Solution of Graph - Composed Markov Models Using

Continuous time Markov chains (CTMCs) are a common mathematical model for studying the dependability of many complex processes and have been especially successful in modeling large computer systems. CTMCs are typically generated from more natural modeling formalisms that often better represent the system they model. This results in the famous state-space explosion problem, in which the number of states in the CTMC depends exponentially on the number of models in the higher-level formalism. The problem affects numerical analysis in two ways: the space needed to represent the transition rate matrix, and the space needed to represent the iteration vectors. The goal of this thesis is to develop new techniques to extend the size of models that can be studied using CTMCs generated from higher-level formalisms. To combat the state-space explosion problem, we present a combination of a largeness-avoidance and a largeness-tolerance technique to address the size of the transition rate matrix. In the first technique, we present a representation of the model called the model composition graph, which separates a model into its public and private state. We then use symmetry detection to generate the smallest CTMC possible using model-level symmetry. In the second technique, we use symbolic data structures to represent the state space and transition rate matrix using minimal memory, leaving the rest for iteration vectors. We combine the techniques for the case of graph-composed Markov models with state-sharing composition. Using several example models, we study the space and time efficiency of the techniques. We present results showing an orders of magnitude decrease in running time due to lumping of modellevel symmetry, and an orders of magnitude decrease in space due to symbolic data structures. Since our techniques are complementary, we are able to get the benefits of both, which greatly improves our ability to solve these types of models.