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This paper studies an M/G/1 repairable queueing system with multiple vacations and N-policy, in which the service station is subject to occasional random breakdowns. When the service station breaks down, it is repaired by a repair facility. Moreover, the repair facility may fail during the repair period of the service station. The failed repair facility resumes repair after completion of its replacement. Under these assumptions, applying a simple method, the probability that the service station is broken, the rate of occurrence of breakdowns of the service station, the probability that the repair facility is being replaced and the rate of occurrence of failures of the repair facility along with other performance measures are obtained. Following the construction of the long-run expected cost function per unit time, the direct search method is implemented for determining the optimum threshold N* that minimises the cost function.

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This paper investigates an EPQ model with the increasing demand and demand dependent production rate involving the trade credit financing policy, which is seldom reported in the literatures. The model considers the manufacturer was offered by the supplier a delayed payment time. It is assumed that the demand is a linear increasing function of the time and the production rate is proportional to the demand. That is, the production rate is also a linear function of time. This study attempts to offer a best policy for the replenishment cycle and the order quantity for the manufacturer to maximum its profit per cycle. First, the inventory model is developed under the above situation. Second, some useful theoretical results have been derived to characterize the optimal solutions for the inventory system. The Algorithm is proposed to obtain the optimal solutions of the manufacturer. Finally, the numerical examples are carried out to illustrate the theorems, and the sensitivity analysis of the optimal solutions with respect to the parameters of the inventory system is performed. Some important management insights are obtained based on the analysis.