A Counterexample to a Conjecture of Harrison and Wein

For two station closed queueing networks, Harrison and Wein studied a Reflected Brownian Motion approximation of the network and arrived at a conjecture for an explicit formula for the asymptotic loss of any buffer priority policy. Here we show by explicit solution of two examples that this formula is not correct if the network is unbalanced, even though the networks are efficient. However, if the example networks are balanced, the true asymptotic loss agrees exactly with the conjecture. Thus, while dispelling the conjecture for the general unbalanced case, the examples bolster the evidence for the conjectured form in the balanced case.