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4395 results in Advances in Applied Probability

Page 180 of 220

  • Some properties of the crossings process generated by a stationary χ2 process

  • Ken Sharpe
  • Journal:
    Advances in Applied Probability / Volume 10 / Issue 2 / June 1978
    Published online by Cambridge University Press:
    01 July 2016, pp. 373-391
    Print publication:
    June 1978

  • The process generated by the crossings of a fixed level, u, by the process Pn(t) is considered, where

    and the Xi(t) are identical, independent, separable, stationary, zero mean, Gaussian processes. A simple formula is obtained for the expected number of upcrossings in a given time interval, sufficient conditions are given for the upcrossings process to tend to a Poisson process as u→∞, and it is shown that under suitable scaling the distribution of the length of an excursion of Pn(t) above u tends to a Rayleigh distribution as u→ ∞.

  • Optimal replacement under a general failure model

  • Bo Bergman
  • Journal:
    Advances in Applied Probability / Volume 10 / Issue 2 / June 1978
    Published online by Cambridge University Press:
    01 July 2016, pp. 431-451
    Print publication:
    June 1978

  • Replacement policies based on measurements of some increasing state variable, e.g. wear, accumulated damage or accumulated stress, are studied in this paper. It is assumed that the state measurements may be regarded as realizations of some stochastic process and that the proneness to failure of an active unit may be described by an increasing state-dependent failure rate function. Average long-run cost per unit time is considered. The optimal replacement rule is shown to be a control limit rule, i.e. it is optimal to replace either at failure or when the state variable has reached some threshold value, whichever occurs first. The optimal rule is determined. Some generalizations and special cases are given.

Page 180 of 220