Optimal replacement policies for systems under sporadic shocks and healing impetus

ABSTRACT

A system is experiencing two kinds of sporadic impacts: valid shocks that cause damage instantaneously, and positive interventions that induce partial healing. Whereas each shock inflicts a fixed magnitude of damage, the accumulated effect of $k$ positive interventions nullify the damaging effect of one shock. The system is said to be in stage 1, when it can possibly heal, until the net count of impacts (valid shocks registered minus valid shocks nullified) reaches a threshold $m_1$. Thereafter, the system enters stage 2, when no more healing is possible. The system fails when the net count of valid shocks reaches another threshold $m_2(>m_1)$. The inter-arrival times between successive valid shocks and those between successive positive interventions are independent and follow arbitrary distributions; thereby we remove the restrictive assumption of exponential distributions often found in the literature. We find the distributions of the sojourn time in stage 1 and the failure time of the system. Finally, we find the optimal values of choice variables that minimize the expected maintenance cost per unit time for three different maintenance policies.

KEYWORDS:

• Reliability
• counting process
• adjusted convolution
• maintenance cost
• cycle time

AMSSubject Classifications:

• 90B25
• 62N05
• 60K10

Acknowledgments

We thank our colleagues for some discussions and feedback. We are indebted to two anonymous reviewers whose comments helped us improve the clarity and readability of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Notes on contributors

Debolina Chatterjee

Debolina Chatterjee is an advanced Doctoral Student in the Department of Mathematical Sciences at Indiana University-Purdue University, Indianapolis (IUPUI). Her research interests include Reliability theory and Stochastic Processes and Statistics in general. She enjoys teaching and reading books.

Jyotirmoy Sarkar

Jyotirmoy Sarkar is a Professor working at Indiana University-Purdue University Indianapolis. His research areas include enumeration, probability, statistics, and reliability theory. He enjoys reading, 'riting, 'rithmetic and R-coding.