We describe the point spectrum of the operator which corresponds to the M/M/1 queueing model with vacations and multiple phases of operation. Then by using this result we prove that the essential growth bound of the C0-semigroup generated by the operator is 0, the C0-semigroup is not compact, not eventually compact, even not quasi-compact. Moreover, we verify that it is impossible that the time-dependent solution of the M/M/1 queueing model with vacations and multiple phases of operation exponentially converges to its steady-state solution. In addition, we obtain the spectral radius and essential spectral radius of the C0-semigroup. Lastly, we discuss other spectrum of the operator and obtain a set which belongs to the union of its continuous spectrum and residual spectrum.

中文翻译：

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M/M/1排队模型对应的算子谱，带假期多阶段运行应用

我们描述了对应于具有假期和多个操作阶段的 M/M/1 排队模型的算子的点谱。然后用这个结果证明算子生成的C0-半群的本质增长界为0，C0-半群不是紧致的，不是最终紧致的，甚至不是拟紧致的。此外，我们验证了具有假期和多个操作阶段的 M/M/1 排队模型的时间相关解不可能指数收敛到其稳态解。此外，我们还得到了C0-半群的谱半径和本质谱半径。最后讨论算子的其他谱，得到一个属于其连续谱和残差谱并集的集合。