Performability of the hypercube (reliability)

Islam, S.M.R.; Ammar, H.H.

doi:10.1109/24.46474

Cited by 12 publications

(3 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In our review, we will restrict our attention to general-purpose methods which, besides (possibly) Ω being finite, do not impose any restrictions on X. Smith et al [28] developed a method with time complexity O(M 3 ), where M = |Ω|, which is based on the inversion of a double Laplace transform. Another method with time complexity O(M 3 ) using Laguerre functions was developed by Islam and Ammar [11]. De Souza e Silva and Gail [29] developed a randomization-based method with time complexity exponential on the number of different reward rates.…”

Section: Crcd(t S)mentioning

confidence: 99%

Two methods for computing bounds for the distribution of cumulative reward for large Markov models

Carrasco

2006

Performance Evaluation

View full text Add to dashboard Cite

Degradable fault-tolerant systems can be evaluated using rewarded continuous-time Markov chain (CTMC) models. In that context, a useful measure to consider is the distribution of the cumulative reward over a time interval [0, t]. All currently available numerical methods for computing that measure tend to be very expensive when the product of the maximum output rate of the CTMC model and t is large and, in that case, their application is limited to CTMC models of moderate size. In this paper, we develop two methods to compute bounds for the cumulative reward distribution of CTMC models with reward rates associated with states: BT/RT (Bounding Transformation/Regenerative Transformation) and BT/BRT (Bounding Transformation/Bounding Regenerative Transformation). The methods require the selection of a regenerative state, are numerically stable and compute the bounds with well-controlled error. For a class of rewarded CTMC models, class C ′′′ 1 , and a particular, natural selection for the regenerative state the BT/BRT method allows to trade off bounds tightness with computational cost and will provide bounds at a moderate computational cost in many cases of interest. For a class of models, class C ′′ 1 , slightly wider than class C ′′′ 1 , and a particular, natural selection for the regenerative state, the BT/RT method will yield tighter bounds at a higher computational cost. Under additional conditions, the bounds obtained by the less expensive version of BT/BRT and BT/RT seem to be tight for any value of t or not small values of t, depending on the initial probability distribution of the model. Class C ′′ 1 and class C ′′′ 1 models with those additional conditions include both exact and bounding typical failure/repair performability models of fault-tolerant systems with exponential failure and repair time distributions and repair in every state with failed components and a reward rate structure which is a non-increasing function of the collection of failed components. We illustrate both the applicability and the performance of the methods using a large CTMC performability example of a fault-tolerant multiprocessor system.

show abstract

Section: Crcd(t S)mentioning

confidence: 99%

Two methods for computing bounds for the distribution of cumulative reward for large Markov models

Carrasco

2006

Performance Evaluation

View full text Add to dashboard Cite

show abstract

“…Other performability evaluation studies have dealt more specifically with aspects of interprocessor communication in multiprocessors or internode communication in local area networks. Contributions here include those of Muralidhar and Pimentel ([87]; token bus LANs), Najjar and Gaudiot ([88]; hypercube multiprocessors), Bisbee and Nelson ( [12]; shuffle exchange networks), Aupperle and Meyer ([4]; balanced multibus networks), Islam and Ammar ( [49]; hypercube multiprocessors), Koren and Koren ([55]; multistage interconnection networks), Meyer et al ([78]; token bus LANs), and Karmarkar and Kuhl ([53]; multibus LANs).…”

Section: -1990mentioning

confidence: 99%

Performability: a retrospective and some pointers to the future

Meyer

1992

Performance Evaluation

122

View full text Add to dashboard Cite

“…Doing so, we clearly have F (s) = Pr[ for t large enough. The distribution function F t (s), often referred to as performability, a more general concept introduced in [7], has received much attention in the last years and there exist currently several methods for the computation of F t (s) [8,9,10,11,12], a method for the computation of Pr[ [13] (see also [14]), two methods for the computation of the distribution function of the timeaveraged cumulative reward, [15,16], a method for the computation of Pr[(1/t) t 0 r X(τ ) dτ > s] = 1 − F t (ts) [16], and a method for the computation of bounds for F t (s) [17]. All these methods require r i ≥ 0, i ∈ Ω, and, thus, to apply them in the case S + = ∅, S − = ∅ one has to replace r i , i ∈ Ω, by r i = r i − min j∈S − r j and use…”

Section: Letmentioning

confidence: 99%

A Numerical Method for the Evaluation of the Distribution of Cumulative Reward till Exit of a Subset of Transient States of a Markov Reward Model

Carrasco

Suñé

2011

IEEE Trans. Dependable and Secure Comput.

View full text Add to dashboard Cite

Markov reward models have interesting modeling applications, particularly those addressing fault-tolerant hardware/software systems. In this paper, we consider a Markov reward model with a reward structure including only reward rates associated with states, in which both positive and negative reward rates are present and null reward rates are allowed, and develop a numerical method to compute the distribution function of the cumulative reward till exit of a subset of transient states of the model. The method combines a model transformation step with the solution of the transformed model using a randomization construction with two randomization rates. The method introduces a truncation error, but that error is strictly bounded from above by a user-specified error control parameter. Further, the method is numerically stable and takes advantage of the sparsity of the infinitesimal generator of the transformed model. Using a Markov reward model of a fault-tolerant hardware/software system, we illustrate the application of the method and analyze its computational cost. Also, we compare the computational cost of the method with that of the (only) previously available method for the problem. Our numerical experiments seem to indicate that the new method can be efficient and that for medium-size and large models can be substantially faster than the previously available method.

show abstract

Performability of the hypercube (reliability)

Cited by 12 publications

References 31 publications

Two methods for computing bounds for the distribution of cumulative reward for large Markov models

Two methods for computing bounds for the distribution of cumulative reward for large Markov models

Performability: a retrospective and some pointers to the future

A Numerical Method for the Evaluation of the Distribution of Cumulative Reward till Exit of a Subset of Transient States of a Markov Reward Model

Contact Info

Product

Resources

About