By Ilya Gertsbakh
The fabric during this e-book was once first awarded as a one-semester path in Relia bility concept and Preventive upkeep for M.Sc. scholars of the economic Engineering division of Ben Gurion college within the 1997/98 and 1998/99 educational years. Engineering scholars are commonly attracted to the utilized a part of this idea. the worth of preventive upkeep thought lies within the risk of its imple mentation, which crucially depends upon how we deal with statistical reliability information. The very nature of the item of reliability thought - procedure lifetime - makes it super tricky to gather quite a lot of information. the information to be had are usu best friend incomplete, e.g. seriously censored. therefore, the need to make the direction fabric extra appropriate led me to incorporate within the direction themes comparable to mod eling procedure lifetime distributions (Chaps. 1,2) and the utmost chance ideas for lifetime info processing (Chap. 3). A path within the conception of records is aprerequisite for those lectures. Stan dard classes frequently pay little or no recognition to the thoughts wanted for our function. a brief precis of them is given in Chap. three, together with known chance plotting. bankruptcy four describes the main worthwhile and well known versions of preventive major tenance and substitute. a few useful features of making use of those types are addressed, similar to treating uncertainty within the info, the position of information contamina tion and the opportunistic scheduling of upkeep activities.
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Extra resources for Reliability Theory: With Applications to Preventive Maintenance
Example text
50 CHAPTER 3. 1 The Maximum Likelihood Method Suppose we observe a sampie {tl, t2, ... (tja,ß). ) The standard maximum likelihood method of estimating the parameters works as follows. (i) Write the likelihood function Lik = Lik(a, ßj tl, ... (tij a, ß) . 1) i=l (ii) Find the values of a and ß (depending on the ohserved sampie values) which maximize the likelihood function. These values & = &(t1, t2, ... , t n ) and ß = ß(t1' t2, . , t n ) are called the maximum likelihood estimates (MLEs) of the parameters.
12) It has been proved in the above cited paper that for a location-scale family, the system of likelihood equations for this case has a unique solution if at least one ''yes'' and one "no" response have been observed. t, all Yi = y. r items have failed on [0, y] and n - r have survived past y. y(n-r) , so logLik = r 10g(1 - e->'Y) - 'xy(n - r). Hence _81_o8::-:g~_~_·k = rye->'Y /(1 - e->'Y) - y(n - r) = 0, which gives A ,X -1 n = Y log (n-r )" Grouped data Another situation often arising in lifetime testing and/or in follow-up studies is the following: the time axis is divided into nonoverlapping time intervals 11 = (to, tl], 12 = (tl, t2], .
18) will be determined by the first summand with the smallest k = r. 3. 21) states that for any t in a fixed interval [0, to], R(t) approaches exp( _artrg(8)) as a -+ O. 10) is the three-parameter Weibull distribution: P(r ~ t) = 1- exp(-(A(t - to))ß), t ~ to. 22) As in the lognormal case, to is called a threshold or a guaranteed time parameter. 21). 9 shows a network with 20 nodes and 30 edges called dodecahedron. The nodes are absolutely reliable. The edges faH independently and have lifetimes r ,....