Asymptotic behavior of the stationary distribution in a finite QBD process with zero mean drift

We consider a finite Quasi-Birth-Death (QBD) process with m levels. Assuming that the mean drift is 0, we obtain an asymptotic behavior πm^(m) ˜(1/m)c as m→∞ in the stationary distribution (π0^(m),…,πm^(m)), by finding an explicit expression for vector c. This solves the problem that was conjectured by Miyazawa et al. [Asymptotic behaviors of the loss probability for a finite buffer queue with QBD structure, Stochastic Models 23 (2007) 79-95].