Asymptotic behavior of a retrial queueing system with server breakdowns

In this paper, we study the asymptotic behavior of an M/G/1 retrial queueing system with server breakdowns, which is described by infinitely many partial integro-differential equations. Through investigating the spectrum of the system operator associated with the system on the imaginary axis, we show that the time-dependent solution of the system is strongly stable in the natural Banach state space. When the server failure rate is equal to zero, we show that the system admits a unique positive time-dependent solution and the solution is strongly convergent to its steady-state solution. In addition, when the service completion rate of server is a constant, the spectrum of the system operator lies on the left real axis. Finally, the corresponding C0-semigroup generated by the system operator is uniformly exponentially stable, irreducible, uniformly mean ergodic, quasi-compact but not compact and not eventually compact. (c) 2022 Elsevier Inc. All rights reserved.

Automated summary

In this paper, the asymptotic behavior of an M/G/1 retrial queueing system with server breakdowns is studied, and it is described by infinitely many partial integro-differential equations. The stability and convergence properties of the system are analyzed through the investigation of the system operator's spectrum. The results show the strong stability of the time-dependent solution in the natural Banach state space and the convergence of the solution to its steady-state solution when the server failure rate is zero. Additionally, the properties of the system operator and the corresponding C0-semigroup are examined.