Sharp results on convergence rates for the distribution of \(\text{GI}/ \text{M}/1/K\) queues as \(K\) tends to infinity (Q2725297)

Consider the \(\text{GI/M}/1/K\)-queue (with a buffer of size \(K=1,2,\ldots \)). The authors study the convergence for \(K\to \infty\) of the stationary distribution \(q^{K}\) of queue length toward \(q\), the stationary queue length distribution of the \(\text{GI/M}/1\)-queue (with infinite buffer). They identify positive constants \(s\) and \(g\) such that \(s^K\cdot (q^{K}-q) \to g\) in \(L_1\)-norm. As an obvious corollary they obtain a result of \textit{F. Simonot} [J. Appl. Probab. 34, No. 4, 1049-1060 (1997; Zbl 0898.60090)] which states that the sequence \(r^K\cdot (q^{K}-q)\) converges to 0 in case \(r<s\) and diverges to infinity for \(r>s\).