The instant system availability $$S_\tau (t)$$ S τ ( t ) of a repairable system with the renewal equation was studied. The starting point monotonicity of $$S_\tau (t)$$ S τ ( t ) was proved and the upper bound of $$S_\tau (t)$$ S τ ( t ) is also derived. It was found that the interval of instant system availability monotonically decreases. In addition, we provide examples that validate the analytically derived properties of $$S_\tau (t)$$ S τ ( t ) based on the Lognormal, Gamma and Weibull distributions and the results show that the value of T is slightly smaller than its value defined in Theorem 2. The procedure of using a bathtub as application for this article is also discussed.

This research aims to explore the use of the educational game "MARIO" in improving the cognitive capacity of young children, especially in number recognition. "MARIO" is an acronym for Marble, Number, Recreational, Innovative, and Observation, each element of which is integrated with the game to create a fun learning experience. This research involved 30 students aged 4-6 years in a kindergarten in Jakarta. Data collection is carried out through observation, interviews, and documentation, known as the OWI method. Data analysis uses the Miles and Huberman model which includes data reduction, data presentation, and conclusion. The results showed that the "MARIO" game significantly improved children's cognitive abilities in recognizing numbers. The three stages of number recognition include recognizing numbers through marbles (Marbel), matching numbers with number cards (Numbers), and movement activities involving numbers (Recreative). In the observation stage, children record the results of their play, which helps in understanding the concepts of comparison and measurement. Of the 30 students, 85% showed improvement in basic numeracy skills, with an average improvement of 30% in the basic math skills test. This research concludes that the integration of educational elements in the "MARIO" game is not only effective in improving children's cognitive abilities but also makes the learning process more interesting and enjoyable. Thus, "MARIO" can be recommended as an innovative learning aid for early childhood.

In this article we develop a model for determining the appropriate level of inspection sampling for any manufacturing process. The model is useful for manufacturers, who naturally are concerned with profits and therefore with minimizing the cost of production. The model design aims to reduce total manufacturing cost and has general applicability to various manufacturing operations. The model considers the interests of consumers, who wish to minimize the cost of production while simultaneously ensuring the final product is of high quality. The cost parameters for production, the acceptance test, and admissible strategy are applied in the model. The cost components are formulated along with the minimization of the expected cost, and we used the repairable systems to guarantee the maintenance and sustainability of the economic system. We also discuss the assumptions and their appropriateness, as well as the application of the model to burn-in of system components.