QBDs with marked time epochs: a framework for transient performance measures

A framework to assess transient performance measures is introduced by generalizing the theory of the quasi birth-and-death (QBD) paradigm to QBDs with marked time epochs (QBD^m). The distinction with the classical QBD process is that certain time epochs get marked according to a specific set of Markovian rules. Our interest lies in obtaining the system state at the n-th marked time epoch. The steady state vector of a so-called reset Markov chain is used to obtain the above-mentioned system state (either by approximation or in an exact manner). A fast algorithm, with limited memory usage, based on solving a single quadratic matrix equation, a set of Sylvester matrix equations and fast Fourier transforms is proposed. The generality and flexibility of our framework is demonstrated on a set of queueing systems and applied to dimensioning a video playout buffer and studying the transient throughput of a wireless random access algorithm.