Abstract
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.
Highlights
The probability generating function approach plays a vital role in the study of queueing problems as it is crucially useful in performance analysis of a wide range of queueing models
Present paper deals a M M 1 : ;GD queueing model with interdependent controllable arrival and service rates wherein customers arrive in the system according to poisson distribution with two different arrivals rates-slower and faster as per controllable arrival policy
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
Summary
The probability generating function approach plays a vital role in the study of queueing problems as it is crucially useful in performance analysis of a wide range of queueing models. It has been enthusiastically observed that most of the previous researchers [2,3,4,5,6,7] and references therein have presumed that the parameters of arrival and service rates in the queueing systems are independent to each other. [9] focused their attention to explore the M X M 1 interdependent queueing model with bulk arrivals and controllable arrival rates. In this sequential work, we consider here an interdependent M M 1 : ;GD queueing model incorporating bivariate Poisson process and controllable arrival rates in order to investigate the probability generating functions in faster and slower arrival rates
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