Calculating Cumulative Operational Time Distributions of Repairable Computer Systems

Silva, De Souza E; Gail,

doi:10.1109/tc.1986.1676765

Cited by 119 publications

(48 citation statements)

References 6 publications

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“…Using the well-known result [3,Theorem 3.3.5] that Q(t) has for t → ∞ an asymptotic normal distribution with mean and variance t, it is easy to realize that for large t and 1 the required N will be ≈ t and, then, the method will be very costly for large models and large time intervals. The other proposed methods able to deal with general ÿnite CTMCs [2,5,9] have similar problems.…”

Section: Review Of Algorithm a Of [6]mentioning

confidence: 96%

“…This is mainly because that distribution tends to achieve its asymptotic shape (interval availability equal to the steady-state availability with probability one) very slowly and, then, the steady-state availability may be a poor measure of the behavior of the system over a ÿnite time interval. Computing the interval availability distribution of a fault-tolerant system modeled by a CTMC is a challenging problem for which methods have been developed quite recently [1][2][3][4][5][6][7][8][9]. The ÿrst e ort is reported in [1], where a closed form integral expression is obtained for a two-state model.…”

Section: Introductionmentioning

confidence: 99%

“…In [3], randomization is used to obtain the distribution of the operational time in a time interval of the same two-state model. The ÿrst method able to deal with general ÿnite CTMC models was developed by de Souza e Silva and Gail using randomization [9]. Goyal and Tantawi [2] developed a numerical approximate method, but they did not obtain bounds for the approximation error.…”

Section: Introductionmentioning

confidence: 99%

“…Rubino and Sericola [4] developed an e cient algorithm to compute the distribution of the interval availability for the particular case in which operational and down periods are independent one by one and of each other, which is based on the analysis of the distributions of the durations of consecutive operational periods. In [5], Rubino and Sericola developed two algorithms which reduce the computational requirements of the randomization-based method developed in [9]. The ÿrst of such algorithms reduces the time requirements; the second one reduces the storage requirements.…”

Section: Introductionmentioning

confidence: 99%

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Solving large interval availability models using a model transformation approach

Carrasco

2004

Computers & Operations Research

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Fault-tolerant systems are often modeled using (homogeneous) continuous time Markov chains (CTMCs).Computation of the distribution of the interval availability, i.e. of the distribution of the fraction of time in a time interval in which the system is operational, of a fault-tolerant system modeled by a CTMC is an important problem which has received attention recently. However, currently available methods to perform that computation are very expensive for large models and large time intervals. In this paper, we develop a new method to compute the distribution of the interval availability which, for large enough models and large enough time intervals, is signiÿcantly faster than previous methods. In the method, a truncated transformed model, which has with some arbitrarily small error the same interval availability distribution as the original model, is obtained from the original model and the truncated transformed model is solved using a previous state-of-the-art method. The method requires the selection of a "regenerative" state and its performance depends on that selection. For a class of models, including typical failure/repair models of coherent fault-tolerant systems with exponential failure and repair time distributions and repair in every state with failed components, a natural selection for the regenerative state exists and theoretical results are available assessing 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 competitive for models in that class. Numerical results are presented showing that the new method can indeed be signiÿcantly faster than a previous state-of-the-art method and is able to deal with some large models and large time intervals in reasonable CPU times. ?

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Section: Review Of Algorithm a Of [6]mentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Solving large interval availability models using a model transformation approach

Carrasco

2004

Computers & Operations Research

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“…We assume that IAVCD(t, p) has to be computed at a single (t, p) pair. First, we determine the truncation point N ′ using (9) and the truncation point C ′′′ using (11). Next, we obtain the transition matrix P = I + Λ Λ Λ −1 U D A of the randomized DTMC X considered in the method.…”

Section: Description and Computational Costmentioning

confidence: 99%

A New General-Purpose Method for the Computation of the Interval Availability Distribution

Carrasco

2013

INFORMS Journal on Computing

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We develop a new randomization-based general-purpose method for the computation of the interval availability distribution of systems modeled by continuous-time Markov chains (CTMCs). The basic idea of the new method is the use of a randomization construct with different randomization rates for up and down states. The new method is numerically stable and computes the measure with well-controlled truncation error. In addition, for large CTMC models, when the maximum output rates from up and down states are significantly different, and when the interval availability has to be guaranteed to have a level close to one, the new method is significantly or moderately less costly in terms of CPU time than a previous randomizationbased state-of-the-art method, depending on whether the maximum output rate from down states is larger than the maximum output rate from up states or vice versa. Otherwise, the new method can be more costly, but a relatively inexpensive for large models switch of reasonable quality can be easily developed to choose the fastest method. Along the way, we show the correctness of a generalized randomization construct, in which arbitrarily different randomization rates can be associated with different states, for both finite CTMCs with infinitesimal generator and uniformizable CTMCs with denumerable state space.

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A unified approach for reliability and performability evaluation of semi-Markov systems

Limnios

Oprian

1999

Appl. Stochastic Models Bus. Ind.

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Calculating Cumulative Operational Time Distributions of Repairable Computer Systems

Cited by 119 publications

References 6 publications

Solving large interval availability models using a model transformation approach

Solving large interval availability models using a model transformation approach

A New General-Purpose Method for the Computation of the Interval Availability Distribution

A unified approach for reliability and performability evaluation of semi-Markov systems

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