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Optimal preventive maintenance for systems having a continuous output and operating in a random environment

  • Ji Hwan ChaEmail author
  • Maxim Finkelstein
Original Paper


We consider systems that are operating in a random environment modeled by an external shock process. Performance of a system is characterized by a quality (output) function that is decreasing (due to degradation) in the absence of shocks. Shocks have a double impact, i.e., they affect the failure rate of a system directly and at the same time, and each shock contributes to the additional decrease in the quality function. The unconditional and conditional (on survival) expectations for the corresponding stochastic quality process are obtained. The system is replaced either on failure or on the predetermined replacement time, whichever comes first. The corresponding optimization problem is considered and illustrated by detailed numerical examples.


Random environment Shocks Shot-noise process Output function Optimal maintenance 

List of symbols

\( \{ N(t),t \ge 0\} \)

The nonhomogeneous Poisson process (NHPP) of shocks

\( \lambda (t) \)

The rate (intensity function) of \( \{ N(t),t \ge 0\} \)

\( 0 \le T_{1} \le T_{2} \le \cdots \)

The sequential arrival times of shocks in \( \{ N(t),t \ge 0\} \)

\( T_{{l}} \)

The lifetime of the system

\( r_{0} (t) \)

The baseline failure rate of the system

\( \{ W_{i} ,i \ge 1\} \)

The sequence of the non-negative i.i.d. random variables

\( F_{W} (t) \), \( f_{W} (t) \), \( M_{W} (t) \)

The common Cdf, pdf and mgf of \( \{ W_{i} ,i \ge 1\} \)

\( Q(t) \)

Deterministic quality of performance function of a system operating without shocks

\( \tilde{Q}(t) \)

The quality at time \( t \) under a shock process

\( I(t) \)

The indicator of the system state (1 if the system is functioning at time t and 0 if it is in the state of failure)

\( Q_{{E}} (t) \)

The expectation of a quality function of a system at time t

\( Q_{{ES}} (t) \)

The conditional expectation of a quality function of a system at time t

\( \lambda_{{S}} (t) \)

The system failure rate function

\( C_{{f}} \)

The cost of the failure

\( C_{{r}} \)

The cost of renewal/replacement (\( C_{{f}} > C_{{r}} \))

\( \kappa \)

The proportionality constant for the reward

\( C(T) \)

The long-run mean cost rate function

Mathematical Subject Classification

90B25 60K10 



The authors sincerely thank the referees for helpful comments and advices. The work of the first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1A2B2014211). The work of the second author was supported by the National Research Foundation (SA) (Grant no: 103613).


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Copyright information

© Sociedad de Estadística e Investigación Operativa 2019

Authors and Affiliations

  1. 1.Department of StatisticsEwha Womans UniversitySeoulRepublic of Korea
  2. 2.Department of Mathematical StatisticsUniversity of the Free State 339BloemfonteinSouth Africa
  3. 3.ITMO UniversitySt. PetersburgRussia

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