SURF-2: A program for dependability evaluation of complex hardware and software systems

2000

Performance Evaluation

In this paper we deal with iterative numerical methods to solve linear systems arising in continuous-time Markov chain (CTMC) models. We develop an algorithm to dynamically tune the relaxation parameter of the successive over-relaxation method. We give a sufficient condition for the Gauss-Seidel method to converge when computing the steady-state probability vector of a finite irreducible CTMC, an a suffient condition for the Generalized Minimal Residual projection method not to converge to the trivial solution 0 when computing that vector. Finally, we compare several splitting-based iterative methods an a variant of the Generalized Minimal Residual projection method.

“…The convergence of both GS and SOR may be extremely slow when they are used to solve (4). In [15] an efficient technique is described which improves the convergence of such methods.…”

Section: Ssrr = Limmentioning

confidence: 99%

“…These are, among others, SAVE [12], SPNP [9], UltraSAN [10] and SURF-2 [4]. For the solution of linear systems of equations SPNP uses Successive over-relaxation (SOR) with dynamic tuning of the relaxation parameter ω [8].…”

Section: Introductionmentioning

confidence: 99%

Numerical iterative methods for Markovian dependability and performability models: new results and a comparison

Suñé

Domingo

2000

Performance Evaluation

“…It was first proposed by Grassman [5] and has been further developed by Gross and Miller [6]. The method is also offered by well-known performance, dependability and performability modeling packages [7,8,9,10]. A major drawback of the (standard) randomization method is that it is expensive when the model is stiff.…”

Section: Introductionmentioning

confidence: 99%

Transient Analysis of Rewarded Continuous Time Markov Models by Regenerative Randomization with Laplace Transform Inversion

2003

The Computer Journal

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.

“…1 It was ÿrst proposed by Grassman [5] and has been further developed by Gross and Miller [6]. The method is also o ered by the well-known performance, dependability and performability modeling packages [7][8][9][10]. The randomization method is based on the following result [11,Theorem 4.19].…”

Section: Introductionmentioning

confidence: 99%

Computation of bounds for transient measures of large rewarded Markov models using regenerative randomization

Computers & Operations Research

2003

In this paper we generalize a method (called regenerative randomization) for the transient solution of continuous time Markov models. The generalized method allows to compute two transient measures (the expected transient reward rate and the expected averaged reward rate) for rewarded continuous time Markov models with a structure covering bounding models which are useful when a complete, exact model has unmanageable size. The method has the same good properties as the well-known (standard) randomization method: numerical stability, well-controlled computation error, and ability to specify the computation error in advance, and, for large enough models and long enough times, is signiÿcantly faster than the standard randomization method. The method requires the selection of a regenerative state and its performance depends on that selection. For a class of models, class C , including typical failure=repair models with exponential failure and repair time distributions and repair in every state with failed components, a natural selection for the regenerative state exists, and results are available assessing approximately the performance of the method for that natural selection in terms of "visible" model characteristics. Those results can be used to anticipate when the method can be expected to be signiÿcantly faster than standard randomization for models in that class. The potentially superior e ciency of the regenerative randomization method compared to standard randomization for models not in class C is illustrated using a large performability model of a fault-tolerant multiprocessor system. Scope and purposeRewarded continuous time Markov models are widely used for performance, dependability and performability analysis of computer and telecommunication systems. Realistic modeling of such systems usually yields Markov models whose size exceeds the available memory resources. An approach to deal with the "largeness" problem is the use of bounding Markov models. However, even those bounding models can have very * Tel.: +34-3-4016652; fax: +34-3-4017785.E-mail address: carrasco@eel.upc.es (J.A. Carrasco). so that it can be used for the computation of bounds for the expected transient reward rate and the expected averaged reward rate transient measures with general reward rate structures including arbitrary non-negative reward rates associated with the states of the model. Examples of such measures are the unavailability of a fault-tolerant system at time t and the expected interval unavailability of a fault-tolerant system at time t. The method has the same good properties as the well-known standard randomization (also called uniformization) method: numerical stability, well-controlled computation error, and ability to specify the computation error in advance, and, for large models and long mission times, can be signiÿcantly faster than that method. Furthermore, for a class of models, class C , including typical failure=repair models with exponential failure and repair time distributions and repair in every s...