A unified approach to the proportional relation for discrete-time single-server queues

It is well known that a simple relation called proportional relation holds for some queueing models, that is, the stationary queue length distribution of one system can be expressed as the product of a constant and the distribution of another system which is different only in the buffer capacity. Recently, the proportional relation has been verified for various discrete-time single-server queues with correlated arrivals, where it has been also shown that the proportional constant can be expressed in terms of the distribution of one system. This implies that the stationary queue length distribution of one system can be completely expressed in terms of the distribution of the other system. In this paper, we consider a generalized model of discrete-time single-server queue, which covers all previous ones, and give a simple and unified proof to the proportional relation as well as the expression of the proportional constant.