Tightness of stationary waiting times in heavy traffic for G1/G1/1 queues with thick tails

Recently, a Heavy Traffic Invariance Principle was proposed by Szczotka and Woyczyński to characterize the heavy traffie limiting distribution of normalized stationary waiting times of G/G/X queues in terms of an appropriate convergence to a Levy process. It has two important assumptions. The first of them deals with a convergence to a Levy process of appropriate processes which is well investigated in the literature. The second one states that the sequence of appropriate normalized stationary waiting times is tight. In the present paper we characterize the tightness condition for the case of GI/GI/1 queues in terms of the first condition.