The MGInf queue system transient probabilities, with time origin at the beginning of a busy period, are determined. It is highlighted the obtained distribution mean and variance study as time functions. In this study, it is determinant the service time hazard rate function and two induced differential equations. The study model has potential applications in various real-world scenarios, such as epidemics and unemployment. The model's application in these areas offers insights into potential mitigating measures, including vaccination and medical care in epidemics, and government support and training in unemployment. Also, we will show how the results obtained can be applied in modeling epidemic and unemployment situations.

This work is partially financed by national funds through FCT - Fundação para a Ciência e Tecnologia, I.P., under the project FCT UIDB/04466/2020. Furthermore, the author thanks ISCTE-Instituto Universitário de Lisboa and ISTAR-IUL, for their support.

• Matemáticas - Ciências Naturais
• Ciências da Computação e da Informação - Ciências Naturais