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We consider an infinite server queueing system. An examination of sample path dynamics allows a straightforward development of integral equations having solutions that give time-dependent occupancy (number of customers) and backlog (unfinished work) distributions (conditioned on the time of the first arrival) for the GI/G/∞ queue. These integral equations are amenable to numerical evaluation and can be generalized to characterize GIX/G/∞ queue. Two examples are given to illustrate the results.

PurposeLiterature shows that the economics of early failures in maintenance and electric utilities have not been deeply analyzed. This study aims to focus on quantifying the economic impact that early failures in current transformers have on total maintenance costs. The empirical study is conducted in a regional transmission division of an electric utility located in Mexico.Design/methodology/approachThe utility's database was accessed to collect 219 maintenance records. Clustering techniques were used to identify early failures from a bimodal distribution of failures. Confirmatory goodness-of-fit procedures followed the analysis, and finally, direct and opportunity costs were estimated by adapting the cost-of-quality (PAF) Model.FindingsAround 11% of all maintenance activities are triggered by early failures, and they account for up to US$2.2m during the eight-year period under study, which represents 16% of total maintenance costs. Additionally, opportunity costs represent close to two-thirds of the total costs due to early failures. This was obtained after finding and validating a clear-cut border of 3.5 months between early failures and the rest.Originality/valueFailures in energy grids and power transmission can have a large economic impact on the power industry and the society in general. Thus, the maintenance function in equipment such as current transformers is a crucial entry of the budget of any electric utility. This study is one of the very few that highlights the magnitude and importance of direct and opportunity costs derived from early failures.