The number of rearrangements for Clos networks – new results

Abstract

Rearrangeable Clos networks have been studied for a long time due to their many applications, as well as their theoretical interest. In a rearrangeable Clos network, a newly requested connection may be blocked by existing connections. However, this blocking is eliminated by adequately rearranging some other existing connections. In this operation, an interesting topic is the sufficient number of rearrangements required to eliminate the blocking. Despite previous studies, the sufficient number of rearrangements has only been found for limited cases and has not been completely determined for generic parameter values. This paper analyses the number of rearrangements using the connection chain concept, which clearly and efficiently represents a sequence of connections to be rearranged. The analysis assumes the employment of a rearrangement algorithm, which eliminates the blocking using the shortest connection chain. The usage of the shortest connection chain results in the minimum number of rearrangements. As a result, this paper determines a new bound on the number of rearrangements for a parameter range that has not been considered in any previous studies. In addition, this paper examines the condition for which the system is unblocked via one rearrangement.