An Aggregation Technique for the Transient Analysis of Stiff Markov Chains

Bobbio, A.; Trivedi,

doi:10.1109/tc.1986.1676840

Cited by 164 publications

(52 citation statements)

References 14 publications

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“…We use the M/M/1 queuing system with server breakdown and repair, explained in [7], as another example to illustrate analytic-numeric epistemic uncertainty propagation. In [7], an approximate expression for the average number of jobs in the system, is derived for an M/M/1 system with breakdown and repair of the server, using an aggregation technique developed to handle the stiffness in the model due to orders of magnitude of difference in parameter values (rate of job arrival and service being much faster than the rate of server breakdown and repair).…”

Section: M/m/1 Queue With Server Breakdown and Repairmentioning

confidence: 99%

“…In [7], an approximate expression for the average number of jobs in the system, is derived for an M/M/1 system with breakdown and repair of the server, using an aggregation technique developed to handle the stiffness in the model due to orders of magnitude of difference in parameter values (rate of job arrival and service being much faster than the rate of server breakdown and repair). The Markov chain used to model the system in [7] has been reproduced in Figure 4.2, for ease of reference (while the figure in [7] is for an M/M/1/m system with server breakdown and repair, here, we have modified the figure by setting m to ∞, for it to model an M/M/1 system with server breakdown and repair). The states of the Markov chain are denoted by (l, k), where l is the number of customers/jobs in the system and k is the number of non-failed servers.…”

Section: M/m/1 Queue With Server Breakdown and Repairmentioning

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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Section: M/m/1 Queue With Server Breakdown and Repairmentioning

confidence: 99%

Section: M/m/1 Queue With Server Breakdown and Repairmentioning

confidence: 99%

Uncertainty Propagation through Software Dependability Models

Mishra

Trivedi

2011

2011 IEEE 22nd International Symposium on Software Reliability Engineering

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“…Integrating system availability and performance in a single model often causes the largeness and stiffness problem [11,12]. We built hierarchical model for system modeling.…”

Section: Hierarchical System Modelingmentioning

confidence: 99%

Analysis Task Scheduling Models based on Hierarchical Timed Marked Graph

Cao²

2010

International Journal of Contents

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“…and ET RRt is approximated with error " by the ET RR a N t given by (2). The computational cost of standard randomization is essentially the cost of performing the N vector-matrix multiplications qk I qkP, k H; I; F F F ; N À I. Qt has, for Ãt 3 I, an asymptotic normal distribution with mean and variance Ãt [20], and, for large Ãt and " ( I, the required N is % Ãt, making standard randomization computationally very expensive if both X and Ãt are large.…”

Section: Review Of Standard and Regenerative Randomizationmentioning

confidence: 99%

“…An approach to deal with stiffness is the aggregation technique proposed in [2]. For class g HH models, that technique could be used to aggregate the states in S À fog, yielding an aggregated CTMC model with a transient state and A absorbing states with symbolic solution.…”

Section: Introductionmentioning

confidence: 99%

Computationally efficient and numerically stable reliability bounds for repairable fault-tolerant systems

Carrasco

2002

IEEE Trans. Comput.

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AbstractÐThe transient analysis of large continuous time Markov reliability models of repairable fault-tolerant systems is computationally expensive due to model stiffness. In this paper, we develop and analyze a method to compute bounds for a measure defined on a particular, but quite wide, class of continuous time Markov models, encompassing both exact and bounding continuous time Markov reliability models of fault-tolerant systems. The method is numerically stable and computes the bounds with wellcontrolled and specifiable-in-advance error. Computational effort can be traded off with bounds accuracy. For a class of continuous time Markov models, class g HH , including typical failure/repair reliability models with exponential failure and repair time distributions and repair in every state with failed components, the method can yield reasonably tight bounds at a very small computational cost. The method builds upon a recently proposed numerical method for the transient analysis of continuous time Markov models called regenerative randomization.

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An Aggregation Technique for the Transient Analysis of Stiff Markov Chains

Cited by 164 publications

References 14 publications

Uncertainty Propagation through Software Dependability Models

Uncertainty Propagation through Software Dependability Models

Analysis Task Scheduling Models based on Hierarchical Timed Marked Graph

Computationally efficient and numerically stable reliability bounds for repairable fault-tolerant systems

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