Fault-Tolerant General Beneš Networks

Beneš networks are well-known rearrangeable nonblocking (RNB) multistage networks. The so-called conventional Benes networks are based on 2×2 switches. In this paper, Beneš network is a general term used to refer to an N × N n -nary Benes network. Such networks, denoted by B( n, t ), where N = n t and t ≥ 2, are based on regular n × n switches, and are RNB as well. A Beneš network is constructed recursively from a 3-stage Clos network. For an N × N RNB Clos network C( n, m, r ), where N = nr and m ≥ n , the maximum allowable number of non-contact faults in each single shell for realizing any permutation has been investigated in an earlier study, where shell k in a network consists of both the k th and the k th-to-last node stages. That study showed that, for a given integer N , an RNB C( n, n, r ) network with larger n × n switches leads to tolerance of more non-contact faults in shell 1. In this paper, for an N × N B( n, t ) network, we study the maximum allowable number, say f k , of non-contact faults for any permutation not only in each single shell k , but also in all shells simultaneously under the fault condition that at most f k non-contact faults are arbitrarily located in the switches in each shell k . We call the former the fault tolerance capability in a single shell, and the latter the fault tolerance capability of the network. We show that a larger switch size, i.e., n × n , in an N × N B( n, t ) network leads to a higher fault tolerance capability of the network and a higher fault tolerance capability in each non-middle shell. An N × N Beneš network B( n, t ) considers only the value of N which is a power of n . To consider a flexible N with N = n s · q , where s ≥ 2, 1 < q < n and q | n , which means that n is divisible by q , we propose in this paper an N × N RNB Beneš-type network using regular n × n switches, which is called an extended Beneš and denoted by B( n, s, q ). Both Beneš and extended Beneš networks are based on regular switches, and they have better scalabilities than a Clos network. We define a network's fault tolerance rate as the ratio of the fault tolerance capability to the total crosspoints in the network. For given integers N and n , we derive that the fault tolerance capability and fault tolerance rate of an N × N Beneš (or extended Beneš) network are higher than or equal to those of an N × N RNB C( n, n, r ) network, and the former outperforms the latter in most cases.