By Alessandro Birolini
Stochastic approaches are strong instruments for the research of reliability and availability of repairable gear and structures. due to the concerned versions, and on the way to be mathematically tractable, those approaches are usually restrained to the category of regenerative stochastic techniques with a finite kingdom area, to which belong: renewal procedures, Markov strategies, semi-Markov approaches, and extra basic regenerative tactics with just one (or a number of) regeneration staters). the article of this monograph is to study those approaches and to exploit them in fixing a few reliability difficulties encountered in useful purposes. Emphasis is given to a entire exposition of the analytical approaches, to the restrictions in volved, and to the unification and extension of. the types recognized within the literature. The types investigated the following imagine. that platforms have just one fix team and that no extra failure can happen at method down. fix and failure premiums are common ized step by step, as much as the case during which the concerned strategy is regenerative with just one (or a number of) regeneration state(s). Investigations take care of other forms of reliabilities and availabilities for series/parallel constructions. Preventive major tenance and imperfect switching are thought of in a few examples.
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Additional info for On the Use of Stochastic Processes in Modeling Reliability Problems
Example text
N t;;(t ) =Zi n t;;(t 1) =Zi 1 n ••• J 1. • >t 1 , and for any i,j,i 1 ,i 2 , ••• ,i n (i,j,i 1 ,i 2 , ... ,i n =O,1, ••. ,m). The conditional probabilities given by equation (75) are the transition probabilities Pij(t,t+a) of the Markov process p .. }. J J In the following it is assumed that 1. (76) 27 O";P .. (t,t+a)"; 1 ~J and m L P .. (t,t+a) = 1 j=O ~J hold for all t,t+a (stochastic matrix). Together with the initial conditions i::; 0, 1, ... ,m p. }, 1 1 (77) the transition probabilities Pij(t,t+a) completely determine the stochastic behaviour of the Markov process.
The conditional probabilities given by equation (75) are the transition probabilities Pij(t,t+a) of the Markov process p .. }. J J In the following it is assumed that 1. (76) 27 O";P .. (t,t+a)"; 1 ~J and m L P .. (t,t+a) = 1 j=O ~J hold for all t,t+a (stochastic matrix). Together with the initial conditions i::; 0, 1, ... ,m p. }, 1 1 (77) the transition probabilities Pij(t,t+a) completely determine the stochastic behaviour of the Markov process. For instance, the state probabilities p. }, i = O,l, ...
Using for approximation an Erlang distribution function, the process is semi-Markovian. As an example, let us consider the case of a two-element series structure and assume that the repair times are arbitrary, with densities g10(x) and g20(X) , and that the failure-free times have densities (195) (196) Equation (195) is the density of the sum of two exponentially distributed random -A x time intervals with density A01 e 01 . Under these assumptions, the two-element series structure corresponds to a 1-out~of-2 standby redundancy, with constant failure rate A01 ' in series with an element with constant failure rate A02 .