Abstract
In this paper, we investigate well-posedness and asymptotic behavior of an M/G/1 retrial queueing system characterized by reserved idle time and setup time. This system is intricately described by an infinite system of integro-partial differential equations. Our methodology is rooted in the feedback theory of well-posed and regular infinite-dimensional linear systems. Firstly, we convert the system into a control framework, incorporating both input and output variables. Subsequently, we establish its well-posedness by leveraging the feedback theory specific to regular infinite-dimensional linear systems. Secondly, by scrutinizing the spectral distribution of the system operator along the imaginary axis, we demonstrate that the time-evolving solution of the system converges robustly to its steady-state solution. Lastly, we also discuss the essential properties of the semigroup induced by the system operator, encompassing stability and hyperbolicity, thereby shedding light on its regularity.
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