A Unified Approach to the Proportional Relation for Discrete-time Single-server Queues
It is known that a simple relation called proportional relation holds for many 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 di#erent only in the bu#er capacity. Recently, the proportional relation has been verified for various discretetime single-server queues with correlated arrivals using the matrix analytic approach, where it has been shown that the proportional constant is also expressed in terms of the distribution in one 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 without using the matrix analysis. Keywords: Discrete-time single-server queues, proportional relation, stationary queue length distribution 1 Introduction It is well known that a simple relation holds between the stationary queue length distribution..