A method to evaluate the spatial extensibility of a switching unit and network

Switching units and networks have been analyzed as extensible fabrics, mostly in terms of their scheduling algorithms. The traditional literature on switching extensibility has provided complexity theory only relating to the total numbers of inputs (or outputs) and exchange lines. This paper analyzes switching extensibility in terms of not only the scheduling algorithm and also the fabric itself. It is found that determining extensibility from soft complexity related to the number of inputs (or outputs) of the scheduling algorithm and the fabric extensibility in previous studies without quantization is a flawed conception. A method is thus proposed to express the spatial extensibility of a switching unit or network in terms of the connections of a switching resource and capacity. The method calculates parameter ES (the efficiency of switching) of an m× n switching unit and obtains two functions of the switching unit to describe spatial extensibility along with the number of unilateral inputs or outputs. It is found that the range of ES is (0, 1] and three types of switching unit and two types of crosspoint networks have ES = 1. ES is calculated for banyan, Clos, parallel packet, fully interconnected and recirculation switching networks. The ES value for the banyan switching network is larger than that for other networks, and switching networks are classified into three types that have absolute/linear/denied spatial extensibility according to the limES value. It is demonstrated that a switching network has the largest ES value when it contains only the five types of switching unit for which ES = 1. Finally, a group-switching-first self-routing banyan switching network with lower blocking probability and time delay is deduced, and the ES method is contrasted with two other methods of evaluating spatial extensibility in terms of their mathematical expressions and intuitive graphics, for the five types of switching network listed above.