具有工作休假及休假中止的$M/M/1$排队模型的主算子的点谱

Point Spectra of the Operator Corresponding to the $M/M/1$ Queueing Model with Working Vacation and Vacation Interruption

中文关键词:  $M/M/1$排队模型  工作休假及休假中止  $C_0$-半群  特征值  本质增长界

英文关键词:$M/M/1$ queueing model  working vacation and vacation interruption  $C_0$-semigroup  eigenvalue  essential spectral bound

中文摘要:

      本文考虑具有工作休假及休假中止的$M/M/1$排队模型的主算子的点谱. 证明该模型主算子在左半轴有不可数无穷多个特征值. 此结果描述了主算子的点谱. 然后证明该主算子生成的$C_0$-半群的本质增长界为0,由此推出该$C_0$-半群不是紧算子、它的本质谱半径等于1. 此外,这些结果蕴含该模型的时间依赖解不可能指数收敛于其稳态解.

英文摘要:

      In this paper, we consider point spectra of the operator corresponding to the $M/M/1$ queueing model with working vacation and vacation interruption. We prove that the underlying operator has uncountable eigenvalues on the left real line and these results describe the point spectra of the operator. Then, we show that the essential growth bound of the $C_0$-semigroup generated by the operator is 0 and therefore it is not quasi compact, the essential spectral bound of the $C_0$-semigroup is equal to 1. Moreover, our results imply it is impossible that the time-dependent solution of the model exponentially converges to its steady-state solution.