Analysis of a k/n(G) Retrial System with Multiple Working Vacations

In this paper, a k/n(G) retrial system with multiple working vacations is considered, and a mathematical model of the system is established by supplementary variable method, and a dynamic analysis of the system is carried out. Firstly, the model is transformed into an abstract Cauchy problem in Banach space by introducing the state space, main operator and its definition domain. Secondly, the C0-semigroup theory in functional analysis and the spectral theory of linear operators are used to prove that the main operator of the model generates a positive contraction C0-semigroup, which leads to the existence of a unique, non-negative time-dependent solution of the system that satisfies the probabilistic properties. Finally, Greiner’s boundary perturbation idea and the spectral properties of the corresponding operators are used to show that the time-dependent solution strongly converges to its steady-state solution.