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

https://izlik.org/JA25DB36LA

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