WDM Split Cross-connects and 3-stage Clos Networks

Abstract

In this paper, we first show that a wavelength division multiplexed (WDM) split crossconnect is strictly, wide-sense, or rearrangeably non-blocking if and only if a corresponding 3-stage Clos network is strictly, wide-sense or rearrangeably non-blocking. This observation implies known results on strictly non-blocking WDM split cross-connects and gives rise to new results on wide-sense and rearrangeably non-blocking WDM split cross-connects. We next introduce a routing algorithm for the Clos networks which show that there are widesense non-blocking Clos networks which are not strictly non-blocking. Hence, the same holds for the WDM split cross-connects. We also consider another demand model for WDM split cross-connects and present associated results. Routing algorithms for WDM split crossconnects could be viewed as on-line and off-line algorithms to edge-color a class of bipartite multi-graphs.

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WDM Split Cross-connects and
?-stage Clos Networks
Hung Q. Ngo
?
July 9, 2002
Abstract
In this paper, we first show that a wavelength division multiplexed (WDM) split cross-
connect is strictly, wide-sense, or rearrangeablynon-blockingif and only if a corresponding
?-stageClos networkis strictly, wide-senseorrearrangeablynon-blocking. Thisobservation
implies known results on strictly non-blocking WDM split cross-connects and gives rise to
new results on wide-sense and rearrangeably non-blocking WDM split cross-connects. We
next introduce a routing algorithm for the Clos networks which show that there are wide-
sense non-blocking Clos networks which are not strictly non-blocking. Hence, the same
holds forthe WDM split cross-connects. We also consideranotherdemandmodelforWDM
split cross-connectsandpresentassociatedresults. RoutingalgorithmsforWDM splitcross-
connects couldbe viewedas on-lineand off-linealgorithms to edge-colora class of bipartite
multi-graphs.
1Introduction
Wavelength division multiplexed (WDM) cross-connects are important in modern WDM optical
networks. We can think of the cross-connects as black boxes with disjoint sets of input and
output fibers, each of which is capable of carrying a set of signals on different wavelengths. We
would like the cross-connect to be able to route any demand from an unused wavelength at an
input fiber to an unused wavelength at an output fiber. Since the input and output wavelength
of a demand might be different, wavelength interchangers (WIs) are needed to design the cross-
connects. We assume the WIs to have the ability to convert any incoming wavelength to any
outgoing wavelength. Since WIs are expensive components, an objective is to design cross-
connects with as few WIs as possible. Obviously, the minimum number of WIs shall depend on
which degree of non-blockingness (strictly, wide-sense, rearrangeably, to be defined) is desired.
The wavelength division multiplexed (WDM) split cross-connects (WSXC) were introduced
by Rasala and Wilfong [13,14] as an optimal architecture for WDM cross-connects in terms of
number of WIs. An
in Figure 1. A set of
?
?
??
?WSXC has
?
?input fibers and
?
?output fibers. The design is shown
?
?wavelengths
??
?
??????
?
?
? are available at each input fiber, and a set of
?
switching fabric
of fibers from the WIs to the output fabric
to denote the WSXC. For convenience, we use
the authors show that
blocking (SNB), given that
the general case of
?wavelengths
??
?
??????
?
?
? are available at each output fiber. The input fibers are inputs to a
?. The fabric
? has
? outputs connected to
? WIs. The output side consists
? of dimension
???
?. We use
???
?
??
?
? ???
?
??
?
?
???????? to denote
????????????. In [13]
???? WIs are necessary and sufficient for
???????? to be strictly non-
? and
? are SNB. In [14], corresponding conditions were derived for
???
?
??
?
????
?
??
?
?. The authors called
???????? the homogeneous WSXC,
?Computer Scienceand Engineering Department, 201 Bell Hall StateUniversity of New York at Buffalo, Amherst,
NY 14260, USA. hungngo@cse.buffalo.edu
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input
fibers
output
fibers
?
?
?
?
??
?
??????
?
?
?
??
?
??????
?
?
?
??
?
??????
?
?
?
??
?
??????
?
?
?
??
??
?
??
?
??
?
Figure 1: The WDM split cross-connect
???
?
??
?
????
?
??
?
?
and
different from ours. The new notations are used to be consistent with well-established notations
for the
The Clos network has been widely used for data communications and parallel computing
systems [1,5,6,8]. The Clos network
where the first stage consists of
dimension
Each input crossbar
???
?
??
?
????
?
??
?
?theheterogeneous WSXC.Thenotations used intheirpaperwereslightly
?-stage Clos networks.
???
?
??
?
????
?
??
?
? is a
?-stage interconnection network,
?
?crossbars of size
?
?
??, the last stage has
?
?crossbars of
???
?, and the middle stage has
? crossbars of dimension
?
?
??
?(see Figure 2).
?
?( ?????????
?) is connected to each middle crossbar
?
?( ?????????).
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
Figure 2: The
?-stage Clos network
???
?
??
?
????
?
??
?
?
Similarly, the middle stage and the last stage are fully connected. When
?
?
??
?
?? and
?
?
?
In this paper, we show that
derive consequences of this fact. In particular, we get for free many results from [13,14] on SNB
WSXC. The observation also gives rise to a few results on wide-sense non-blocking (WSNB)
and rearrangeably non-blocking (RNB) WSXC. Next, a routing algorithm for Clos networks and
WSXCs called the degree-counting algorithm is presented in terms of an online algorithm for
edge-coloring a class of bipartite multi-graphs. We also introduce a new demand model for the
WSXC and present associated results under different blocking properties.
?
?
??, the network is called the symmetric
?-stage Clos network, denoted by
???? ????.
???
?
??
?
????
?
??
?
? is “equivalent” to
???
?
??
?
????
?
??
?
?, and
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2Preliminaries
2.1The classical demand model
In this section, we present basic definitions of different degrees of non-blockingness for the
WSXCs and Clos networks under the so-called classical demand model. In the WSXC case, this
model assumes that each link can carry a number of channels each of which is on a different
wavelength. As for Clos networks, this model assumes that each link can carry at most one
channel. In the next section, another demand model is introduced, in which a wavelength in
WSXC or a link in Clos networks can be further divided into sub-channels. Practically, the sub-
channels could be thought of as representing time division multiplexed (TDM) channels within
one wavelength.
2.1.1 WDM Split Cross-connects
Recall the general architecture of
?????
?
??
?
? ???
?
??
?
? as shown in Figure 1. A demand
? is defined to be a
?-tuple
??? ??????, where
??????????
?
? is an input fiber to the fabric
?,
?
The fibers
and
an input of
?????????
?
? is an output fiber from the fabric
?,
????
?
??????
?
?
?, and
????
?
??????
?
?
?.
? and
? are called input fiber and output fiber for demand
?, respectively. Similarly,
?
? are called the input wavelength and output wavelength of
?. A route
? in
? is a path from
? to an output of
? to the corresponding
?? to an input of
? and exits at an output of
?. Along each fiber (or link),
assigned the same wavelength unless the fibers are connected through a WI. Straightforwardly,
a route
assigned with wavelengths
any incoming wavelength to any outgoing wavelength as long as there is no conflict at the input
of the output.
A valid demand set is a set
input fiber and input wavelength, and no two demands share both the output fiber and output
wavelength. A valid demand set
one for each demand, such that any two routes which use the same fiber is assigned different
wavelengths on that fiber. In this case, we say that
new demand
A WSXC
routes realizing
? is assigned a wavelength such that consecutive fibers in
? are
? for a demand
??????????? is a route from
? to
? such that the fibers
? and
? are
? and
?, respectively. The WIs are able to simultaneously convert
? of demands such that no two demands in
? share both the
? is realizable (or satisfied) by
? if there is a set
? of routes,
? realizes
?. Given a valid demand set
?, a
???? is said to be compatible with
? iff
????? is valid.
? is said to be strictly non-blocking if given a valid demand set
? and a set
? of
?, for any new demand
? compatible with
? there exists a new route
? so that
????
As demands come and go, a strategy to pick a route for a new demand is called a routing
algorithm. A WSXC is said to be wide-sense non-blocking (WSNB) with respect to a routing
algorithm
compatible with
A WSXC is WSNB with respect to a class
to every algorithm in
algorithm.
Lastly, a WSXC is rearrangeably non-blocking if for any valid demand set
set
Note that the switching fabrics
non-blocking. Note also that SNB implies WSNB which in turn implies RNB. Unless men-
tioned otherwise, we will assume that
minimizing the number of WIs. It is straightforward to extend the results in this paper to the
cases when
? realizes
?????.
? if given a route set
? realizing a valid demand set
?, for any new demand
?
? the algorithm
? can choose a new route
? so that
????? realizes
?????.
? of routing algorithms if it is WSNB with respect
?. In general, a WSXC is WSNB if it is WSNB under some routing
? there exists a
? of routes realizing
?.
? and
? can also be strictly, wide-sense or rearrangeably
? and
? are SNB in order to simplify the discussion on
? and/or
? are non-blocking in other senses.
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2.1.2Three-stage Clos Networks
The inlets (outlets) of the input (output) crossbars are the inputs (outputs) of the network. Inputs
and outputs are referred to as external links, while links between crossbars are referred to as
internal links. As in the case of WSXC, we shall assume that all crossbars are SNB to shorten
our presentation.
A demand
inlet of an input crossbar and
valid if no two demands in
inlet
a link from
? for the Clos network
?????
?
??
?
????
?
??
?
? is a pair
????? where
? is an
? is an outlet of an output crossbar. A set
? of demands for
? is
? share the same inlet nor the same outlet. A route
? consists of an
?, an input crossbar
? one of whose inlets is
?, a link from
? to a middle crossbar
?,
?,
? to an output crossbar
?,
?, and an outlet
? of
?. A route
? realizes a demand
??????? if
? begins from
? and ends at
?. A set
? of
? routes realizes a valid demand set
? of
? demands if each route in
a link.
The definitions of compatibility, routing algorithms, SNB, WSNB, and RNB are similar to
those in the previous section.
? realizes a different demand in
? so that no two routes in
? share
2.2 The multi-rate demand model
The multi-rate demand models were initially introduced in [10]. Other works on these models
include those in [3,4,9,11,12]. In the multi-rate demand model, each demand
a
parameters remain the same as before. For simplicity, we shall assume that all wavelengths have
unit capacity. A rate
A set
having
also hold for every pair of output fiber and output wavelength.
All the definitions are similar to the classical demand model except that now many routes
can share the same wavelength on the same fiber as long as the total weight of the corresponding
demands does not exceed unity. The concepts of SNB, WSNB, and RNB are now called multi-
rate SNB, multi-rate WSNB, and multi-rate RNB.
The corresponding definitions for the Clos networks is similar, and thus omitted here.
The multi-rate model for the Clos networks given here is often referred to as the continuous
multi-rate model [8]. It is more general than the discrete multi-rate model where the weights are
required to be chosen from a given set
the smallest weight. The smallest weight is meant to model the smallest granularity of a TDM
channel.
? for a WSXC is
? tuple
??? ????????, where
?? ????℄ is the requested rate (or weight) of the demand. Other
? can be thought of as the fraction of bandwidth on a particular wavelength.
? of demands is valid if for each fiber
? and wavelength
?, the total weight of demands
? as the input fiber and
? as the input wavelength does not exceed unity. The same must
? of
? weights where all weights are integer multiples of
3The equivalence of WSXCs and Clos networks
3.1The classical demand model
Lemma 3.1. Assume that switching fabrics
bars of
wide-sense non-blocking (under any algorithm), and rearrangeably non-blocking if and only if
is strictly non-blocking, wide-sense non-blocking, and rearrangeably non-blocking, respectively.
? and
? of
?????
?
??
?
? ???
?
??
?
? and the cross-
?????
?
??
?
????
?
??
?
? are are strictly non-blocking, Then,
? is strictly non-blocking,
?
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Proof. We first establish a one-to-one correspondence between the following two sets:
?????????? realizes
? in
??
?????
?
??
?
???
?realizes
?
?in
???
Consider
and
??????? where
??????
?
? ???
?
?. Note that
?????
?,
?????
?,
?????
?,
?????
?. Further assume that the route
? routes
? through the wavelength interchanger
??
?. The corresponding pair
??
?
??
?
??? is defined as follows. The demand
?
?is from
?
?to
?
The route
can be thought of as representing the WIs of
and hence the mapping is bijective.
Next, observe that two routes
that fiber if and only if the corresponding two routes for
routing of
follows.
?, where
?
?is the
?th input to input crossbar
?
?, and
?
?is the
?th output from output crossbar
?
?.
?
?is the sequence
?
?
??
?
??
?
??
?
??
?. Thus, the middle-stage crossbars of
?
?. It is easy to see that the construction is invertible,
?
?and
?
?for
? share a fiber and the same wavelength on
? share the same link. In essence, any
? induces a routing of
? and vice versa. The equivalence of
? and
? now readily
We are now ready to derive some results on WSXCs for free. All the corollaries follow from
the corresponding results about
Clos networks (see [8] and [1]). We will quote here a few more important corollaries.
The following corollary includes all the results about SNB WSXCs in [13,14].
?-stage Clos networks. There are numerous results on
?-stage
Corollary 3.2 (Rasala-Wilfong [13,14]). The WDM split cross-connect
strictly non-blocking if and only if the number of wavelength interchangers
where
???
?
??
?
????
?
??
?
? is
? is at least
?
?
?
?
?
?
?
?
?
?
?
?
?
?
if
if
otherwise.
?
?
?
?
??
?
?
?
?
?
?
?
?
?
??
?
?
?
??
?
??
Corollary 3.3. The WDMsplit cross-connect
if and only if the number of wavelength interchangers
???
?
??
?
????
?
??
?
? is rearrangeably non-blocking
? is at least
?????
?
??
?
?.
Unlike the cases of SNB and RNB, very little is known on WSNB properties of the Clos
networks. Particularly, the difficulty comes from the fact that WSNB is algorithm dependent,
and that there are very few known routing algorithms for the network except some common-
sense strategies. We first give a negative result, which basically says that when
large,
argument by Moore credited by Kurshan and Beneˇ s.
?
?and
?
?are
???
?
??
?
????
?
??
?
? can not be WSNB without being SNB. The argument given is from an
Corollary 3.4. If one of the following two conditions holds:
?
?
???
?
? ??
?
?
?
??
?
??
?
?
??
?
??? and
?
?
?
?
?
??
?
?
?
?
??
(1)
?
?
???
?
???
?
?
?
??
?
??
?
?
??
?
??? and
?
?
?
?
?
??
?
?
?
?
??
(2)
Then, the WDM split cross-connect
if it is strictly non-blocking.
???
?
??
?
????
?
??
?
?. is wide-sense non-blocking if and only
5