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Interval Markovian Models in Dependability Evaluation
Abstract
Model-based dependability evaluation is based on abstractions of the real system. When uncertainties or variabilities are associated with system parameters, single point characterization of parameters is inadequate. Interval arithmetic has been applied to model uncertainties and variabilities. An interval model is a space, family or class of models in which there are parameters represented by intervals, instead of real numbers. This paper describes the Interval Generalized Stochastic Petri Net (IGSPN) as an interval extension to the GSPN model. The IGSPN analysis takes into account the effects of variability on exponential transition rates and weights when calculating dependability measures. IGSPN analysis may be useful as a tool for decision-making. A case study related to availability evaluation of two network devices widely used in communication networks, namely a multiplexer ADM and a multiplexer SDH.
... Users specify their dependability model using some highlevel modeling language in which the underlying mathematical model is automatically generated and analyzed. We propose the adoption of intervals to represent the uncertainties related to the parameters of ISPN (Interval Stochastic Petri Net) models [5], [6], [7]. Therefore, the set of methods considered for steady-state analysis has to be adapted for taking into account interval arithmetic. ...
... ISPN extends the GSPN (Generalized Stochastic Petri Nets) model in order to introduce interval analysis [5] and it is already being used for performance and dependability evaluations [5], [6], [7]. A GSPN is a particular timed PN (Petri Net) that incorporates both stochastic timed transitions (represented as white boxes) and immediate transitions (represented as thin black bars). ...
... ISPN is a model that might be considered for both stochastic simulation and analysis. In this work in particular, ISPN is considered to be a high-level formalism for ICTMC (Interval Continuous Time Markov Chain) generation [7]. The classical algorithms found in literature [3] are adapted to take into account the interval coefficients of the ISPN model [5], [6], [7]. ...
Interval methods have been applied to the outer estimation of solution sets for real interval linear systems of equations. This work presents the evaluation of results relating to a significant set of algorithms for solving interval linear systems of equations applied to the steady state solution of dependability models. These algorithms have been implemented using the Matlab Intlab framework. The ISPN analysis takes into account the effects of variabilities in exponential transition rates and immediate transition weights when calculating dependability indices. Case studies were conducted and an evaluation is presented comparing the algorithms which have been applied to solving steady state dependability studies.
... We propose the adoption of intervals to represent the uncertainties related to the parameters RDB (Reliability Block Diagram) models. In early works [11], [12], [13], therefore, the set of methods considered for Markov chain steady-state analy sis has to be adapted for taking into account interval arithmetic. In ISPN [13], the exponential transition rates and immediate transition weights are represented by intervals. ...
... In early works [11], [12], [13], therefore, the set of methods considered for Markov chain steady-state analy sis has to be adapted for taking into account interval arithmetic. In ISPN [13], the exponential transition rates and immediate transition weights are represented by intervals. ISPN extends the GSPN (Generalized Stochastic Petri Nets) model in order to introduce interval analysis [11] and it is already being used for performance and dependability evaluations [11], [12], [13]. ...
... In ISPN [13], the exponential transition rates and immediate transition weights are represented by intervals. ISPN extends the GSPN (Generalized Stochastic Petri Nets) model in order to introduce interval analysis [11] and it is already being used for performance and dependability evaluations [11], [12], [13]. ...
In availability predictive analysis not every model input parameter is known exactly when the model is developed. The sources of uncertainties include lack of data and lack of knowledge about physical systems. In this paper, we present a reliable mechanism to help improve evaluation robustness when significant uncertainties exist. The mechanism incorporates variabilities and uncertainties based on imprecise distribution parameters in the availability evaluations are intervals instead of precise real numbers. The analysis takes into account the effects of variabilities in MTTR and MTTF when calculating interval availabilities. Interval studies of availability models should be used to show the influence of input parameters uncertainties on resultant availability evaluation. Interval methods and interval arithmetics have been applied in Scilab toolbox Int4Sci framework for steady state solution of availability models.
... ISPN is considered to be a high-level formalism for ICTMC (Interval Continuous Time Markov Chain) generation Galdino et al. (2007b). The classical algorithms found in literature (Bolch et al., 2006) are adapted to take into account the interval coefficients of the ISPN model. ...
... ISPN is considered to be a high-level formalism for ICTMC (Interval Continuous Time Markov Chain) generation Galdino et al. (2007b). The classical algorithms found in literature (Bolch et al., 2006) are adapted to take into account the interval coefficients of the ISPN model. ...
This paper discusses fundamental aspects of inference with imprecise prob- abilities from the decision theoretic point of view. It is shown why the equiv- alence of prior risk and posterior loss, well known from classical Bayes- ian statistics, is no longer valid under imprecise priors. As a consequence, straightforward updating, as suggested by Walley's Generalized Bayes Rule or as usually done in the Robust Bayesian setting, may lead to suboptimal decision functions. As a result, it must be warned that, in the framework of imprecise probabilities, updating and optimal decision making do no longer coincide.
Conventional solution techniques for analytic performance models of computer and telecommunication systems use single values
as inputs. Uncertainties or variabilities in model parameters may exist in many types of systems. Using models with a single
aggregated mean value for each parameter for such systems can produce inappropriate and misleading results. This chapter presents
intervals and extended histograms for characterizing system parameters that are associated with uncertainty and variability.
Adaptation of existing analytic performance evaluation methods to this interval-based parameter characterization is described.
The application of this approach is illustrated with two examples: a hierarchical model of a multicomputer system and a queueing
network model of an EJB server implementation.