排队模型与可靠性模型的时间依赖解的结构研究

项目名称： 排队模型与可靠性模型的时间依赖解的结构研究

项目编号： No.10861011

项目类型： 地区科学基金项目

立项/批准年度： 2009

项目学科： 金属学与金属工艺

项目作者： 艾尼.吾甫尔

作者单位： 新疆大学

项目金额： 11万元

中文摘要： 运用 Hille-Yosida 定理,Phillips 定理与Fattorini定理证明了几个排队模型概率瞬态解的存在唯一性。运用 Greiner 的边界扰动思想证明了几个排队模型的时间依赖解强收敛于其稳态解。当服务率为常数时，证明了几个排队模型的主算子在左半复平面中有不可数无穷多个特征值。由此说明了排队模型的主算子生成的算子半群不是紧算子，甚至不是最终紧算子，排队模型的时间依赖解不可能指数收敛于其稳态解。 证明了由有限多个偏微分积分方程描述的可靠性模型的主算子生成的算子半群是拟紧算子，它们的时间依赖解指数收敛于它们的稳态解。运用 Greiner 的边界扰动思想证明了由无穷多个偏微分积分方程描述的可靠性模型的时间依赖解强收敛于其稳态解。当修复率为常数时，证明了由无穷多个偏微分积分方程描述的可靠性模型的主算子在左半复平面中有不可数无穷多个特征值。由此说明了此类可靠性模型的主算子生成的算子半群不是紧算子，甚至不是最终紧算子，时间依赖解不可能指数收敛于其稳态解。从而，指出了由有限多个偏微分积分方程描述的可靠性模型与由无穷多个偏微分积分方程描述的可靠性模型和排队模型的本质区别。

中文关键词： 排队模型;可靠性模型;时间依赖解;算子半群;谱

英文摘要： By using the Hille-Yosida theorem, the Phillips theorem and the Fattorini theorem we have proved that several queueing models have unique time-dependent solutions which satisfy the proability condition. By using Greiner's idea to perturb boundary condition we have proved that the time-dependent solutions of several queueing models strongly converge their steady-state solutions. When the service rates are constants, we have proved that the underlying operators corresponding to several queueing models have uncountably infinitly many eigenvalues in the left half complex plane. This results show that the semigroups generated by the underlying operators are not compact, even not eventually compact. Moreover, we conlcude: it is impossible that their time-dependent solutions exponentially converge to their steady-state solutions. We have proved that the semigroups generated by the underlying operators corresponding to several reliability models which were described by finitely many partial differential equations are quasi-compact operators and the time-dependent solutions of the models exponentially converge to their steady-state solutions. By using Greiner's idea to perturb boundary condition we have proved that the time-dependent solution of a reliability model which was described by infinitely many partial differential equations strongly converge to its steady-state solution. When the repair rate is a contant, we have proved that the underlying operator corresponding to the reliability model has uncountably infinitely many eigenvalues in the left half complex plane. Thus, we showed that the semigroup generated by the underlying operator which corresponds to the model is not compact, even not eventually compact and it is impossible that the time-dependent solution of the model exponentially converges to its steady-state solution. And we showed that the essential diffirence between the reliability models described by finitely many partial differential equations and the queueing models and relaibility models which were described by infinitely many partial differential equations.

英文关键词： Queueing Models;reliability models; time-dependent solution;semigroup of linear operators; spectrum