On the cusp catastrophe of slotted ALOHA systems with capture and multi-packet reception

Abstract

The stability of slotted ALOHA systems with various types of capture phenomena and multiple packet reception capability is discussed in terms of the cusp catastrophe. The slotted ALOHA systems considered are classified into; (1) single packet reception with geometric capture, (2) independent multiple packet reception with geometric capture, (3) single packet reception with partially-perfect capture, and (4) multiple packet reception with partially-perfect capture. First, general expressions for the cusp points and the bifurcation sets are derived. Then, we present an explicit formula for the stability of slotted ALOHA systems for the four types of capture and multi-packet reception capability and demonstrate how the bistable behavior is mitigated due to capture effect and multi-packet reception capability.