A continuous approximation for batch arrival queues with threshold

In this paper, the system size of batch arrival queues with threshold is approximated by a diffusion process and the accuracy is evaluated. Firstly, the busy period process is approximated by a diffusion process with an instantaneous return boundary. Secondly, the idle period process is approximated by applying the renewal theory. Then we combine the two approximations to derive the system size distribution at an arbitrary time point. The accuracy of the approximation is evaluated for a number of interarrival and service time distributions.