American Journal of Applied Mathematics 2018; 6(1): 8-14 http://www.sciencepublishinggroup.com/j/ajam doi: 10.11648/j.ajam.20180601.12 ISSN: 2330-0043 (Print); ISSN: 2330-006X (Online) Mathematical Model of Corrective Maintenance Based on Operability Checks for Safety Critical Systems Ahmed Raza Department of Electronics, National Aviation University, Kiev, Ukraine Email address: To cite this article: Ahmed Raza. Mathematical Model of Corrective Maintenance Based on Operability Checks for Safety Critical Systems. American Journal of Applied Mathematics. Vol. 6, No. 1, 2018, pp. 8-14. doi: 10.11648/j.ajam.20180601.12 Received: January 20, 2018; Accepted: February 1, 2018; Published: March 5, 2018 Abstract: Maintenance based on equipment operability checks is widely used for technical systems of various physical nature. For commercial and military aircraft such checks are carried-out after a certain amount of time according to specific maintenance programs. Therefore, great attention in the research literature is paid to the mathematical modeling of maintenance on the basis of equipment operability checks. In this study, a mathematical model of corrective maintenance with operability checks at discrete times for the safety critical systems is considered. The criterion of the corrective maintenance effectiveness is proposed to provide a given level of operational reliability with minimum maintenance costs. A finite time interval is considered for modeling the moments of the system operability checks. The graph of decision making is analyzed for imperfect operability checks and the probabilities of possible decisions are determined. Analytical equations for the operational reliability and expected maintenance costs are derived for an arbitrary distribution of time to failure. The criteria of determining optimal policies of sequential checks are formulated. Numerical examples illustrate the developed theory. For the first time it has been shown that conditional probabilities of correct and incorrect decisions when checking system operability are dependent on the time of failure and parameters of the degradation model. Numerical calculations have shown that in the case of mixing deteriorating systems with different initial time points of operation, the interval between operability checks converges to a constant periodicity. Keywords: Corrective Maintenance, Imperfect Checks, Operational Reliability, Expected Costs, Sequential Checks 1. Introduction At present, corrective maintenance based on operability checks is widely used to maintain the operational reliability of various technical systems. Evidence of this is a large number of publications on periodic and sequential plans of operability checks. Mathematical models of corrective maintenance based on operability checks can be conditionally divided into the two groups: models with perfect checks and models with imperfect checks. Models with perfect checks were considered in a large number of publications, for example, in [1-5]. In these studies, the problems of determining the optimal moments of checks are considered. The criterion of optimization is the minimum of expected maintenance costs, which includes the cost of checks, losses due to the unrevealed failure and cost of the system repair. The plans of checks can be sequential and periodic. Let us now turn to the analysis of maintenance models with imperfect checks. A typical inspection model with two imperfect inspection probabilities is analyzed in [6]. The system under testing may be judged as failed even if it is operable or the system may be incapable of detecting its failure due to imperfect inspection. The optimal policy that minimizes the total expected cost up to the detection of system failure is considered. An imperfect-inspection model in which failures can only be detected with probability p < 1 is considered in [7]. The exponential distribution of time to system failure is supposed. The asymptotic distribution of the test statistic is obtained under the null hypothesis as well as under the alternative. In [8], a maintenance model with periodic checks is examined to detect and eliminate unrevealed failures. Imperfect periodic checks are conducted with periodicity τ in the finite time interval [0, (n + 1) τ]. For any of the checks, the failure of the system is detected with probability p∈ (0, 1). After detecting failure, corrective repair is performed, which is equivalent to replacing the system with a new one. If there was no failure on the interval [0, (n + 1) τ] or it was not detected, then at the time (n + 1) τ the system is