Data structures for the analysis of large structured markov models

Abstract

High-level modeling formalisms are increasingly popular tools for studying complex systems. Given a high-level model, we can automatically verify certain system properties or compute performance measures about the system. In the general case, measures must be computed using discrete-event simulations. In certain cases, exact numerical analysis is possible by constructing and analyzing the underlying stochastic process of the system, which is a continuous-time Markov chain (CTMC) in our case. Unfortunately, the number of states in the underlying CTMC can be extremely large, even if the high-level model is “small”. In this thesis, we develop data structures and techniques that can tolerate these large numbers of states. First, we present a multi-level data structure for storing the set of reachable states of a model. We then introduce the concept of event “locality”, which considers the components of the model that an event may affect. We show how a state generation algorithm using our multi-level structure can exploit event locality to reduce CPU requirements. Then, we present a symbolic generation technique based on our multi-level structure and our concept of event locality, in which operations are applied to sets of states. The extremely compact data structure and efficient manipulation routines we present allow for the examination of much larger systems than was previously possible. The transition rate matrix of the underlying CTMC can be represented with Kronecker algebra under certain conditions. However, the use of Kronecker algebra introduces several sources of CPU overhead during numerical solution. We present data structures, including our new data structure called matrix diagrams, that can reduce this CPU overhead. Using our techniques, we can compute measures for large systems in a fraction of the time required by current state-of-the-art techniques. Finally, we present a technique for approximating stationary measures using aggregations of the underlying CTMC. Our technique utilizes exact knowledge of the underlying CTMC using our compact data structure for the reachable states and a Kronecker representation for the transition rates. We prove that the approximation is exact for models possessing a product-form solution.