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

  • Maxim Finkelstein
  • Gregory Levitin


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.


Premature mission termination External shocks Expected profit Minimal repair Optimization 



homogeneous Poisson process


non-homogeneous Poisson process


mission success probability


cumulative distribution function

Acronyms Notation


mission completion time


time of premature mission termination


failure rate with respect to internal failures


failure rate for the combined model


failure rate with respect to major internal failure


probability that the internal failure is major


probability that internal failure is minor


survival function with respect to the major internal failure


survival function with respect to major failure caused by shocks


survival function for the combined model


rate of the NHPP of shocks


rate of the NHPP of minimal repairs in the combined model


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


probability that a shock results in a major failure


probability that a shock results in a minor failure

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

probability that a shock is harmless


profit associated with completion of a mission


reward for completing a mission


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


per time unit operational cost


cost of a single minimal repair


penalty associated with system failure


penalty associated with premature mission termination


profit comparison function


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

Mathematics Subject Classification (2010)



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


  1. Aven T, Jensen U (1999) Stochastic models in reliability. Springer-Verlag, New YorkCrossRefzbMATHGoogle Scholar
  2. Block HW, Borges W, Savits TH (1985) Age-dependent minimal repair. J Appl Probab 22:370–386MathSciNetCrossRefzbMATHGoogle Scholar
  3. Boland PJ (1982). Periodic replacement when minimal repair costs vary with time, Nav Res Logist, 29: 541–546Google Scholar
  4. Brown M, Proschan F (1983) Imperfect repair. J Appl Probab 20(4):851–859MathSciNetCrossRefzbMATHGoogle Scholar
  5. Caballé NC, Castro IT, Pérez CJ, Lanza-Gutiérrez JM (2015) A condition-based maintenance of a dependent degradation-threshold-shock model in a system with multiple degradation processes. Reliab Eng Syst Saf 134:98–109CrossRefGoogle Scholar
  6. Cha JH, Finkelstein M (2011) On new classes of extreme shock models and some generalizations. J Appl Probab 48:258–270MathSciNetCrossRefzbMATHGoogle Scholar
  7. Finkelstein M (2008) Failure rate modelling for reliability and risk. Springer, LondonzbMATHGoogle Scholar
  8. Finkelstein M (2007) Shocks in homogeneous and heterogeneous populations. Reliab Eng Syst Saf 92:569–575CrossRefGoogle Scholar
  9. Finkelstein M, Cha JH (2013) Stochastic modelling for reliability. Shocks, Burn-in and Heterogeneous Populations. Springer, LondonzbMATHGoogle Scholar
  10. Finkelstein M, Marais F (2010) On Terminating Poisson processes in some shock models. Reliab Eng Syst Saf 95:874–879CrossRefGoogle Scholar
  11. Jiang L, Feng Q, Coit D (2015) Modeling zoned shock effects on stochastic degradation in dependent failure processes. IIE Trans 47:460–470CrossRefGoogle Scholar
  12. Kenzin M, Frostig E (2009) M out of n inspected systems subject to shocks in random environment. Reliab Eng Syst Saf 94:1322–1330CrossRefzbMATHGoogle Scholar
  13. Levitin G, Xing L, Dai Y (2017) Mission abort policy in heterogeneous non-repairable 1-out-of-N warm standby systems, to appear in IEEE Transactions on ReliabilityGoogle Scholar
  14. Montoro-Cazorla D, Pérez-Ocón R, del Carmen Segovia M (2009) Replacement policy in a system under shocks following a Markovian arrival process. Reliab Eng Syst Saf 94:497–502CrossRefGoogle Scholar
  15. Myers A (2009) Probability of loss assessment of critical k-Out-of-n: G systems having a mission abort policy. IEEE Trans Reliab 58(4):694–701CrossRefGoogle Scholar
  16. Noorossana R, Sabri-Laghaie K (2015) Reliability and maintenance models for a dependent competing-risk system with multiple time-scales. Proc Institution Mech Eng Part O-J Risk Reliab 229:131–142Google Scholar
  17. Ruiz-Castro JE (2014) Preventive maintenance of a multi-state device subject to internal failure and damage due to external shocks. IEEE Trans Reliab 63:646–660CrossRefGoogle Scholar
  18. Song S, Coit D, Feng Q (2014) Reliability for systems of degrading components with distinct component shock sets. Reliab Eng Syst Saf 132:115–124CrossRefGoogle Scholar
  19. van der Weide JAM, Pandey MD (2011) Stochastic analysis of shock process and modeling of condition-based maintenance. Reliab Eng Syst Saf 96:619–626Google Scholar

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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