R. E. BARLOW AND F. PROSCHAN, Statistical Theory of Reliability and Life Testing, Holt, Rinehart and Winston, New York, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....4.2 Hu man algebras for random variables In order to extend these Hu man algebras to nondegenerate random variables, we have to introduce the de nitions for several commonly used partial orderings for random variables. Refer to Stoyan [24] Ross ( 22] Chapter 8) and Barlow and Proschan [1]. De nition 4.7 Suppose X and Y are two independent random variables with density functions f( and g( and distribution functions F ( and G( i) Nonoverlapping ordering: X Y if their distributions are nonoverlapping, that is, supfx : f(x) 0g inffx : g(x) 0g. ii) Likelihood ....

....Likelihood Ratio (ILR) distributionsg, i.e. g(x) g(x a) is increasing in x for all a 0. S logILR = findependent r.v. s with log(X) being ILRg. In addition, S def = S fnonnegative r.v. sg. We note that the ILR property is also known as the PF2 property in the literature [1]. Also, ILR ) IFR ) logIFR and logILR ) logIFR. Examples of ILR r.v. s are normal and Erlang. Examples of logILR r.v. s are lognormal and Weibull. For more examples of these distributions, we refer to [1, 16] The following proposition regarding the property of these sets of r.v. s is shown in ....

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R.E. Barlow and F. Proschan. Statistical Theory of Reliability and Life Testing Probabiltity Models. Holt, Reinhart and Winston, Inc., 1981.

....the distribution of the completion time as a first marking problem is provided in [10] 10.2. TIME SPENT IN A MARKING 27 Let S be the subset of markings in which a particular condition is fulfilled. The expected time # S (t) spent in the markings s S in the interval 0 t is given by [8]: # S (t) q s (z) dz (13) Moreover, it is well known from the theory of Markov chains that as t approaches infinity the proportion of the time spent in states s S equals the asymptotic probability: Q S (#) q s (#) 14) If S is the set of working states, # S (t) is the expected ....

R.E. Barlow and F. Proschan. Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York, 1975.

....already been reached by the virus, being infected or immune respectively with i and i . R i [k] is instead equal to 0 if node i has not yet been reached by the virus. We can easily prove that R i [k] and R j [k] where nodes i and j are neighbors) are associated random variables in the sense of [2] (see [1] for further details) We conclude that j [k] i [k] j [k] 11) which provides an upper bound to the joint probabilities P I i S j [k] In words, if we assume that the probabilities to be reached by the virus at any given time are independent from node to node, we overestimate the ....

R. E. Barlow, F. Proschan, "Statistical Theory of Reliability and Life Testing," Holt, Rinehart and Winston, Inc. , New York, 1975

...., EXRtdtMTTF (2.41) which is the expression for the Mean Time to Failure for a general system configuration. This direct relationship between MTTF and the system failure rate is one reason the constant failure rate assumption is often made when the supporting reliability data is scanty [Barl75] Appendix G describes the analysis of the variance for this distribution. Using an exponential failure distribution implies two important behaviors for the Since a used subsystem is stochastically as good as a new subsystem, a policy of scheduled replacement of used subsystems which are know ....

Barlow, R. E. and Proschan, F., Statistical Theory of Reliability and Life Testing: Probability Models, Holt, Rinehart and Winston, 1975.

....dual problems. PAS allows to consider one common strategy for computing answers to the questions in the di#erent situations. 1 Introduction and Overview One main problem for technical systems is the computation of the reliability of a system. This is studied in reliability theory (see for example [7, 8]) The reliability depends on various factors like the quality and the age of components, complexity of the system, etc. The reliability of a system conveys some information about the behavior of the system in the future, based on information about the components, for example probabilistic ....

R. E. Barlow and R. Proschan, Statistical Theory of Reliability and Life Testing, New York, 1975. IAUTOM 3.9.4-10.

....been proposed, with approximations focusing on the special case of equal link reliability values [13, 5] While these approximate algorithms are e#ective, the central assumption that all links are equally reliable makes them unsuitable for practical use. Other methods for algebraic calculation [3, 28] have prohibitive costs. On the other hand, edge packing methods o#er an e#cient means for calculating a two terminal reliability lower bound. The simplest and very e#ective approach for calculating an edge packing bound, which can readily be employed for the selection of multipaths, is the ....

R.E. Barlow and F. Proschan. Statistical theory of reliability and life testing. Silver Spring, 1981.

....models taking into account the ageing aspect of the lifetime distributions, independence of system components, and a lack of satisfactory data. We introduce the new non parametric life distribution classes which generalize the well known increasing and decreasing failure rate distributions [1] and can represent various judgements related to the lifetime distributions. In this paper we apply the theory of imprecise probabilities to reliability analysis of simple unrepairable systems. 2 Natural extension Consider a system consisting of n components. Let f ij (x i ) be the functions of ....

....In this paper we apply the theory of imprecise probabilities to reliability analysis of simple unrepairable systems. 2 Natural extension Consider a system consisting of n components. Let f ij (x i ) be the functions of the i th component lifetime x i , j = 1; m i . According to [1], the system lifetime can be uniquely determined by the component lifetimes. Denote X = x 1 ; xn ) Then there exist a function g(X) of the components lifetimes characterizing the system reliability behavior. The functions f ij (x i ) and g(X) can be regarded as gambles. Suppose that partial ....

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R.E. Barlow and F. Proschan. Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York, 1975.

....= st X n for all n =1, 2, V. POSITIVE DEPENDENCE Positive dependence in a collection of rvs can be captured in several ways. The association of rvs is one of the most useful such characterizations; it was introduced by Esary, Proschan and Walkup [11] and has proved useful in various settings [3], 9] 16] are said to be associated if, with X = X 1 , X n ) the inequality E [f(X)g(X) E [f(X) E [g(X) holds for all increasing functions f,g :IR IR for which the expectations exist and are finite. Here, we focus on a stronger notion of positive dependence: are said ....

R.E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing, International Series in Decision Processes, Holt, Rinehart and Winston, New York (NY), 1975.

.... has probability R(G) The problem of evaluating R(G) is usually called the K terminal reliability problem, and it has received considerable attention from the research community (see [8] and [17] for many references) Series parallel reductions are frequently used in the exact evaluation context [2]. They allow to deal in linear time with sp reducible networks [25] In the general case, they can be used as a precomputing task of any algorithm in order to reduce the size of the network without changing its reliability. Also, series parallel reductions have been combined with a recursive ....

....Readers Authors at rear of each issue. 3 2 Series parallel reductions and exact evaluation In the evaluation of network (un)reliability, simplification techniques are frequently used to reduce the size of considered networks. A well known technique is the use of series parallel reductions [2], 18] which can be performed in time linear on the network size and do not change the (un)reliability measure (Q(G) Q( G) 8] 25] In this section, we recall how series parallel reductions can be applied in the exact evaluation context. When the network studied is a K tree, its ....

B.E. Barlow and F. Proschan. Statistical Theory of Reliability and Life Testing. Holt, Rinehart & Winston, New York, 1975.

.... Poisson distributed random shocks, is that under only the hypothesis that X is positive, the survival probability D T (t) is increasing hazard rate in average (IHRA) Hence, independently on the nature of the cdf of a single shock, the overall process represents a true degradation on the average [4]. 3. Rejuvenation policies Rejuvenation is defined as a preventive maintenance action that prevents the system from reaching the crash condition. Hence, based on the fine grained degradation model introduced in the previous section, a rejuvenation policy is a rule (or a set of rules) to clean up ....

R.E. Barlow, F. Proschan, Statistical Theory of Reliability and Life Testing, Holt, Rinehart and Winston, New York, 1975.

....2 2 Explicit Calculation of the cdf FL for Series Parallel Systems 4 3 Estimating the cdf FL from Simulation 6 4 Optimization 13 5 Concluding Remarks 16 1 1 Introduction Systems with redundancy are abundant in real life. Most books on reliability engineering (e.g. Barlow and Proshan [1], Gertzbakh [3] Kozlov and Ushakov [6] Mann, Shaffer and Singpruvalla [7] Zacks [18] and Ushakov and Harrison s [15] recent handbook on reliability (which can serve as a good source of references) include a chapter on redundancy models and redundancy optimization. In this work we extend the ....

Barlow, R. and F. Proshan (1975). "Statistical Theory of Reliability and Life Testing Probability Models". Holt, Reimont and Winston, New York. 16

....on the processor it is allocated to. The reliability of this system is bounded from above by the reliability of a 2 out of 4 system. Let x i be 1 if p i is alive and let it be 0 if p i has failed, for i = 1,2,3,4. Let x i denote the complement of x i . Then the structure function for the system [3] is s(x 1 ; x 2 ; x 3 ; x 4 ) x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 . From this, the upper bound on the reliability of the ....

R.E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing, New York: Holt, Rinehart and Winston, 1975.

.... 1)du] 36) Letting t ##, we obtain the desired expression in (2) Proof of Proposition 2 To show (i) note that the inequality in (15) implies # # 0 # j#T G c j (u)du # # # 0 # j#T G c j (u)du. This can be directly verified via integration by parts. Also see Theorem 7. 3 of [2]) From (14) we observe that E[ # j#T X j ] is a linear combination, with positive coe#cients, of # # 0 # j#T # G c j (u)du, for T # # T ; hence, the desired inequality. To show the orderings in (ii) we first consider unit arrivals. In this case, X follows a multivariate Poisson ....

....) # #(# 1,T , # m,T ) n ##. Now, the convex order implies that, for any U # 1, m , the following holds: # # 0 # # i#U G i (u) 1 # du # # # 0 # # i#U G i (u) 1 # du. The above can again be directly verified via integration by parts; or see Theorem 7. 4 of [2]. Hence, we have, for any T , #(# 1,T , # m,T ) exp # # KK # K # # 0 # # i#T#K G i (u) 1 # du # # #(# 1,T , # m,T ) exp # # KK # K # # 0 # # i#T#K G i (u) 1 # du # . Consequently, for n large enough, we have, # n (# 1,T , # m,T ) # # ....

Barlow, R.E. and Proschan, F., Statistical Theory of Reliability and Life Testing: Probability Models, Hol, Rinehart and Winston, Inc, 1975.

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R. E. BARLOW AND F. PROSCHAN, Statistical Theory of Reliability and Life Testing, Holt, Rinehart and Winston, New York, 1975.

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Barlow, R.E., and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Probability Models. New York: Holt, Rinehart and Winston.

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R.E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing, Holt, Rinehart and Winston, Reading, (MA), 1975.

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Richard E. Barlow and Frank Proschan. Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, Inc., New York, 1975.

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R.E. Barlow, F. Proschan, Statistical Theory of Reliability and Life Testing: Probability Models, New York: Holt, Rinehart and Winston, 1975.

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R. E. Barlow, F. Proschan, Statistical Theory of Reliability and Life Testing. Probability Models (Holt, Rinehart and Winston, NewYork, 1975).

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R. E. Barlow, F. Proschan, \Statistical Theory of Reliability and Life Testing", Holt, Rinehart, Winston, 1975. 10

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R. E. Barlow, F. Proschan, "Statistical Theory of Reliability and Life Testing," Holt, Rinehart and Winston, Inc. , New York, 1975

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R.E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Test- ing: Probability Models (To begin with, Silver Spring, 1981).

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Barlow R. and Proschan F. (1981), Statistical Theory of Reliability and Life Testing. To Begin With: Silver Spring, Maryland.

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Barlow R. and F. Proschan (1975). Statistical theory of reliability and life testing probability models, Holt, Rinehart and Winston Inc..

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Barlow, R. E., F. Proschan (1975). Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York.

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