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.
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Abbreviations
- HPP:
-
homogeneous Poisson process
- NHPP:
-
non-homogeneous Poisson process
- MSP:
-
mission success probability
- Cdf:
-
cumulative distribution function
- T :
-
mission completion time
- τ :
-
time of premature mission termination
- λ(t):
-
failure rate with respect to internal failures
- λ c (t):
-
failure rate for the combined model
- λ m f (t):
-
failure rate with respect to major internal failure
- p int(t):
-
probability that the internal failure is major
- q i n t (t):
-
probability that internal failure is minor
- S i n t (t):
-
survival function with respect to the major internal failure
- S s h (t):
-
survival function with respect to major failure caused by shocks
- S c (t):
-
survival function for the combined model
- v(t):
-
rate of the NHPP of shocks
- r m (t):
-
rate of the NHPP of minimal repairs in the combined model
- R m (t):
-
cumulative rate of the NHPP of minimal repairs in the combined model
- p s h (t):
-
probability that a shock results in a major failure
- q s h (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
- C R :
-
reward for completing a mission
- c p :
-
per time unit profit for the failure-free performance (cost of product supplied in time unit)
- c o :
-
per time unit operational cost
- c m :
-
cost of a single minimal repair
- C f :
-
penalty associated with system failure
- C t e r :
-
penalty associated with premature mission termination
- A(τ):
-
profit comparison function
- R τ :
-
number of minimal repairs performed by time τ from the mission beginning
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Finkelstein, M., Levitin, G. Optimal Mission Duration for Partially Repairable Systems Operating in a Random Environment. Methodol Comput Appl Probab 20, 505–516 (2018). https://doi.org/10.1007/s11009-017-9571-6
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DOI: https://doi.org/10.1007/s11009-017-9571-6