Improving efficiency of markov chain analysis of complex distributed systems

Abstract

Abstract: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 perturbed, are likely to adversely affect system performance. Results are presented for both methods to show that they can be used to identify the same failure scenarios presented in our earlier paper (as well as additional scenarios, using the first method), while reducing the number of perturbations needed. We argue that these methods provide a basis for creating practical tools for analysis of complex systems behavior in distributed systems.

Identifying Failure Scenarios in Complex Systems by Perturbing Markov Chain Models

“…The transitions in the cut set can thus be identified as critical transitions, which serve as a basis for describing potential failure scenarios. In previous work [12], we reported the results of experiments which showed that minimal s-t cut set analysis could be used to find all critical state transitions in an absorbing DTMC for a much smaller grid computing system at 1/100 th the computation cost of large-scale simulations. This exhaustive analysis need not be repeated for the problem described in this paper, as there is not the space for it.…”

“…These savings increase even more dramatically if combinations of three critical transitions are considered. Though further research is necessary, it is our belief that both in [12] and in this study, we have described an analytical approach that can aid in understanding where and how catastrophic failures may occur in complex systems. The results to date have shown that the approach is tractable for the types of problems we have examined.…”

“…Cut set enumeration algorithms are known to be computationally expensive for large problems. For instance in [12], a Markov chain with 50 states, though sparse, was found to contain over 10 8 minimal s-t cut sets on paths between the initial and absorbing states. Further, computational characteristics of directed graphs are not well understood and remain a topic for future work.…”

“…This DTMC is based on a discrete event large-scale simulation model described in [10], which we also use as a proxy for a real-world system in this study. This 39-state model is far larger than previous models we have analyzed [11][12][13], which consisted of only seven states. To this DTMC, we apply our combined method to find likely failure scenarios, which we verify in the large-scale model.…”

“…VII), and conclusions (Sec VIII). While the approach presented here employs minimal s-t cut set analysis as its basis, elsewhere we describe the use of spectral methods for eigendecomposition to identify critical state transitions [13] and algorithms for exhaustive search of a Markov chain transition probability matrix [12].…”