The Number of Rearrangements Performed via Extended Connection Chains for Rearrangeable Clos Networks

Abstract

Various switching networks have been investigated because of their practical importance and theoretical interests. Among these networks, this study focuses on the Clos network. A Clos network is constructed by placing switches in three stages. In the first and third stages, r (r > 1) switches are aligned, whereas m (m > 1) switches are aligned in the second stage. There are n inputs and m outputs in the first stage. Symmetrically, the third stage switch has m inputs and n outputs. For this configuration, if n  m 2n  2, the network is rearrangeable. Though existing connections in a rearrangeable network may block a newly requested connection, the blocking is always removed by rerouting existing connections. An interesting problem arose during this process is how many existing connections must be rearranged: the number of rearrangements. Although the problem has been studied for a long time, the number of rearrangements is not completely clarified for arbitrary combinations of parameters m, n, and r. This study presents a new upper bound on the number of rearrangements for 2  n m 2n  2 . This bound is derived from the extended connection chain concept proposed in a previous study. Using this concept, the paper first derives from the case where a parameter, s, represents the load on a second-stage switch. Then, the paper presents another new upper bound, which is independent of parameter s. The study shows that the presented upper bound is smaller than the previously known bounds for a certain range of m.