing networks [23].…”

Simultaneous Transient Analysis of QBD Markov Chains for all Initial Configurations using a Level Based Recursion

“…The underlying ideas presented in [14] gave rise to the development of a new framework, termed QBDs with marked time epochs [15], to assess transient measures in a plug-andplay manner for any QBD, where a significant improvement over [14] was also achieved in terms of the required computational resources. An extension to the class of tree-like processes [2], a generalization of the QBD paradigm, was discussed in [16] and applied to analyze a set of random access algorithms.…”

“…Both [15] and [16] were limited to the case where the initial state of the Markov chain was part of the boundary level. That is, the i-th component of some vector αini gave the initial probability of being in the i-th boundary state at time 0.…”