On the sensitivity of transient solutions of Markov models

Ramesh, A. V.; Trivedi, Kishor S.

doi:10.1145/166955.166998

Cited by 19 publications

(8 citation statements)

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“…The sensitivity analysis of the aforementioned measures to changes in the parameters may reveal performance or reliability bottlenecks and help designers in achieving desired performance measures and robustness values. Formally, the gradients ∇ θ ER, ∇ θ EY or ∇ θ MTFF are the solutions of the equations arising after taking derivatives of both sides of (1), (3) and (4), respectively [2]. Sensitivities of the performance measures to the individual parameters θ[i] are the components of the gradient vectors.…”

Section: B Analysis Tasksmentioning

confidence: 99%

Configurable numerical analysis for stochastic systems

Marussy

Klenik

Molnár³

et al. 2016

2016 International Workshop on Symbolic and Numerical Methods for Reachability Analysis (SNR)

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Abstract-Stochastic aspects of complex systems require more and more involved analysis approaches. Answering reachability and related analysis questions can often be reduced to steadystate, transient, reward or sensitivity value analysis of stochastic models. In this paper we introduce a configurable stochastic analysis framework which supports the user to combine explicit, symbolic and numerical algorithms to efficiently compute the measures of stochastic models. Beyond the well-known algorithms from the field, we also developed an experimental version of an Induced Dimensionality Reduction Stabilized numerical solver to compute steady-state probabilities of Markovian models. As far as we know, this is the first attempt to exploit this algorithm in stochastic analysis. We have conducted experiments on different combinations of the algorithms on various models to assess their advantages and disadvantages in the analysis. This information can be used by the user to choose the combination of algorithms being efficient solving the analysis problem.

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Section: B Analysis Tasksmentioning

confidence: 99%

Configurable numerical analysis for stochastic systems

Marussy

Klenik

Molnár³

et al. 2016

2016 International Workshop on Symbolic and Numerical Methods for Reachability Analysis (SNR)

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“…While intrusive epistemic uncertainty propagation can be easily performed through aleatory models with simulative solutions, 2 they are harder to perform through analytic aleatory models. Perturbation methods, which formulate and solve a version of the original model with the model parameters incorporating the epistemic uncertainty, are examples of intrusive methods of epistemic uncertainty propagation through analytic aleatory models [65]. On the other hand, non-intrusive methods use repeated solutions of the existing model with different parameter values.…”

Section: Epistemic Uncertainty Propagationmentioning

confidence: 99%

“…al. derive analytic bounds for model output for variations (perturbations) in value of a single parameter or values of multiple parameters [65] for Markov reliability models, with the help of differential equations based on the transient solution of state probabilities (obtained by solving the Kolmogorov differential equations) [86].…”

Section: E[r(t)] ≈ R(t)|mentioning

confidence: 99%

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Uncertainty Propagation through Software Dependability Models

Mishra

Trivedi

2011

2011 IEEE 22nd International Symposium on Software Reliability Engineering

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Systems in critical applications employ various hardware and software fault-tolerance techniques to ensure high dependability. Stochastic models are often used to analyze the dependability of these systems and assess the effectiveness of the fault-tolerance techniques employed. Measures like performance and performability of systems are also analyzed using stochastic models. These models take into account randomness in various events in the system (known as aleatory uncertainty) and are solved at fixed parameter values to obtain the measures of interest. However, in real life, the parameters of the stochastic models themselves are uncertain as they are derived from a finite (limited) number of observations or are simply based on expert opinions.Solving the stochastic models at fixed values of the model input parameters result in estimates of model output metrics which do not take into account the uncertainty in model input parameters (known as epistemic uncertainty).In this research work, we address the computation of uncertainty in output metrics of stochastic models due to epistemic uncertainty in model input parameters, with a focus on dependability and performance models of current computer and communication systems. We develop an approach for propagation of epistemic uncertainty in input parameters through stochastic dependability and performance models of varying complexity, to compute the uncertainty in the model output measures. and provides more robust estimates of uncertainty in the model output metrics than previous sampling based methods.We demonstrate the applicability of the uncertainty propagation approach by applying it to analytic stochastic dependability and performance models of computer systems, ranging from simple non-state-space models with a few input parameters to large state-space models and even hierarchical models with more than fifty input parameters. We further apply the uncertainty propagation approach to stochastic models with not only analytic or analytic-numeric solutions but also those with simulative solutions. We also consider a wide range of model output metrics including reliability and availability of computer systems, response time of a web service, capacity oriented availability of a communication system, security (probability of successful attack) of a network routing session, expected number of jobs in a queueing system with breakdown and repair of servers and call handoff probability of a cellular wireless communication cell.

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“…Namely, the (i, j) entry of exp(Qt) is the probability that the chain is at state j at time t if it is at state i initially. The sensitivity of the transition matrix has been studied in [12] and it is proved that in the ∞-norm, ν(Q, t) = t Q ∞ , demonstrating benign perturbation properties for this problem. In Markov chains, however, one is often more interested in smaller entries of exp(Qt), which represent small probability events.…”

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confidence: 99%