On the Accuracy of the OPC Approximation for a Symmetric Overflow Loss Model

Abstract

The overflow priority classification approximation (OPCA) and Erlang's fixed-point approximation (EFPA) are distinct methods for estimating blocking probabilities in overflow loss networks. Mounting numerical evidence has indicated that OPCA provides superior accuracy than EFPA in many circumstances. Furthermore, it has been proven that P EFPA ≤ P OPCA for a symmetric overflow loss network called the distributed server model, where P x is the blocking probability estimate yielded by approximation x ∈ {EFPA, OPCA}. The distributed server model is an ideal “proving ground” because the exact blocking probability, P exact, can be calculated with the Erlang B formula, yet the state dependencies caused by mutual overflow are retained. The present article proves the OPC Conjecture, P OPCA ≤ P exact. This new result establishes that OPCA is always equally or more accurate relative to EFPA for the distributed server model and suggests OPCA has utility in more general overflow loss networks. The proof of P OPCA ≤ P exact turned out to be challenging, since OPCA is remarkably close to the exact solution, requiring delicate inequality arguments on the Maclaurin series of both solutions. Another contribution is the derivation of new tight bounds and limiting regimes for the blocking probabilities yielded by OPCA and EFPA in the case of critical loading. The simple and tight lower bound derived for OPCA can serve as a tight lower bound for the Erlang B blocking probability in the case of critical loading.