Upper bound for the decay rate of the joint queue-length distribution in a two-node Markovian queueing system

This paper studies the geometric decay property of the joint queue-length distribution {p(n1, n2)} of a two-node Markovian queueing system in the steady state. For arbitrarily given positive integers c1, c2, d1 and d2, an upper bound η(c1, c2) of the decay rate is derived in the sense exp © lim supn→∞ n −1 log p(c1n+ d1, c2n+ d2) a ≤ η(c1, c2) < 1. It is shown that the upper bound coincides with the exact decay rate in most systems for which the exact decay rate is known. Moreover, as a function of c1 and c2, η(c1, c2) takes one of eight types, and the types explain some curious properties reported in the paper [1].