Probability and reliability

Kottegoda, N. T. and Rosso, R. 1997 Statistics, Probability and Reliability for Civil and Environmental Engineers. NY McGraw-Hill. 

Figure 8 compares the failure probability and reliability functions for an exponential distribution. Whereas the reliability of the device is initially unity, it falls off exponentially with time and asymptotically approaches zero. The failure probability, on the other hand, does the reverse. Thus new devices start life with high reliability and end with a high failure probability. 

T3rpical non-electrical applications are fluid flow systems including oil flow, and general material transport. Presumably also traffic and communication systems can be analyzed by this method. In these cases the power flow model is replaced by the actual system model. The principle based on unit models can also be used in general probability and reliability calculations to build extremely large models. 

The target minimum annual exceedence probabilities Pfannual, for each performance level, is shown in Table 4. This table also shows the corresponding exceedence probabilities and reliability indices for the seismic event, P/e and Pe-... 

The proposed approaches allow us to simplify the mathematical format, if the prediction of survival probabilities and reliability indices of sustainable series, parallel and series-parallel systems are based on the additional concepts of Transformed Conditional Probabilities (TCP) and Conventional Correlation Vectors (CCV). They help us avoid of complicated multidimensional integrations both in a safety analysis of general systems and their autosystem components. 

Song J, Der Kiureghian A (2006) Joint first-passage probability and reliability of systems under stochastic excitation. ASCE J Eng Mech 132(l) 65-77 Soong TT (1973) Random differential equations in science and engineering. Academic, New York Soong TT, Grigoriu M (1993) Random vibration of mechanical and structural systems. Prentice Hall, New Jersey... 

A meehanieal eomponent is eonsidered safe and reliable when the strength of the eomponent, S, exeeeds the value of loading stress, L, on it (Rao, 1992). When the loading stress exeeeds the strength, failure oeeurs, the reliability of the part, R, being related to this failure probability, P, by equation 4.26 ... 

Ayyub, B. M. and McCuen, R. H. 1997 Probability, Statistics and Reliability for Engineers. Boca Raton CRC Press. 

Typical events that are considered are fire, explosion, ship collision, and the failure of pressurized storage vessels for which historical data established the failure frequencies. Assessment of consequences was based partly on conservative treatment of past experience. For example ilic assessment of the number of casualties from the release of a toxic material was based on past histoiy conditioned by knowledge of the toxicology and the prevailing weather conditions. An altemati. e used fault trees to estimate probabilities and identify the consequences. Credit is taken in this process for preventative measures in design, operation, and maintenance procedures. Historical data provide reliability expected from plant components and humans. 

Vescly, V. E., 1977, Estimating Common Cause Failure Probability in Reliability and Risk Analyses Marshall-Olkin Specializations, Proc. Int. Conf. Nucl. Systems Rel. Eng. F ment, Gatlinburg, TN, June. 

The book contains, in alphabetical order, failure rates, event rates and probabilities, and descriptive information which has been collected since 1970 in the course of doing risk and reliability assessments. Twenty appendices contain results of surveys on bursting discs, pipes, valves, relief valves, pump failures and information on human error, international fire losses, and blast effects. 

In this study detailed fault trees with probability and failure rate calculations were generated for the events (1) Fatality due to Explosion, Fire, Toxic Release or Asphyxiation at the Process Development Unit (PDU) Coal Gasification Process and (2) Loss of Availability of the PDU. The fault trees for the PDU were synthesized by Design Sciences, Inc., and then subjected to multiple reviews by Combustion Engineering. The steps involved in hazard identification and evaluation, fault tree generation, probability assessment, and design alteration are presented in the main body of this report. The fault trees, cut sets, failure rate data and unavailability calculations are included as attachments to this report. Although both safety and reliability trees have been constructed for the PDU, the verification and analysis of these trees were not completed as a result of the curtailment of the demonstration plant project. Certain items not completed for the PDU risk and reliability assessment are listed. 

Hazard, risk, failure, and reliability are interrelated concepts concerned witli uncertain events and tlierefore amenable to quantitative measurement via probability. "Hazard" is defined as a potentially dangerous event. For example, tlie release of toxic fumes, a power outage, or pump failure. Actualization of the potential danger represented by a hazard results in undesirable consequences associated with risk. 

Risk is defined as tlie product of two factors (1) tlie probability of an undesirable event and (2) tlie measured consequences of the undesirable event. Measured consequences may be stated in terms of financial loss, injuries, deatlis, or Ollier variables. Failure represents an inability to perform some required function. Reliability is the probability that a system or one of its components will perform its intended function mider certain conditions for a specified period. Tlie reliability of a system and its probability of failure are complementary in tlie sense tliat the sum of these two probabilities is unity. This cluipler considers basic concepts and llieorenis of probability tliat find application in tlie estimation of risk and reliability. 

Hazard, risk, failure, and reliability are interrelated concepts concerned with uncertain events and tlierefore amenable to quantitative measurement via probability. 

The parameters obtained here from measurements of B12 and Dn over wide ranges of temperature are probably as reliable as any that have been proposed for the interaction of molecules of different species. Unfortunately they do not provide an adequate test of Eqs. 9 and 10, since each of the systems has as one of its components either helium or hydrogen (for which there are significant quantum corrections) or carbon dioxide (which does... 

The simultaneous determination of a great number of constants is a serious disadvantage of this procedure, since it considerably reduces the reliability of the solution. Experimental results can in some, not too complex cases be described well by means of several different sets of equations or of constants. An example would be the study of Wajc et al. (14) who worked up the data of Germain and Blanchard (15) on the isomerization of cyclohexene to methylcyclopentenes under the assumption of a very simple mechanism, or the simulation of the course of the simplest consecutive catalytic reaction A — B —> C, performed by Thomas et al. (16) (Fig. 1). If one studies the kinetics of the coupled system as a whole, one cannot, as a rule, follow and express quantitatively mutually influencing single reactions. Furthermore, a reaction path which at first sight is less probable and has not been therefore considered in the original reaction network can be easily overlooked. 

Preceding papers. h Preliminary values obtained through redetermination of parameters in crystals (cal-cite and sodium nitrate) by Mr. Norman Elliot. The values in parentheses are based on older parameter determinations. c L. Pauling and L. O. Brockway, Proc. Nat. Acad. Sci., 20, 336 (1934). The value 1.25 A. reported in crystals of oxalic acids and oxalates is probably less reliable. 

The results of this procedure are shown in Figure 11-14. The symbol P represents the probability and R represents the reliability. The failure probabilities for the basic events were obtained from Example 11-2. I... 

Efforts to tune the reactivity of rhodium catalysts by altering structure, solvent, and other factors have been pursued.49,493 50 Although there is (justifiably) much attention given to catalysts which provide /raor-addition processes, it is probably underappreciated that appropriate rhodium complexes, especially cationic phosphine complexes, can be very good and reliable catalysts for the formation of ( )-/3-silane products from a air-addition process. The possibilities and range of substrate tolerance are demonstrated by the two examples in Scheme 9. A very bulky tertiary propargylic alcohol as well as a simple linear alkyne provide excellent access to the CE)-/3-vinylsilane products.4 a 1 In order to achieve clean air-addition, cationic complexes have provided consistent results, since vinylmetal isomerization becomes less competitive for a cationic intermediate. Thus, halide-free systems with... 

Overall, this study supports the existence of an eating disorder taxon. However, the evidence is not particularly strong. The consistency of the mixed-sample analyses is fairly high (acceptably low base rate variability), but the sample composition confounds the interpretation of these findings. To resolve this issue, the authors performed analyses in the student-only sample and found evidence of taxonicity, but some of these analyses were not consistent. Interestingly, the high base rate variability was mostly a problem for analyses using the empirical indicators, which consisted of only a few items and hence were probably less reliable than theoretical indicators. The observed inconsistency may be due to low reliability of the empirical indicators. 

Modem LP solvers can solve very large LPs very quickly and reliably on a PC or workstation. LP size is measured by several parameters (1) the number of variables n, (2) the number of constraints m, and (3) the number of nonzero entries nz in the constraint matrix A. The best measure is the number of nonzero elements nz because it directly determines the required storage and has a greater effect on computation time than n or m. For almost all LPs encountered in practice, nz is much less than mn, because each constraint involves only a few of the variables jc. The problem density 100(nz/mn) is usually less than 1%, and it almost always decreases as m and n increase. Problems with small densities are called sparse, and real world LPs are always sparse. Roughly speaking, a problem with under 1000 nonzeros is small, between 1000 and 50,000 is medium-size, and over 50,000 is large. A small problem probably has m and n in the hundreds, a medium-size problem in the low to mid thousands, and a large problem above 10,000. 

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