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 that of
another system which is different only in the buffer capacity and a
constant.
Recently, the proportional relation has been verified for various
discrete-time single-server queues using the matrix analytic approach,
where it has been shown that the proportional constant is also
expressed by the distribution in one system.
In this paper, we consider a unified model of discrete-time
single-server queue which covers all previous ones and give a simple
proof to the proportional relation without using the matrix analysis.