Abstract
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.
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Abbreviations
- \( \{ N(t),t \ge 0\} \) :
-
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
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Acknowledgements
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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Appendix
Appendix
Proof of Proposition 1
Observe that
where, given the shock process, \( \tilde{Q}(t) \) becomes just constant and
The joint distribution of \( (T_{1} ,T_{2} , \ldots ,T_{N(t)},N(t); W_{1},W_{2} , \ldots ,W_{N(t)})\) is
Thus, \( Q_{{E}} (t) \) can be obtained as
where, in obtaining the third equality, the following property is used: for any integrable function \( \delta (x) \),
□
Proof of Proposition 2
Observe that \( \tilde{Q}(t) \) is a function of \( (T_{1} ,T_{2} , \ldots ,T_{N(t)} ,N(t)) \); therefore, to obtain \( Q_{{ES}} (t) \), we need the joint conditional distribution of \( (T_{1} ,T_{2} , \ldots ,T_{N(t)} ,N(t)|T_{l} > t) \). The conditional survival function given the shocks and random increment history is
Thus, the joint distribution of \( (T_{1} ,T_{2} , \ldots ,T_{N(t)} ,N(t);W_{1} ,W_{2} , \ldots ,W_{N(t)} ,T_{l} > t) \) is given by
The survival probability \( P(T_{l} > t) \) is obtained as
Then, the joint conditional distribution of \( (T_{1} ,T_{2} , \ldots ,T_{N(t)} ,N(t);W_{1} ,W_{2} , \ldots ,W_{N(t)} |T_{l} > t) \) is given by
and, thus, the joint conditional distribution of \( (T_{1} ,T_{2} , \ldots ,T_{N(t)} ,N(t)|T_{{l}} > t) \) is
Finally, using the joint conditional distribution of \( (T_{1} ,T_{2} , \ldots ,T_{N(t)} ,N(t)|T_{l} > t) \),
where, as in the previous proof, the property
was used.□
Proof of Proposition 4
In a renewal cycle, the expected cost due to preventive replacement and the replacement upon failure are given by \( C_{{r}} \bar{F}(T) + C_{{f}} F(T) \). Thus, now we derive the average gain in a cycle. For this, we have to derive the conditional average quality at time \( s \) given \( T_{l} > T \)(where \( s < T \)) or \( T_{{l}} = t \), \( t \le T \) (where \( s < t \)).
(Case I) \( T_{l} > T \).
Define \( Q_{\text{ES}} (s|T + ) \equiv E[\tilde{Q}(s)|T_{{l}} > T] \), \( s < T \). Observe that \( \tilde{Q}(s) \) is a function of \( (T_{1} ,T_{2} , \ldots ,T_{N(s)} ,N(s)) \), and to obtain \( Q_{\text{ES}} (s|T + ) \), we need the joint conditional distribution of \( (T_{1} ,T_{2} , \ldots ,T_{N(s)} ,N(s)|T_{{l}} > t) \). For convenience, define
Observe that the conditional probability of \( (T_{{l}} > t|H_{{{\text{s}},t}} ) \) is given by
and thus the conditional distribution of \( (H_{s,t} |T_{l} > t) \) is
Thus, the conditional joint distribution of \( (T_{1} = t_{1} ,T_{2} = t_{2} , \ldots ,T_{n(s)} = t_{n(s)} ,N(s) = n(s)|T_{{l}} > t) \) is
Thus,
Given \( T_{l} > T \), the conditional total average gain during the cycle is
(Case II) \( T_{l} \le T \).
For \( t \le T \), define \( Q_{\text{ES}} (s|t) \equiv E[\tilde{Q}(s)|T_{{l}} = t] \), \( s < t \). In this case, to obtain \( Q_{\text{ES}} (s|t) \), we need the joint conditional distribution of \( (T_{1} ,T_{2} , \ldots ,T_{N(s)} ,N(s)|T_{{l}} = t) \).
Observe that
where
When \( \Delta t \approx 0 \),
Thus,
and, accordingly, the conditional pdf of \( (T_{{l}} |H_{{{\text{s}}:t}} ) \) is given by
Thus, the joint distribution of \( (T_{{l}} ,H_{{{\text{s}}:t}} ) \) is
From this, the conditional distribution of \( (H_{{{\text{s}}:t}} |T_{{l}} ) \) can be obtained as
The joint conditional distribution of \( (T_{1} ,T_{2} , \ldots ,T_{N(s)} ,N(s)|T_{{l}} = t) \) is given by
Thus,
Thus, given \( T_{{l}} \le T \), the conditional total average gain during the cycle is obtained as
Combining Case I and Case II, the average gain during a renewal cycle is given by
The proof is completed.□
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Cha, J.H., Finkelstein, M. Optimal preventive maintenance for systems having a continuous output and operating in a random environment. TOP 27, 327–350 (2019). https://doi.org/10.1007/s11750-019-00508-2
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DOI: https://doi.org/10.1007/s11750-019-00508-2