Implicit ODE solvers with good local error control for the transient analysis of Markov models

Obtaining the transient probability distribution vector of a continuous-time Markov chain (CTMC) using an implicit ordinary differential equation (ODE) solver tends to be advantageous in terms of run-time computational cost when the product of the maximum output rate of the CTMC and the largest time of interest is large. In this paper, we show that when applied to the transient analysis of CTMCs, many implicit ODE solvers are such that the linear systems involved in their steps can be solved by using iterative methods with strict control of the 1-norm of the error. This allows the development of implementations of those ODE solvers for the transient analysis of CTMCs that can be more efficient and more accurate than more standard implementations.Peer ReviewedPostprint (published version

A comparison of numerical splitting-based methods for Markovian dependability and performability models

Iterative numerical methods are an important ingredient for the solution of continuous time Markov dependability models of fault-tolerant systems. In this paper we make a numerical comparison of several splitting-based iterative methods. We consider the computation of steady-state reward rate on rewarded models. This measure requires the solution of a singular linear system. We consider two classes of models. The first class includes failure/repair models. The second class is more general and includes the modeling of periodic preventive test of spare components to reduce the probability of latent failures in inactive components. The periodic preventive test is approximated by an Erlang distribution with enough number of stages. We show that for each class of model there is a splitting-based method which is significantly more efficient than the other methods.Postprint (published version

A Parameterized multi-step Newton method for solving systems of nonlinear equations

We construct a novel multi-step iterative method for solving systems of nonlinear equations by introducing a parameter. to generalize the multi-step Newton method while keeping its order of convergence and computational cost. By an appropriate selection of theta, the new method can both have faster convergence and have larger radius of convergence. The new iterative method only requires one Jacobian inversion per iteration, and therefore, can be efficiently implemented using Krylov subspace methods. The new method can be used to solve nonlinear systems of partial differential equations, such as complex generalized Zakharov systems of partial differential equations, by transforming them into systems of nonlinear equations by discretizing approaches in both spatial and temporal independent variables such as, for instance, the Chebyshev pseudo-spectral discretizing method. Quite extensive tests show that the new method can have significantly faster convergence and significantly larger radius of convergence than the multi-step Newton method.Peer ReviewedPostprint (author's final draft

On the computation of poisson probabilities

The Poisson distribution is a distribution commonly used in statistics. It also plays a central role in the analysis of the transient behaviour of continuous-time Markov chains. Several methods have been devised for evaluating using floating-point arithmetic the probability mass function (PMF) of the Poisson distribution. Restricting our attention to published methods intended for the computation of a single probability or a few of them, we show that neither of them is completely satisfactory in terms of accuracy. With that motivation, we develop a new method for the evaluation of the PDF of the Poisson distribution. The method is intended for the computation of a single probability or a few of them. Numerical experimentation illustrates that the method can be more accurate and slightly faster than the previous methods. Besides, the method comes with guaranteed approximation relative error.Postprint (author's final draft

Cumulative dominance and heuristic performance in binary multi-attribute choice

Working paper 895, Department of Economics and Business, Universitat Pompeu FabraSeveral studies have reported high performance of simple decision heuristics in multi-attribute decision making. In this paper, we focus on situations where attributes are binary and analyze the performance of Deterministic-Elimination-By-Aspects (DEBA) and similar decision heuristics. We consider non-increasing weights and two probabilistic models for the attribute values: one where attribute values are independent Bernoulli randomvariables; the other one where they are binary random variables with inter-attribute positive correlations. Using these models, we show that good performance of DEBA is explained by the presence of cumulative as opposed to simple dominance. We therefore introduce the concepts of cumulative dominance compliance and fully cumulative dominance compliance and show that DEBA satisfies those properties. We derive a lower bound with which cumulative dominance compliant heuristics will choose a best alternative and show that, even with many attributes, this is not small. We also derive an upper bound for the expected loss of fully cumulative compliance heuristics and show that this is moderate even when the number of attributes is large. Both bounds are independent of the values of the weights.Postprint (author’s final draft

Simulation of steady-state availability models of fault-tolerant systems with deferred repair

This paper targets the simulation of continuous-time Markov chain models of fault-tolerant systems with deferred repair. We start by stating sufficient conditions for a given importance sampling scheme to satisfy the bounded relative error property. Using those sufficient conditions, it is noted that many previously proposed importance sampling schemes such as failure biasing and balanced failure biasing satisfy that property. Then, we adapt the importance sampling schemes failure transition distance biasing and balanced failure transition distance biasing so as to develop new importance sampling schemes which can be implemented with moderate effort and at the same time can be proved to be more efficient for balanced systems than the simpler failure biasing and balanced failure biasing schemes. The increased efficiency for balanced and unbalanced systems of the new adapted importance sampling schemes is illustrated using examples.Preprin

Transient analysis of rewarded continuous time Markov models by regenerative randomization with Laplace transform inversion

In this paper we develop a variant, regenerative randomization with Laplace transform inversion, of a previously proposed method (the regenerative randomization method) for the transient analysis of rewarded continuous time Markov models. Those models find applications in dependability and performability analysis of computer and telecommunication systems. The variant differs from regenerative randomization in that the truncated transformed model obtained in that method is solved using a Laplace transform inversion algorithm instead of standard randomization. As regenerative randomization, the variant requires the selection of a regenerative state on which the performance of the method depends. For a class of models, class C’, including typical failure/repair models, a natural selection for the regenerative state exists and, with that selection, theoretical results are available assessing the performance of the method in terms of “visible” characteristics. Using dependability class C’ models of moderate size of a RAID 5 architecture we compare the performance of the variant with those of regenerative randomization and randomization with steady-state detection for irreducible models, and with those of regenerative randomization and standard randomization for models with absorbing states. For irreducible models, the new variant seems to be about as fast as randomization with steady-state detection for models which are not too small when the initial probability distribution is concentrated in the regenerative state, and significantly faster than regenerative randomization when the model is stiff and not very large. For stiff models with absorbing states, the new variant is much faster than standard randomization and significantly faster than regenerative randomization when the model is not very large. In addition, the variant seems to be able to achieve stringent accuracy levels safely.Postprint (author's final draft

Adapted importance sampling schemes for the simulation of dependability models of Fault-tolerant systems with deferred repair

This paper targets the simulation of continuous-time Markov chain models of fault-tolerant systems with deferred repair. We start by stating sufficient conditions for a given importance sampling scheme to satisfy the bounded relative error property. Using those sufficient conditions, it is noted that many previously proposed importance sampling techniques such as failure biasing and balanced failure biasing satisfy that property. Then, we adapt the importance sampling schemes failure transition distance biasing and balanced failure transition distance biasing so as to develop new importance sampling schemes which can be implemented with moderate effort and at the same time can be proved to be more efficient for balanced systems than the simpler failure biasing and balanced failure biasing schemes. The increased efficiency for both balanced and unbalanced systems of the new adapted importance sampling schemes is illustrated using examples.Postprint (published version

Failure distance-based simulation of repairable fault-tolerant systems

This paper presents a new importance sampling scheme called failure biasing for the efficient simulation of Markovian models of repairable fault-tolerant systems. The new scheme enriches the failure biasing scheme previously proposed by exploiting the concept of failure distance. This results in a much more efficient simulation with speedups over failure biasing of orders of magnitude in typical cases. The paper also discusses the efficient implementation of the new importance sampling scheme and presents a practical method for the optimization of the biasing parameters.Postprint (author’s final draft

An efficient and numerically stable method for computing interval availability distribution bounds

The paper develops a method, called bounding regenerative transformation, for the computation with numerical stability and well-controlled error of bounds for the interval availability distribution of systems modeled by finite (homogeneous) continuous-time Markov chain models with a particular structure. The method requires the selection of a regenerative state and is targeted at a class of models, class C'_1, with a “natural” selection for the regenerative state. For class C'_1 models, bounds tightness can be traded-off with computational cost through a control parameter D_C, with the option D_C = 1 yielding the smallest computational cost. For large class C'_1 models and the selection D_C = 1, the method will often have a small computational cost relative to the model size and, with additional conditions, seems to yield tight bounds for any time interval or not small time intervals, depending on the initial probability distribution of the model. Class C'_1 models with those additional conditions include both exact and bounding failure/repair models of coherent fault-tolerant systems with exponential failure and repair time distributions and repair in every state with failed components with failure rates much smaller than repair rates.Preprin