This paper studies the geometric decay property of the joint queue-length distribution {p(n_{1},n_{2})} of a two-node Markovian queueing system in the steady state. For arbitrarily given positive integers c_{1}, c_{2}, d_{1} and d_{2}, an upper bound eta(c_{1},c_{2}) of the decay rate is derived in the sense exp{limsup{n rightarrow infty} n^{-1}log p(c_{1}n+d_{1},c_{2}n+d_{2})} <= eta(c_{1},c_{2})<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 c_{1} and c_{2}, eta(c_{1},c_{2}) takes one of eight types, and the types explain some curious properties reported in the paper.
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