In this study, we investigate a single server architecture in which the customer service system is not aware of the length of the line that was sent to them. Thus, the system-controlling equations here become what are known as the vacation differential equations. Here, we argue specifically that this vacation is what is causing the oscillations in the performance measurements of the system. The symbolic structure of the differential equation of the non-Markovian queuing problem is introduced in this study. A procedure of maintenance work is included in this model in terms of the vacation stage to support this minimal non-interrupted service system. The findings offer a thorough analysis of the system that enables it to operate more profitably even if it is interrupted by any associated activities. The supplemental variable method solves the queuing problem caused by the aforementioned subsequent outcomes. Estimates are made for the queue size, server idle time, use, and probability generating factors for each operating method. Numerical analysis was performed on specific examples using mathematical software. This strategy is perfectly acceptable because it is regularly employed and makes use of a statistical demarcation method. The graphical representation of this perspective provides precise calculations of the apparent constraints.

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