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

Abstract

A new algorithm to assess transient performance measures for every possible initial configuration of a quasi-birth-and-death (QBD) Markov chain is introduced. We make use of the framework termed QBDs with marked time epochs that transforms the transient problem into a stationary one by applying a discrete Erlangization and constructing a reset Markov chain. To avoid the need to repeat ail computations for each initial configuration, we propose a level based recursive algorithm that uses intermediate results obtained for initial states belonging to levels 0,hellip, tau - 1 to compute the transient measure when the initial state is part of level tau. Also, the computations for all states belonging to level tau are performed simultaneously. A key property of our approach lies in the exploitation of the internal structure of the block matrices involved, avoiding any need to store large matrices. A flexible Matlab implementation of the proposed algorithm is available online.