<s>
In	O
the	O
field	O
of	O
telecommunications	O
,	O
a	O
Clos	B-Device
network	I-Device
is	O
a	O
kind	O
of	O
multistage	O
circuit-switching	B-Protocol
network	O
which	O
represents	O
a	O
theoretical	O
idealization	O
of	O
practical	O
,	O
multistage	O
switching	O
systems	O
.	O
</s>
<s>
By	O
adding	O
stages	O
,	O
a	O
Clos	B-Device
network	I-Device
reduces	O
the	O
number	O
of	O
crosspoints	O
required	O
to	O
compose	O
a	O
large	O
crossbar	O
switch	O
.	O
</s>
<s>
A	O
Clos	B-Device
network	I-Device
topology	O
(	O
diagrammed	O
below	O
)	O
is	O
parameterized	O
by	O
three	O
integers	O
n	O
,	O
m	O
,	O
and	O
r	O
:	O
n	O
represents	O
the	O
number	O
of	O
sources	O
which	O
feed	O
into	O
each	O
of	O
r	O
ingress	O
stage	O
crossbar	O
switches	O
;	O
each	O
ingress	O
stage	O
crossbar	O
switch	O
has	O
m	O
outlets	O
;	O
and	O
there	O
are	O
m	O
middle	O
stage	O
crossbar	O
switches	O
.	O
</s>
<s>
Circuit	B-Protocol
switching	I-Protocol
arranges	O
a	O
dedicated	O
communications	O
path	O
for	O
a	O
connection	O
between	O
endpoints	O
for	O
the	O
duration	O
of	O
the	O
connection	O
.	O
</s>
<s>
This	O
sacrifices	O
total	O
bandwidth	O
available	O
if	O
the	O
dedicated	O
connections	O
are	O
poorly	O
utilized	O
,	O
but	O
makes	O
the	O
connection	O
and	O
bandwidth	O
more	O
predictable	O
,	O
and	O
only	O
introduces	O
control	O
overhead	O
when	O
the	O
connections	O
are	O
initiated	O
,	O
rather	O
than	O
with	O
every	O
packet	O
handled	O
,	O
as	O
in	O
modern	O
packet-switched	B-Protocol
networks	I-Protocol
.	O
</s>
<s>
When	O
the	O
Clos	B-Device
network	I-Device
was	O
first	O
devised	O
,	O
the	O
number	O
of	O
crosspoints	O
was	O
a	O
good	O
approximation	O
of	O
the	O
total	O
cost	O
of	O
the	O
switching	O
system	O
.	O
</s>
<s>
Upon	O
the	O
advent	O
of	O
complex	O
data	O
centers	O
,	O
with	O
huge	O
interconnect	O
structures	O
,	O
each	O
based	O
on	O
optical	O
fiber	O
links	O
,	O
Clos	B-Device
networks	I-Device
regained	O
importance	O
.	O
</s>
<s>
A	O
subtype	O
of	O
Clos	B-Device
network	I-Device
,	O
the	O
Beneš	O
network	O
,	O
has	O
also	O
found	O
recent	O
application	O
in	O
machine	O
learning	O
.	O
</s>
<s>
Clos	B-Device
networks	I-Device
have	O
three	O
stages	O
:	O
the	O
ingress	O
stage	O
,	O
the	O
middle	O
stage	O
,	O
and	O
the	O
egress	O
stage	O
.	O
</s>
<s>
Clos	B-Device
networks	I-Device
are	O
defined	O
by	O
three	O
integers	O
n	O
,	O
m	O
,	O
and	O
r	O
.	O
n	O
represents	O
the	O
number	O
of	O
sources	O
which	O
feed	O
into	O
each	O
of	O
r	O
ingress	O
stage	O
crossbar	O
switches	O
.	O
</s>
<s>
The	O
relative	O
values	O
of	O
m	O
and	O
n	O
define	O
the	O
blocking	O
characteristics	O
of	O
the	O
Clos	B-Device
network	I-Device
.	O
</s>
<s>
If	O
m≥2n1	O
,	O
the	O
Clos	B-Device
network	I-Device
is	O
strict-sense	O
nonblocking	O
,	O
meaning	O
that	O
an	O
unused	O
input	O
on	O
an	O
ingress	O
switch	O
can	O
always	O
be	O
connected	O
to	O
an	O
unused	O
output	O
on	O
an	O
egress	O
switch	O
,	O
without	O
having	O
to	O
re-arrange	O
existing	O
calls	O
.	O
</s>
<s>
If	O
m	O
≥	O
n	O
,	O
the	O
Clos	B-Device
network	I-Device
is	O
rearrangeably	O
nonblocking	O
,	O
meaning	O
that	O
an	O
unused	O
input	O
on	O
an	O
ingress	O
switch	O
can	O
always	O
be	O
connected	O
to	O
an	O
unused	O
output	O
on	O
an	O
egress	O
switch	O
,	O
but	O
for	O
this	O
to	O
take	O
place	O
,	O
existing	O
calls	O
may	O
have	O
to	O
be	O
rearranged	O
by	O
assigning	O
them	O
to	O
different	O
centre	O
stage	O
switches	O
in	O
the	O
Clos	B-Device
network	I-Device
.	O
</s>
<s>
To	O
prove	O
this	O
,	O
it	O
is	O
sufficient	O
to	O
consider	O
m	O
=	O
n	O
,	O
with	O
the	O
Clos	B-Device
network	I-Device
fully	O
utilised	O
;	O
that	O
is	O
,	O
rn	O
calls	O
in	O
progress	O
.	O
</s>
<s>
The	O
proof	O
shows	O
how	O
any	O
permutation	O
of	O
these	O
rn	O
input	O
terminals	O
onto	O
rn	O
output	O
terminals	O
may	O
be	O
broken	O
down	O
into	O
smaller	O
permutations	O
which	O
may	O
each	O
be	O
implemented	O
by	O
the	O
individual	O
crossbar	O
switches	O
in	O
a	O
Clos	B-Device
network	I-Device
with	O
m	O
=	O
n	O
.	O
</s>
<s>
In	O
the	O
context	O
of	O
a	O
Clos	B-Device
network	I-Device
,	O
each	O
boy	O
represents	O
an	O
ingress	O
switch	O
,	O
and	O
each	O
girl	O
represents	O
an	O
egress	O
switch	O
.	O
</s>
<s>
If	O
this	O
middle-stage	O
switch	O
is	O
now	O
removed	O
from	O
the	O
Clos	B-Device
network	I-Device
,	O
m	O
is	O
reduced	O
by	O
1	O
,	O
and	O
we	O
are	O
left	O
with	O
a	O
smaller	O
Clos	B-Device
network	I-Device
.	O
</s>
<s>
The	O
Jacobaeus	O
approximation	O
is	O
more	O
accurate	O
,	O
and	O
to	O
see	O
how	O
it	O
is	O
derived	O
,	O
assume	O
that	O
some	O
particular	O
mapping	O
of	O
calls	O
entering	O
the	O
Clos	B-Device
network	I-Device
(	O
input	O
calls	O
)	O
already	O
exists	O
onto	O
middle	O
stage	O
switches	O
.	O
</s>
<s>
There	O
are	O
i	O
input	O
calls	O
entering	O
via	O
the	O
same	O
ingress	O
switch	O
as	O
the	O
free	O
input	O
terminal	O
to	O
be	O
connected	O
,	O
and	O
there	O
are	O
j	O
calls	O
leaving	O
the	O
Clos	B-Device
network	I-Device
(	O
output	O
calls	O
)	O
via	O
the	O
same	O
egress	O
switch	O
as	O
the	O
free	O
output	O
terminal	O
to	O
be	O
connected	O
.	O
</s>
<s>
Clos	B-Device
networks	I-Device
may	O
also	O
be	O
generalised	O
to	O
any	O
odd	O
number	O
of	O
stages	O
.	O
</s>
<s>
By	O
replacing	O
each	O
centre	O
stage	O
crossbar	O
switch	O
with	O
a	O
3-stage	O
Clos	B-Device
network	I-Device
,	O
Clos	B-Device
networks	I-Device
of	O
five	O
stages	O
may	O
be	O
constructed	O
.	O
</s>
