Exact PMF Estimation Of System Indices In Boundary Crossing Problem

Year 1987, Volume: 36 Issue: 02, 0 - 0, 01.01.1987

M. Şahinoğlu

https://doi.org/10.1501/Commua1_0000000544

Cited By: 2

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 (hour. day, year), the discrete randonı variables of interest are defined as zero level (0^^ + or Oj^-) and level-crossing = 1 otZ^ = O). Hence provided that the probabilities of transitions from po- sitive to negative (P^^) and from positive to positive (l-P^) from negative to positive (P^F) and from negative to negative (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 (O^^-) = 1-P (O^+) and the probability of becoming defective P (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

References

• Ankara Üniversitesi – Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics Dergisi