Before revision:  This is in contrast with the classic version of the power-of-d-choice algorithm, where queue lengths are unbounded. Our upper bound on the size of the most loaded server, j^*+1, is tight and increases slowly when \lambda approaches its critical value from below. For instance, when \lambda= 0.995 and d=2 (respectively, d=3), we find that no server will contain more than just 5 (3) jobs\ge  in equilibrium.
After revision:  This is in contrast with the classic version of the power-of-d-choice algorithm, where at the fluid scale a strictly positive proportion of servers containing i jobs exists for all i\ge in equilibrium.