Skip to main content
SpringerLink
Log in

Sharp bounds for NBUE distributions

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

Let F be a new better than used in expectation (NBUE) distribution function with mean μ. In a previous paper (Brown in Probab. Eng. Inf. Sci. 20:195–230, 2006), the author derived the following bound. For any t≥μ,

$$\overline{F}(t) = \mathit{Pr}(X \ge t) \le e^{-[{t\over\mu}-1]}. $$

The main result of this paper is to show that this bound is sharp. Other sharp bounds for NBUE distributions are also derived.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Barlow, R. E., & Proschan, F. (1975). Statistical theory of reliability and life testing. New York: Holt, Rinehart & Winston.

    Google Scholar 

  • Brown, M. (2006). Exploiting the waiting time paradox: applications of the size-biasing transformation. Probability in the Engineering and Informational Sciences, 20, 195–230.

    Article  Google Scholar 

  • Daley, D. J. (1988). Tight bounds on the exponential approximations of some aging distributions. The Annals of Probability, 16, 414–423.

    Article  Google Scholar 

  • Diaconis, P., & Miclo, L. (2007). On times to quasi-stationarity for birth and death processes. Journal of Theoretical Probability, 22(3), 558–586.

    Article  Google Scholar 

  • Fill, J. A. (2009a). The passage time distribution for a birth-and-death chain: strong stationary duality gives a first stochastic proof. Journal of Theoretical Probability, 22(3), 547–557.

    Google Scholar 

  • Fill, J. A. (2009b). On hitting times and fastest strong stationary times for skip-free and more general chains. Journal of Theoretical Probability, 22(3), 587–600.

    Article  Google Scholar 

  • Karlin, S., & McGregor, J. (1959). Coincidence properties of birth and death properties. Pacific Journal of Mathematics, 9, 1109–1140.

    Article  Google Scholar 

  • Karlin, S., & Taylor, N. M. (1975). A first course in stochastic processes (4th ed.). San Diego: Academic Press.

    Google Scholar 

  • Katehakis, M. N., & Derman, C. (1989). On the maintenance of systems composed of highly reliable components. Management Science, 35(5), 551–560.

    Article  Google Scholar 

  • Katehakis, M. N., & Derman, C. (1984). Optimal repair allocation in a series system. Mathematics of Operations Research, 9(4), 615–623.

    Article  Google Scholar 

  • Marshall, A. W., & Olkin, I. (2007). Life distributions. Structure of nonparametric, semiparametric, and parametric families. New York: Springer.

    Google Scholar 

  • Ross, S. M. (1979). Multivalued state component systems. The Annals of Probability, 7, 379–383.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, City College, The City University of New York, New York, NY, USA

    Mark Brown

Authors
  1. Mark Brown
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mark Brown.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brown, M. Sharp bounds for NBUE distributions. Ann Oper Res 208, 245–250 (2013). https://doi.org/10.1007/s10479-012-1151-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-012-1151-0

Keywords

Search

Navigation