Heavy-traffic Asymptotic Expansions for the Asymptotic Decay Rates in the Bmap/g/1 Queue

In great generality, the basic steady-state distributions in the BMAP / G /1 queue have asymptotically exponential tails. Here we develop asymptotic expansions for the asymptotic decay rates of these tail probabilities in powers of one minus the traffic intensity. The first term coincides with the decay rate of the exponential distribution arising in the standard heavy-traffic limit. The coefficients of these heavy-traffic expansions depend on the moments of the service-time distribution and the derivatives of the Perron-Frobenius eigenvalue δ(z) of the BMAP matrix generating function D(z) at z = 1. We give recursive formulas for the derivatives δ(k) ( 1 ). The asymptotic expansions provide the basis for efficiently computing the asymptotic decay rates as functions of the traffic intensity, i.e., the caudal characteristic curves. The asymptotic expansions also reveal what features of the model the asymptotic decay rates primarily depend upon. In particular, δ(z) coincides with the limiting time-average of the factorial cumulant generating function (the logarithm of the generating function) of the arrival counting process, and the derivatives δ(k) ( 1 ) coincide with the asymptotic factorial cumulants of the arrival counting process. This insight is important for admission control schemes in multi-service networks based in part on asymptotic decay rates. The interpretation helps identify appropriate statistics to compute from network traffic data in order to predict performance.