Unifying aspect of Liapunov's direct and indirect methods in the stability analysis of Aloha systems

This paper explores the stability aspects of a multipacket heterogeneous slotted Aloha system by both the direct and the indirect methods due to Liapunov, and establishes the equivalence between them. With the identification of a Liapunov function in terms of the flow balance function, it is found that both of the approaches essentially converge to the assessment of the property of the same matrix - the system matrix. If the system is stable with respect to one of the approaches, it is also simultaneously stable with respect to the other method. This unifying aspect encompasses simultaneously the computability of the indirect method and the greater extent of stability implicit in the direct method. Sufficient conditions, computable in terms of the user and system parameters, for the stability of the equilibrium states are obtained. Numerical examples are presented to illustrate the analytical results obtained.