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 η(c 1 , c 2 ) of the decay rate is derived in the senseIt 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 , η(c 1 , c 2 ) takes one of eight types, and the types explain some curious properties reported in Fujimoto and Takahashi (J. Oper. Res. Soc. Jpn. 39:525-540 1996).

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