Present paper deals a M/M/1:(∞;GD) queueing model with interdependent controllable arrival and service rates where- in customers arrive in the system according to poisson distribution with two different arrivals rate...Present paper deals a M/M/1:(∞;GD) queueing model with interdependent controllable arrival and service rates where- in customers arrive in the system according to poisson distribution with two different arrivals rates-slower and faster as per controllable arrival policy. Keeping in view the general trend of interdependent arrival and service processes, it is presumed that random variables of arrival and service processes follow a bivariate poisson distribution and the server provides his services under general discipline of service rule in an infinitely large waiting space. In this paper, our central attention is to explore the probability generating functions using Rouche’s theorem in both cases of slower and faster arrival rates of the queueing model taken into consideration;which may be helpful for mathematicians and researchers for establishing significant performance measures of the model. Moreover, for the purpose of high-lighting the application aspect of our investigated result, very recently Maurya [1] has derived successfully the expected busy periods of the server in both cases of slower and faster arrival rates, which have also been presented by the end of this paper.展开更多
本文以复变函数论中的 Rouche 定理为基础,给出了有关多项式根的分布规律。Rouche 定理:若 f(Z)与 g(Z)在封闭曲线 C 内及 C 上都解析,又在 C 上有|g(Z)|<f(Z),则 f(Z)与 f(Z)+g(Z)在 C 内的根的个数相等。根据代数基本定理,多项式(?...本文以复变函数论中的 Rouche 定理为基础,给出了有关多项式根的分布规律。Rouche 定理:若 f(Z)与 g(Z)在封闭曲线 C 内及 C 上都解析,又在 C 上有|g(Z)|<f(Z),则 f(Z)与 f(Z)+g(Z)在 C 内的根的个数相等。根据代数基本定理,多项式(?)(Z)=a_nZ^n+a_(n-1)Z^(n-1)+…+a_1Z+a_0(1)取封闭曲线 C 是一个半径为 R 的圆,即|Z|=R,其中 R>max{1,(|a_(n-1)|+|a_(n-2|+…+|a_1|+|a_0|)/|a_n|}令 f(Z)=a_nZ^n,g(Z)=a_(n-1)Z^(n-1)+a_(n-2))Z^(n-2)+…+a_1Z+a_0 由有关 R 的假设可得:|a_(n-1|+|a_(n-2|+…+|a_1|+|a_0|<|a_n|R 即(|a_(n-1)|+|a_(n-2)|+…+|a_1|+|a_0|)<|a_n|R^n由于 R>1及在 C 上|Z|=R,所以,|a_(n-1)Z^(n-1)+a_(n-2)Z^(n-1)+……+a_1Z+a_0|<|a_nZ^n|也就是说,|g(Z)|<|f(Z)|,因此 f(Z)与 f(Z)+g(Z)在 C 内(|Z|<R)的根的个数相同。展开更多
文摘Present paper deals a M/M/1:(∞;GD) queueing model with interdependent controllable arrival and service rates where- in customers arrive in the system according to poisson distribution with two different arrivals rates-slower and faster as per controllable arrival policy. Keeping in view the general trend of interdependent arrival and service processes, it is presumed that random variables of arrival and service processes follow a bivariate poisson distribution and the server provides his services under general discipline of service rule in an infinitely large waiting space. In this paper, our central attention is to explore the probability generating functions using Rouche’s theorem in both cases of slower and faster arrival rates of the queueing model taken into consideration;which may be helpful for mathematicians and researchers for establishing significant performance measures of the model. Moreover, for the purpose of high-lighting the application aspect of our investigated result, very recently Maurya [1] has derived successfully the expected busy periods of the server in both cases of slower and faster arrival rates, which have also been presented by the end of this paper.
文摘本文以复变函数论中的 Rouche 定理为基础,给出了有关多项式根的分布规律。Rouche 定理:若 f(Z)与 g(Z)在封闭曲线 C 内及 C 上都解析,又在 C 上有|g(Z)|<f(Z),则 f(Z)与 f(Z)+g(Z)在 C 内的根的个数相等。根据代数基本定理,多项式(?)(Z)=a_nZ^n+a_(n-1)Z^(n-1)+…+a_1Z+a_0(1)取封闭曲线 C 是一个半径为 R 的圆,即|Z|=R,其中 R>max{1,(|a_(n-1)|+|a_(n-2|+…+|a_1|+|a_0|)/|a_n|}令 f(Z)=a_nZ^n,g(Z)=a_(n-1)Z^(n-1)+a_(n-2))Z^(n-2)+…+a_1Z+a_0 由有关 R 的假设可得:|a_(n-1|+|a_(n-2|+…+|a_1|+|a_0|<|a_n|R 即(|a_(n-1)|+|a_(n-2)|+…+|a_1|+|a_0|)<|a_n|R^n由于 R>1及在 C 上|Z|=R,所以,|a_(n-1)Z^(n-1)+a_(n-2)Z^(n-1)+……+a_1Z+a_0|<|a_nZ^n|也就是说,|g(Z)|<|f(Z)|,因此 f(Z)与 f(Z)+g(Z)在 C 内(|Z|<R)的根的个数相同。
