On the expected busy beriod of an interdependent M/M/1:(∞;Gd) queueing model using bivariate Poisson process and controllable arrival rates

Abstract

Busy period analysis plays a vital role in the study of queueing problems for forecasting the behaviour of the queueing systems. Performance analysis including determination of the average length of busy period is entirely based on the busy period distribution of the queueing model. In the queueing literature, it has been observed that most of the previous researchers [2,5,7,8 & 9] have presumed that the parameters of arrival and service rates in the system are independent to each other. However, it is not so in general because we find many queueing situations in our real life where the arrival and service rates are correlated with an elevated extent. A large number of previous noteworthy researchers [2,5,7,8& 9] and references therein confined their study to analyze a variety of queueing models taking into account that the arrival and service processes are independent. A very few number of worth mentioning researchers [11 & 13] of the current decade are found in the literature of queueing theory who have contributed their devotion to analyze an M/M/1 queueing model with consideration of interdependent arrival and service rates. In this context, Srinivasa Rao et al [13] have focused their attention to investigate the average system size and average waiting time of an M/M/1/∞ interdependent queueing model using controllable arrival rates under steady state conditions. Recently, Pal [11] considered the same queueing model with a version of its limited waiting space and he succeeded to investigate the cost per unit time of a served customer in the system. In our current study, we consider the same queueing model already analyzed by Srinivasa Rao et al [13] with a version that the arrival and service processes follow a bivariate Poisson distribution and our keen interest is to investigate the average length of busy periods in two different cases of slower and faster arrival rates. By the end of the present paper, a special stress on practical aspect of the investigated average busy period has also been given in our conclusion which reveals the application of both the realistic modeling enhancing as well as regulatory techniques.