Before revision:  When the future is unknown, the optimal average queue length diverges at rate \log_{1/(1-p) 1/(1-\lambda)} 1{1-\lambda}} , as \lambda\to 1.
After revision:  When the future is unknown, the optimal average queue length diverges at rate \log_{1/(1-p) 1/(1-\lambda)} \sim\log_{1/(1-p)1{1-\lambda}} , as \lambda\to 1.