Structural stability aspects of alternate routing in non-hierarchical networks under reservation

This paper presents a behavioral characterization of non-hierarchical alternate routing schemes in which a certain number of trunks are reserved for direct routed traffic. It is found that the fundamental behavioral characteristic of the alternate routing schemes with trunk reservation control remains the same as that of the schemes without reservation. With no reservation, the system has two equilibrium states separated wide apart [10]. Under the trunk reservation, the spatial distance between the equilibrium states of the system is considerably reduced. Even for a small number of trunks under reservation, the spatial distribution of the two equilibrium states is so close to each other that both the equilibrium states can be considered to coincide. The approach of this paper can explain why the conventional numerical analysis, under the assumption that the arrival rate of both the direct and the alternate routed traffic are Poisson processes, does not indicate instability of reservation schemes, and, why the empirical results indicate that the optimal number of trunks under reservation is not unique. A sufficient condition, which can be utilized to find the minimum number of trunks to be reserved for direct routed traffic so as to avoid the instability, is obtained. Though the sufficient condition ensures absolute stability of the system from the point of view of the catastrophe theory, the designer may select a lower number of trunks to be reserved, provided that the stable equilibrium states remain close to each other. A performance index to compare various levels of reservations among themselves, is proposed.