TY - JOUR
TI - On successive lumping of large scale systems
DO - https://doi.org/doi:10.7282/T3QZ288X
PY - 2014
AB - The general area of research of this dissertation concerns large systems with random aspects to their behavior that can be modeled and studied in terms of the stationary distribution of Markov chains. As the state spaces of such systems become large, their behavior gets hard to analyze, either via mathematical theory, computational algorithms or via computer simulation. In this dissertation a class of Markov chains that we call successively lumpable is specified for which we show that the stationary probabilities can be obtained by successively computing the stationary probabilities of a propitiously constructed se- quence of Markov chains. Each of the latter chains has a (typically much) smaller state space and this yields significant computational improvements. In a successively lumpable Markov chain, we denote the states by tuples of the form (m,i), where m represents the “current” level of the state and i the current phase of the state. A Markov process is called quasi skip free (QSF) when its transition probability matrix does not permit one step transitions to states that are two or more levels away from the current state in one direction of the level variable m. We study the class of QSF processes for which in all levels m transitions from level m can only go “down” to a single state in level m − 1 while “upward” transitions are not restricted. Furthermore, we study the class of QSF processes for which in all level m transitions from level m can go “down” to any state in level m − 1 while “upward” transitions go only to one state in the highest level. We derive explicit solutions and bounds for the steady state probabilities for both classes of processes, when the process is ergodic. These two classes of QSF processes have applications in many areas of applied proba- bility comprising computer science, queueing theory, inventory theory, reliability and the theory of branching processes. To elaborate the applicability of the method we present explicit solutions for well known queueing models. In addition we will give examples of inventory models and restart models that also fit in the framework of successively lumpable QSF processes.
KW - Management
KW - Markov processes
KW - Stationary processes
KW - Probabilities
LA - eng
ER -