TAIL BEHAVIOR OF THE STEADY-STATE DISTRIBUTION IN TWO-STAGE TANDEM QUEUES : NUMERICAL EXPERIMENT AND CONJECTURE

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This paper is concerned with the geometric decay property of the steady-state probability x(n_l,n_2;i_0,i_1,i_2) in a tandem queuerng system PH/PH/1 → /PH/1. First we observe results of an extensive numerical experiment and see two types of geometric decay for the tail of the joint queue-1ength distribution depending on the traffic intensities of the first and second stages. Then, based on the obser-vation, we give a conjecture on the geometric decay property. The conjecture is roughly summarized as follows. For a given traffic intensity ρ-1 of the first stage, there exists a threshold ρ^^〜_2 for the traffic intensity ρ_2 of the second stage, and if ρ_2 < ρ^^〜_2, then the joint queue-length probability p(n_l, n_2) 〜 Gη^<n_1>_1 η^<n_2>_2 as n_l, n_2 → ∞. If ρ_2 > ρ^^〜_2, then p(n_l, n_2) decays in a similar manner, but the coefficient G and the decay rates η_l η_2 are different between the cases with n_2 < a^^〜n_1 and with n_2 > a^^〜n_1 for a certain positive value a^^〜. Moreover, the conditional probability of phases y(i_0, i_1, i_2 | n_1, n_2) = x(n_1, n_2 ; i_0, i_1, i_2)/p(n_l, n_2) is asymptotically independent of n_1 and n_2 in each case, and hence the steady-state distribution has geometric tail. Equations are also given for determining characteristic values ρ^^〜_2, η_1, η_2 and a^^〜.
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