## Date Awarded

2013

## Document Type

Dissertation -- Access Restricted On-Campus Only

## Degree Name

Doctor of Philosophy (Ph.D.)

## Department

Computer Science

## Advisor

Peter Kemper

## Abstract

Real world systems are frequently of interest to researchers because the fluctuation in the status of the system makes it important to explore the causes and effects of that fluctuation. These systems usually include stochastic and dynamic processes because the conditional changes occur over time. Such systems can also include spatial aspects, such as the location of units or some notion of distance between units. Modeling real systems allows researchers to explore these dynamics via mathematical models by describing the states and actions. Model exploration can also better explain the overall behavior as well as to predict future behavior. If these states can be described as a discrete set having a defined rule set for state transitions, continuous-time Markov chains are a common mathematical model for studying systems in many application areas. For example, a wireless sensor network is comprised of many individual sensors, which, at a basic level, can be considered to have two states, on and off, and a definable probability that a sensor will move from its current state to the opposite state at any given moment. A spatial aspect that can be included is the distance between sensors and its affect on the transition probability. Then the continuous-time Markov chain could consist of all the possible states of the system in order to study how many and/or which sensors are on. Other example applications include explaining dynamic behavior in biological/biochemical systems or exploring the behavior of interactions in social network systems.;One goal of this research is to consider how to represent the spatial aspects of real-world systems, and how to take advantage of the regularities that are linked to the spatial aspects. It is necessary that this be defined in a way that is reasonable for the modeler to specify. This problem is approached through a compositional modeling formalism based on sharing state variables. Another aim is to use a symmetry detection mechanism to gain reduction of the Markov chain through lumpability, with the symmetries defined by the modeler, in order to expedite analysis of the model. Finally, the ability to do this detection in an automated manner is discussed and explored.;A result of this research is that the spatial information in a model can be useful to include. It is shown that a new methodology makes it easier to do so, in terms of both specifying the extra information required and minimizing the size of the resulting state space. to illustrate the usefulness of the spatial information, a new user interface is presented here to facilitate greater inclusion of the spatial aspects into the model, implemented in the Mobius modeling framework as an extension of existing work on state space reduction. Evaluation is done using three main applications: calcium release sites (biology), wireless sensor networks (networking), and disease propagation (medicine).

## DOI

https://dx.doi.org/doi:10.21220/s2-avn5-8s26

## Rights

© The Author

## Recommended Citation

Lamprecht, Ruth Elizabeth, "Translating Spatial Problems into Lumpable Markov Chains" (2013). *Dissertations, Theses, and Masters Projects.* William & Mary. Paper 1539720328.

https://dx.doi.org/doi:10.21220/s2-avn5-8s26