In this research, we advanced the optimal control theory for queuing systems that are characterized by integro-differential equations. Our primary goal was to identify an optimal service rate that minimizes a performance criterion, which is a composite of the system state at the final time and the cost associated with the optimal service rate. The optimal service rate was defined by an optimality system, and this formulation essentially translated the problem into a bilinear control problem within a nonreflexive Banach space, utilizing $L^1$-optimization techniques. We provided a rigorous proof of the existence of an optimal controller and offered a detailed characterization of the optimal control. Additionally, a comparison was made with traditional steady-state results to highlight the differences and improvements. Finally, numerical analysis was conducted on the theoretical results.

In this paper, we investigate the asymptotic behavior of the time-dependent solution for the M/G/1 stochastic clearing queueing system operating in a three-phase environment. The mathematical model of this system is characterized by an infinite set of integro-partial differential equations, with boundary conditions that incorporate integral equations. Initially, we employ probability generating functions to demonstrate that 0 is an eigenvalue of the system operator, possessing a geometric multiplicity of one. Subsequently, by invoking Greiner's boundary perturbation method, we establish that all points on the imaginary axis, with the exception of 0, reside within the resolvent set of the system operator. Furthermore, we highlight that 0 also serves as an eigenvalue of the adjoint operator of the system operator, with a geometric multiplicity of unity. As a result, we conclude that the time-dependent solution of the system converges strongly to its steady-state solution.

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.

In this paper, we studied the spectrum and semigroup properties of the M/G/1 queueing model with exceptional service time for the first customer in each busy period. First, we described the point spectrum of the system operator that corresponds to the model, and we prove that the system operator has an uncountable infinite number of eigenvalues on the left-half complex plane. Second, by using the spectrum analysis and semigroup theory, we obtained that the spectrum-determined growth (SDG) condition holds and the semigroup is not asymptotically stable, compact, eventually compact, or even quasi-compact. Finally, in order to clarify the results of spectral distribution, some numerical analysis were conducted.