Improving the Efficiency of Markov Chain Analysis of Complex Distributed Systems
Christopher Dabrowski, Fern Hunt, Katherine Morrison · 2010
In large-scale distributed systems, the interactions of many independent components may lead to emergent global behaviors with unforeseen, often detrimental, outcomes. The increasing importance of distributed systems such as clouds and computing grids will require analytical tools to understand and predict, complex system behavior to ensure system reliability. In previous work, we described how a piecewise homogeneous Discrete Time Markov chain representation of a computing grid can be systematically perturbed to predict situations that lead to performance degradations. While the execution time of this approach compared favorably with detailed large-scale simulation, a sizable number of perturbations of the model were needed to identify scenarios in which system performance degraded. Here, we evolve our original approach and describe two novel methods for quickly identifying portions of the Markov chain that are sensitive to perturbation. The first method involves finding minimal s-t cut sets, consisting of state transitions that disconnect all paths in a Markov chain from the initial to a desired end state. By perturbing state transitions in the cut set, it is possible to quickly identify scenarios in which system performance is adversely affected. We show this method can be applied to larger Markov models than the approach described in our earlier work. We then present a second method, in which the Spectral Expansion Theorem is used to analyze the eigensystem of a set of Markov transition probability matrices to predict which state transitions, if