Dynamic analysis of the M/G/1 queueing system with multiple phases of operation

In this paper, we considered the M/G/1 queueing system with multiple phases of operation. First, we have proven the existence and uniqueness of the time-evolving solution for this queueing system. Second, by calculating the spectral distribution of the system operator, we proved that the solution converged at most strongly to its steady-state (static) solution. We also discussed the compactness of the system's corresponding semigroup. Additionally, we investigated the asymptotic behavior of dynamic indicators. Finally, to demonstrate the exponential convergence of the solution, we conducted some numerical analysis.