Abstract:In large-scale grid systems with decentralized control, the interactions of many service providers and consumers will likely lead to emergent global system behaviors that result in unpredictable, often detrimental, outcomes. This possibility argues for developing analytical tools to allow understanding, and prediction, of complex system behavior in order to ensure availability and reliability of grid computing services. This paper presents an approach for using piece-wise homogeneous Discrete Time Markov chains to provide rapid, potentially scalable, simulation of large-scale grid systems. This approach, previously used in other domains, is used here to model dynamics of largescale grid systems. A Markov chain model of a grid system is first represented in a reduced, compact form. This model can then be perturbed to produce alternative system execution paths and identify scenarios in which system performance is likely to degrade or anomalous behaviors occur. The expeditious generation of these scenarios allows prediction of how a larger system will react to failures or high stress conditions. Though computational effort increases in proportion to the number of paths modelled, this cost is shown to be far less than the cost of using detailed simulation or testbeds. Moreover, cost is unaffected by size of system being modelled, expressed in terms of workload and number of computational resources, and is adaptable to systems that are non-homogenous with respect to time. The paper provides detailed examples of the application of this approach and discusses future work. 6 7

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Velocity fields for Poiseuille flow through tubes having general cross section are calculated using a path integral method involving the first-passage times of random walks in the interior of the cross sectional domain B of the pipe. This method is applied to a number of examples where exact results are available and to more complicated geometries of practical interest. These examples include a tube with "fractal" cross section and open channel flows. The calculations demonstrate the feasibility of the probabilistic method for pipe flow and other applications having an equivalent mathematical description (e.g., torsional rigidity of rods, membrane deflection). The example of flow through a fractal pipe shows an extended region of diminished flow velocity near the rough boundary which is similar to the suppressed vibration observed near the boundaries of fractal drums.

We consider the problem of identifying a subset of nodes in a network that will enable the fastest spread of information in a decentralized environment.In a model of communication based on a random walk on an undirected graph, the optimal set over all sets of the same or smaller cardinality minimizes the sum of the mean first arrival times to the set by walkers starting at nodes outside the set. The problem originates from the study of the spread of information or consensus in a network and was introduced in this form by V.Borkar et al. in 2010. More generally, the work of A. Clark et al. in 2012 showed that estimating the fastest rate of convergence to consensus of so-called leader follower systems leads to a consideration of the same optimization problem.The set function F to be minimized is supermodular and therefore the greedy algorithm is commonly used to construct optimal sets or their approximations. In this paper, the problem is reformulated so that the search for solutions is restricted to optimal and near optimal subsets of the graph. We prove sufficient conditions for the existence of a greedoid structure that contains feasible optimal and near optimal sets. It is therefore possible we conjecture, to search for optimal or near optimal sets by local moves in a stepwise manner to obtain near optimal sets that are better approximations than the factor (1 − 1/e) degree of optimality guaranteed by the use of the greedy algorithm. A simple example illustrates aspects of the method. Random Walk Consensus ModelGiven a connected graph G = (V, E), with vertices or nodes V and edges E, we imagine a random walker situated at a node i ∈ V , moving to another node j ∈ V in a single discrete time step. The choice of j is random and has probability,The matrix P = (p ij ) i,j=1···N is the transition matrix of a Markov chain which in this paper, is assumed to be irreducible and aperiodic ([7]). N is the number of nodes and as in [1] the spread of information is described in terms of a process that is dual to the movement from informed to uninformed nodes. A random walk begins outside a pre-determined set A of informed target nodes 1

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.

We investigate the trade-o between utility and path diversity in a model of congestion control where there can be multiple routes between two locations in a network. The model contains a random route allocation scheme for each source s where the degree of randomness and therefore path diversity is controlled by hs, the entropy of the route distribution. After deriving the model from a network utility maximization problem we analyze it in detail for two sample topologies. We conclude that, starting from an allocation with maximum robustness and path diversity, one can always increase the utility by decreasing hs. However it can only decrease until a critical value of the entropy is reached. The value depends on the topology and link capacities of the network and can be explicity computed for our examples.

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Asymptotically stable attractors supporting an invariant measure, for which the ergodic theorem holds almost everywhere with respect to Lebesgue measure, can be approximated by a space discretization procedure called Ulam's method. As an application of this result we propose the use of this method to approximate the 'chaotic' attractors of flows in lower dimensions. A Monte Carlo implementation makes this feasible. The approximation method can be extended to attractors whose neighbourhoods contain positively invariant compact sets called blocks. Note that such attractors can fail to have open basins of attraction. When the attractor is uniquely ergodic, we also prove the weak convergence of the approximate measures constructed by the method and as an application, we show the weak convergence of Ulam's method for the logistic map at the Feigenbaum parameter value. More generally, using the work of Buescu and Stewart on transitive attractors of continuous maps, we prove weak convergence of the approximate measures and convergence of their supports to classes of Lyapounov stable attracting Cantor sets.