Transient Behavior of M|G|∞ Systems: Mean and Variance from the Start of Busy Periods with Applications to Epidemics and Unemployment

The M|G|∞ queue system transient probabilities, with time origin at the beginning of a busy period, are determined. It is highlighted that the obtained distribution mean and variance study as time functions. In this study, it is a determinant of 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.