Dependability analysis of systems modeled by non-homogeneous Markov chains

Platis, Agapios Ν.; Limnios, Nikolaos; Du, Marc Le

doi:10.1016/s0951-8320(97)00073-2

Cited by 39 publications

(27 citation statements)

References 12 publications

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“…The accumulated reward, W T k ð Þ, the expected accumulated reward, E W T k ð Þ ½ , and the expected time-averaged accumulated reward, E Y T k ð Þ ½ , over the period (T 1 , T k ) is given by [48]:…”

Section: Performabilitymentioning

confidence: 99%

Performance evaluation of corrosion-affected reinforced concrete bridge girders using Markov chains with fuzzy states

Anoop

Rao

2016

Sādhanā

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A methodology for performance evaluation of reinforced concrete bridge girders in corrosive environments is proposed. The methodology uses the concept of performability and considers both serviceability-and ultimate-limit states. The serviceability limit states are defined based on the degree of cracking (characterized by crack width) in the girder due to chloride induced corrosion of reinforcement, and the ultimate limit states are defined based on the flexural load carrying capacity of the girder (characterized in terms of rating factor using the load and resistance factor rating method). The condition of the bridge girder is specified by the assignment of a condition state from a set of predefined condition states. Generally, the classification of condition states is linguistic, while the condition states are considered to be mutually exclusive and collectively exhaustive. In the present study, the condition states of the bridge girder are also represented by fuzzy sets to consider the ambiguities arising due to the linguistic classification of condition states. A non-homogeneous Markov chain (MC) model is used for modeling the condition state evolution of the bridge girder with time. The usefulness of the proposed methodology is demonstrated through a case study of a severely distressed beam of the Rocky Point Viaduct. The results obtained using the proposed approach are compared with those obtained using conventional MC model. It is noted that the use of MC with fuzzy states leads to conservative decision making for the problem considered in the case study.

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Section: Performabilitymentioning

confidence: 99%

Performance evaluation of corrosion-affected reinforced concrete bridge girders using Markov chains with fuzzy states

Anoop

Rao

2016

Sādhanā

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“…The transition function (p~(i,j); n E N, i,j E E) is called cyclic of period d (d > 1), if d is the smallest integer verifying Prnd+r : Pr for m,r E N. The major benefit of these chains is that an asymptotic analysis is possible due to their eventual weak ergodicity. It has been shown in [18], if Y~ is an embedded homogeneous Markov chain from X~ such as Yn : Xnd and if p~ is the transition probability matrix of Xn, then P0,d is the constant transition probability matrix of Yn. If additionally Y~ is irreducible and aperiodic then its transition probability from time 0 to time n tends to an ergodic matrix, therefore lim~_~ [P0,d] n --1.rr and asymptotically, the steady state probabilities inside a cycle are given by 7rr(j) = ~r0,r(j).…”

Section: Computation Of Thementioning

confidence: 99%

“…Indeed, for fault-tolerant systems, it is necessary to evaluate the performance under degraded states where the system may somehow be available but nevertheless not fully operational. In the same way, it can be applied to other highly available systems such as computer networks [16,17] and electrical systems [18]. In this kind of system, users or customers play the most significant part, and their satisfaction has to be taken into account.…”

Section: Introductionmentioning

confidence: 99%

“…{Xk_l r Xk =i, Xk+l =i .... , Xk+s-1 =i, Xk+s #i} = ~(xk-~ # i}~{xk = i I xk-, # i}~{xk+~ = i ] xk = i} 9 "I? {Xk+~ # i I Xk+s-1 = i}.The last probability can be computed as follows:P{Xk+s~i[Xk+s-l=i}= ~ P{Xk+s=jlXa+s_I =i}=l-p(i,i),(9) jeE--{i}whereas the first can be evaluate as follows:P{Xk-1 r =i IXk_l #i}= E P{Xk-1 =j}P{Xk =ilXk-1 =j} j6E-{i} = E C~Pk-l(J)P(J'i) iCE-{i} = ~p~(i) -~pk-l(i)p(i, z).Since limk~o rxpk(i) = ~r(i) thenlim Pr {Xk-1 # i}Pr{Zk = i l Xk-1 # i} = 7r(i)[1 --p(i,i)].Let us give the formulation for another interesting case, the case of nonhomogeneous Markov chains where the transition probabilities are time-dependent[17,18]. Let {Xn; n E N} now be a nonhomogeneous Markov chain in discrete time, with c~ the initial distribution of the probabilities and P.n (pn(i,j) = IP{Xn+l = j ] Xn i}; n E N, i,j E E) the time-dependent transition probability matrix, thenI?…”

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confidence: 99%

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A generalized formulation for the performability indicator

Platis

2006

Computers & Mathematics with Applications

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The main objective of this paper is to propose a generalized form of the performability measure, which has been initially defined for the purpose of studying the performance and reliability analysis of fault tolerant systems. This generalized form takes into account more detailed rewards and can be used in general for maintenance cost analysis as well as in the modeling of the websitc users behavior. We give different formulations by means of a homogeneous Markov chain and a cyclic nonhomogeneous Markov chain and their asymptotic expression. (~

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“…According to Platis et al [1] Electricité De France (EDF) used this for the homogeneous Markov approach to model an electrical substation in order to evaluate some measures of system performance. However, in this model, the underlying distributions are all non-Markovian, in the sense that, all the underlying distributions in this model are arbitrarily distributed.…”

Section: Introductionmentioning

confidence: 99%