Fault-Tolerant General Beneš Networks

Beneš networks are well-known rearrangeable nonblocking (RNB) multistage networks. The so-called conventional Beneš networks are based on
$2\times 2$
switches. In this paper, Beneš network is a general term used to refer to an
$N\times {N}~n$
-nary Beneš network. Such networks, denoted by B(
$n$
,
$t)$
, where
$N = n^{t}$
and
$t \ge2$
, are based on regular
$n\times n$
switches, and are RNB as well. A Beneš network is constructed recursively from a 3-stage Clos network. For an
$N\times N$
RNB Clos network C(
$n$
,
$m$
,
$r)$
, where
$N = nr$
and
$m \ge n$
, 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
$n\times n$
switches leads to tolerance of more non-contact faults in shell 1. In this paper, for an
$N\times N$
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.,
$n\times n$
, in an
${N} \times N$
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
$N\times N$
Beneš network B(
$n$
,
$t)$
considers only the value of
$N$
which is a power of
$n$
. To consider a flexible
$N$
with
$N = n^{s}\cdot \textit {q}$
, where
$s\ge2$
,
$1 < {q} < n$
and
${q}\vert n$
, which means that
$n$
is divisible by
${q}$
, we propose in this paper an
$N\times N$
RNB Beneš-type network using regular
$n\times n$
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
${N} \times N$
Beneš (or extended Beneš) network are higher than or equal to those of an
$N\times N$
RNB C(
$n$
,
$n$
,
$r)$
network, and the former outperforms the latter in most cases.