Before revision:  If this condition is not satisfied, we show that queue lengths are bounded by j^\star+1, where j^\star\in\mathbb{N\left\lceil {\log (\lambda d +1)} }\right\rceil .
After revision:  If this condition is not satisfied, we show that queue lengths are tightly bounded by\left\lceil {\log (\lambda d +1)} }\right\rceil .