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高校应用数学学报
Applied Mathematics A Journal of Chinese Universities  2016, Vol. 31 Issue (3): 281-293    DOI:
    
Asymptotic property of the time-dependent solution of the repairable closed queueing model with server of Erlangian distributed life time
Alim Mijit
School of Distance Education, Xinjiang Radio & TV Univ., Urumqi 830049, China
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Abstract  By using the $C_0$-semigroup theory, this paper studies the repairable closed queueing system with server of Erlangian distributed life time. First, by using the Hille-Yosida theorem, Phillips theorem and Fattorini theorem in functional analysis, the existence and uniqueness of nonnegative time dependent solution of system model has been proved. Next, the spectral properties of the operator corresponding to system model are investigated, which show that all points on the imaginary axis except zero belong to the resolvent set of the operator and zero is an eigenvalue of the operator and its adjoint operator with geometric multiplicity one. Thus, the above results give that the time-dependent solution of the system model converges strongly to its steady state solution.

Key words： $C_0$-semigroup      dispersive operator      eigenvalue      resolvent set      geometric multiplicity     
Received: 17 March 2016      Published: 16 May 2018
CLC:  O177.92  
  O177.7  
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Alim Mijit. Asymptotic property of the time-dependent solution of the repairable closed queueing model with server of Erlangian distributed life time. Applied Mathematics A Journal of Chinese Universities, 2016, 31(3): 281-293.

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http://www.zjujournals.com/amjcua/     OR     http://www.zjujournals.com/amjcua/Y2016/V31/I3/281


寿命为爱尔兰分布的可修闭路排队模型时间依赖解的渐近性质

利用$C_0$-半群理论研究寿命为爱尔兰分布的可修闭路排队系统. 首先利用泛函分析中的Hille-Yosida定理, Phillips定理和Fattorini定理证明此排队系统模型正时间依赖解的存在唯一性. 然后通过研究该模型相应主算子的谱的特征, 分别得到虚轴上除了0外其他所有点都属于该模型主算子的豫解集与0是其主算子及其共轭算子的几何重数为1的特征值. 最后将上述结果结合在一起推出该模型的时间依赖解强收敛于其稳态解.

关键词： $C_0$-半群,  dispersive算子,  特征值,  豫解集,  几何重数 
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