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

Abstract

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 $\overline{\eta}(c_{1},c_{2})$ of the decay rate is derived in the sense

$$\exp\Bigl\{\limsup_{n\rightarrow\infty}n^{-1}\log p(c_{1}n+d_{1},c_{2}n+d_{2})\Bigr\}\leq\overline{\eta}(c_{1},c_{2})<1.$$

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, $\overline{\eta }(c_{1},c_{2})$ 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]).