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Optimal Mission Duration for Partially Repairable Systems Operating in a Random Environment

  • Maxim Finkelstein
  • Gregory Levitin
Article

Abstract

As a system failure during a mission can result in considerable penalties, at some instances it is more cost-effective to terminate operation of a system than to attempt to complete its mission. This paper analyzes the optimal mission duration for systems that operate in a random environment modeled by a Poisson shock process and can be minimally repaired during a mission. Two independent sources of failures are considered and for both cases, the failures are classified as minor or terminal in accordance with the Brown-Proschan model. Under certain assumptions, an optimal time of mission termination is obtained. It is shown that, if for some reason a termination is not technically possible at this optimal time, the mission should be terminated within a specific time interval and, if this is not possible, it should not be terminated beyond this interval. Illustrative examples are presented. The influence of mission and system parameters on the mission termination interval is demonstrated.

Keywords

Premature mission termination External shocks Expected profit Minimal repair Optimization 

Acronyms

HPP

homogeneous Poisson process

NHPP

non-homogeneous Poisson process

MSP

mission success probability

Cdf

cumulative distribution function

Acronyms Notation

T

mission completion time

τ

time of premature mission termination

λ(t)

failure rate with respect to internal failures

λc(t)

failure rate for the combined model

λmf(t)

failure rate with respect to major internal failure

pint(t)

probability that the internal failure is major

qint(t)

probability that internal failure is minor

Sint(t)

survival function with respect to the major internal failure

Ssh(t)

survival function with respect to major failure caused by shocks

Sc(t)

survival function for the combined model

v(t)

rate of the NHPP of shocks

rm(t)

rate of the NHPP of minimal repairs in the combined model

Rm(t)

cumulative rate of the NHPP of minimal repairs in the combined model

psh(t)

probability that a shock results in a major failure

qsh(t)

probability that a shock results in a minor failure

\(q_{sh}^{0} (t)\)

probability that a shock is harmless

C(T)

profit associated with completion of a mission

CR

reward for completing a mission

cp

per time unit profit for the failure-free performance (cost of product supplied in time unit)

co

per time unit operational cost

cm

cost of a single minimal repair

Cf

penalty associated with system failure

Cter

penalty associated with premature mission termination

A(τ)

profit comparison function

Rτ

number of minimal repairs performed by time τ from the mission beginning

Mathematics Subject Classification (2010)

90B25 

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Notes

Acknowledgments

The authors want to thank the Editor and the referees for helpful comments and constructive suggestions.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.University of the Free StateBloemfonteinSouth Africa
  2. 2.ITMO UniversitySt. PetersburgRussia
  3. 3.The Israel Electric CorporationHaifaIsrael

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