Exact PMF Estimation Of System Indices In Boundary Crossing Problem

Authors : M ŞAHİNOĞLU

Pages : 0-0

Doi:10.1501/Commua1_0000000544

View : 15 | Download : 7

Publication Date : 1987-01-01

Article Type : Research Paper

Abstract :The objective of this research is to derive an exact probability mass function for the Ber- noulli random variables indicating the guality of performance reliability at any instant of time in a discrete-time two-state maintainable-repairable physical system that fluctuates between operating and defective States; that is, above and below a fictitious level where zero level is the boundary between good and bad States in a Markovian or Nonmarkovian chain. Note that since these Bernoulli random variables do not necessarily possess identical success probabilities, we speak of nonstationary processes. Given any discrete instant of time n where the total length of study is N time units insert ignore into journalissuearticles values(hour. day, year);, the discrete randonı variables of interest are defined as zero level insert ignore into journalissuearticles values(0^^ + or Oj^-); and level-crossing = 1 otZ^ = O);. Hence provided that the probabilities of transitions from po- sitive to negative insert ignore into journalissuearticles values(P^^); and from positive to positive insert ignore into journalissuearticles values(l-P^); from negative to positive insert ignore into journalissuearticles values(P^F); and from negative to negative insert ignore into journalissuearticles values(l~Pj^B); are specified earlier by sampling system data or simply given; the analyst can estimate the probability of being at negative State P insert ignore into journalissuearticles values(O^^-); = 1-P insert ignore into journalissuearticles values(O^+); and the probability of becoming defective P insert ignore into journalissuearticles values(Z^^ = 1); — 1-P O); given the system was operative at time increment n-1. Further an analytical expression is derived for the two random variables of interest for general n time increments with examples for up to n = 5. A simple digital Fortran written Com­ puter program evaluates the probabilities at a given n for both random variables, hence comple- tely specifying the probability mass function as illustrated in the tree-diagram. A system example is given to show the applicability of the algorithm vitlı reference to both types of system, namely special and general case.
Keywords : Exact PMF, Bernoulli Process, Markov Chain