Students build an differential equation model using a convolution for machine replacement strategies for two different machine failure models: (1) exponentially distributed failure (student exercise) and (2) fixed time replacement. We discuss all the necessary probability notions which support the building of the model and solve it for (2) while asking students to solve a general situation and then apply their findings to address (1).
We seek the replacement function R(t) for the number of machines to be replaced in the time interval [0, t] to meet a set demand function, and N(t), for the number of machines in operation at time t. We are given probability information on F(t), the fraction of the machines that were new at time to which are still in operation at time to + t, for the two different scenarios (1) and (2) above.
We offer a separate Appendix with some introductory probability notions.
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