Wavelength Partitioning in WDM Ring Networks
by
Kayi Lee
Submitted to the Department of Electrical Engineering and Computer
Science
in partial fulfillment of the requirements for the degrees of
Bachelor of Science in Computer Science and Engineering
and
Master of Engineering in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2000
@ Massachusetts Institute of Technology 2000. All rights reserved.
Author .............
Department of Electrical Engineering and Computer Science
February 2, 2000
Certified by .........
Accepted by.
............
Kai-Yeung Siu
Associate Professor
hesis Supervisor
.........
Arthur C. Smith
Chairman, Department Committee on Graduate Theses
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
ENJUL
2 7 2000
LIBRARIES
Wavelength Partitioning in WDM Ring Networks
by
Kayi Lee
Submitted to the Department of Electrical Engineering and Computer Science
on February 2, 2000, in partial fulfillment of the
requirements for the degrees of
Bachelor of Science in Computer Science and Engineering
and
Master of Engineering in Electrical Engineering and Computer Science
Abstract
The ONRAMP architecture is proposed for future wavelength-division multiplexed
(WDM) access networks. Its main function is to provide communication infrastructure to regional end users. It employs wavelength partitioning to multiplex user
traffic in the wavelength channels. In this thesis, we present a simple model for the
ONRAMP architecture and mathematically formulate the wavelength partitioning
problem, where the objective is to maximize the network throughput. We show that
the wavelength partitioning problem is a generalized form of the NP-complete Multiprocessor Scheduling Problem. We propose a fast approximation algorithm Least
Load (LL). The algorithm is designed for ONRAMP to handle traffic between distribution networks and the internet backbone. We show that Least Load always
guarantees to achieve at least 1 of the optimal throughput performance.
4
Thesis Supervisor: Kai-Yeung Siu
Title: Associate Professor
2
Acknowledgments
This thesis is written when I am working as a research assistant in the Research Group
on Communications and Networking. First of all, I would like to thank professor
Sunny Siu for his supervision of my work. Since I joined the group, he has been very
helpful and patient. Without his guidance and advice, I would not be able to find
out such an interesting topic, which leads to this thesis.
Also, I would like to thank Ching Law for proofreading this thesis. He has been
my great roommate for more than one year and our late-night conversation certainly
provokes interesting thoughts on my research. Also, many thanks to Ada Cheung for
her support. She is the person who reminds me to work hard all the time.
Last but not least, I also want to express my deepest gratitude to my parents for
their unwavering support all these years. I was brought up under their love and care.
Nothing would be possible for me without them.
I
3
Contents
1
7
Introduction
1.1
B ackground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.1.1
First-Generation Optical Networks
. . . . . . . . . . . . . . .
8
1.1.2
Second-Generation Optical Networks . . . . . . . . . . . . . .
9
1.2
Previous Work
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.3
Contribution of this work . . . . . . . . . . . . . . . . . . . . . . . . .
13
14
2 The ONRAMP Architecture
2.1
The Access Node Architecture .....
15
......................
3 Model and Formulation of the Wavelength Partitioning Problem
17
. . . . . . . . . . . . . . . . . . . . .
17
. . . . . . . . . . . . . . . . . . . . . . .
18
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.1
Architecture and Traffic Model
3.2
Formulation of the problem
3.3
Com plexity
22
4 The Least Load Algorithm
4.1
Terminology . . . . . . . . . . . .
. . . . . . . .
22
4.2
Description
. . . . . . . . . . . .
. . . . . . . .
23
4.3
4.2.1
High Level Description
. . . . . . . .
23
4.2.2
Detailed Description
.
. . . . . . . .
23
4.2.3
Complexity
. . . . . . . .
. . . . . . . .
24
Proof of Performance Gaurantee .
. . . . . . . .
25
Notation . . . . . . . . . .
. . . . . . . .
25
4.3.1
4
4.3.2
Intuitive Argument . . . . . . . . . . . . . . . . . . . . . . . .
26
4.3.3
Formal Proof . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
33
5 Conclusion
5
List of Figures
1-1
(a) The physical topology of a WDM network and the lightpaths set
up. (b) The logic topology of the network. . . . . . . . . . . . . . . .
12
2-1
The ONRAMP Architecture . . . . . . . . . . . . . . . . . . . . . . .
14
2-2
A Simple ONRAMP Access Node . . . . . . . . . . . . . . . . . . . .
16
4-1
All unidirectional traffic destined to the hub must pass through the
hub link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
29
Chapter 1
Introduction
In the past few years, optical network with wavelength division multiplexing (WDM)
has been considered an attractive candidate for future wide area networks. A variety
of architectures are being studied for future networks of different scale [11, 8, 20]. The
Optical Network for Regional Access with Multiwavelength Protocols (ONRAMP) is
one of the projects designed for future access networks using WDM technologies.
The role of access networks is to provide communication infrastructure to end
users in a region. Some issues come up when an access network architecture is being
designed. First of all, the number of users of the network is small compared with that
of the backbone networks. Therefore, the cost of building and operating an access
network should be kept small in order to be affordable to users. Second, the traffic
from each user is small. Their traffic needs to be multiplexed in order to utilize the
channel bandwidth efficiently. In short, the challenge is to efficiently carry numerous
small flows given a limited amount of resources.
In this thesis, we propose a wavelength scheduling algorithm Least Load (LL) for
the ONRAMP architecture.
The problem we try to solve can be summarized as
follows: given the aggregated traffic demand from each access node in the network,
how do we multiplex their traffic so as to maintain high utilization? We focus the
problem on ring topology, which is the topology employed for feeder network in the
ONRAMP architecture. With a limited number of wavelength channels and Wavelength add/drop multiplexers (WADMs), our goal is to determine how the channel
7
bandwidth is allocated to the traffic at the access nodes.
In section 1.1, we first give some background on the development of current WDM
technologies. We then review some previous work in section 1.2. In chapter 2, we
introduce the ONRAMP architecture. In chapter 3, we give a model for the ONRAMP architecture and formulate our problem mathematically. We also study the
complexity of the problem. In Chapter 4, we propose the algorithm LL and analyze
its performance. We conclude the thesis and suggest related future work in chapter 5.
1.1
Background
Briefly speaking, the evolution of optical networking can be split into two generations, namely the first-generation optical networks and the second-generation optical
networks.
1.1.1
First-Generation Optical Networks
The first-generation optical networks use optical fibers merely as a transmission
medium in place of copper wires. All other components in the networks, such as
switches, process data electronically. In those networks, when a switch forwards data
from one optical fiber to another, the optical signal from the incoming fiber is converted into electronic form (i.e., the signal is dropped). The electronic data is then
buffered and subsequently forwarded to the appropriate output port, where the data
is converted back to optical signal and transmitted to the outgoing fiber (i.e., the
signal is added). This switching process is slow compared to the bandwidth of the
fibers. Therefore, the bandwidth supported by the first-generation optical networks
is mostly limited by the processing power of the electronic equipments.
The limitation can be overcome using the technique called Wavelength Division
Multiplexing (WDM). WDM is a form a frequency division multiplexing where the
passband within an optical fiber is divided into multiple non-overlapping wavelength
channels to carry signal. Each wavelength channel corresponds to an independent
spectrum of frequency (or wavelength) in the passband. These non-overlapping chan8
nels can simultaneously carry different signal over the optical fiber without interference. As a result, a switch can be equipped with multiple sets of optical transmitters
and receivers. Each set of transmitter and receiver independently handles a particular wavelength channel. As a result, multiple channels are created in a single fiber,
with each channel carrying data at the maximum electronic speed. This technique
substantially increases the rate of data transmitted over the fibers.
A wavelength channel in WDM resembles a traditional physical link in that it
provides point-to-point bit transmission service to its upper layer. The bandwidth
of a wavelength channel is in the order of Gigabits per second [16]. For economic
reasons, these high speed channels are usually shared by many users. By installing
Wavelength add/drop multiplexers (WADMs) on both ends of a fiber, multiple low
rate digital channels can be multiplexed together into a high rate channel. In firstgeneration optical networks, every switch is equipped with a separate WADM for
each wavelength channel. All traffic is dropped when it arrives at the switch.
Currently, the WADMs are still expensive to install. In fact, they become the
dominant cost for a WDM network. In some cases, a wavelength channel may not
need to be dropped at a switch, simply because the switch is not the destination of
the channel traffic. It makes sense to allow such traffic to "bypass" the node without
termination. This eliminates the need for installing a WADM for the wavelength
channel at the switch. Not only does it help to reduce cost but it also improves
efficiency as it reduces unnecessary optoelectronic conversions. The above factors
motivate the development of the second-generation optical networks.
1.1.2
Second-Generation Optical Networks
In second-generationoptical networks, switches are equipped with optical components
to form the optical layer, which provides new functionalities. One of the most important functionalities is optical bypass. By employing optical crossconnect, a wavelength
channel at an input port of a node can be switched directly to the appropriate output port without undergoing optoelectronic conversion. Such a wavlength channel is
called a bypassing channel. On the other hand, if a channel needs to be terminated
9
at the node in order to add (or drop) traffic, a WADM for the channel is installed at
the node. Such a channel is called a terminating channel.
By installing a fixed number of WADMs at each node, the wavelength channels
can then be partitioned into two sets, the bypassing channels and the terminating
channels. If the WADMs are non-reconfigurable, the set of terminating wavelength
channels are fixed when the access node is installed. On the other hand, if the WADMs
are reconfigurable, the set of terminating channels can be altered when necessary. It
is more flexible to have a reconfigurable set of terminating channels. However, the
switch is more sophisticated and costly to implement.
Much effort has been invested on different second-generation WDM network architectures [8, 11, 20]. Today's architectures can be broadly classified into two categories,
the broadcast and select networks and the wavelength routed networks.
Broadcast and Select Networks
Broadcast and select networks are mainly designed for local and metropolitan areas.
In a broadcast and select network, all the stations are connected to a passive broadcast device. To transmit signal, the stations send signal to the broadcast device at
different wavelength channels. The broadcast device then broadcasts the signal to
all stations in the network. At the receiving end, the receiving station selects the
appropriate wavelength channel to listen. To avoid interference, two stations cannot
simultaneously transmit signal at the same channel. Therefore, the number of simultaneous transmissions is limited by the number of wavelength channels supported by
the network. This limits the scalability of broadcast and select networks.
Wavelength Routed Networks
Wavelength routed networks are primarily designed for communications in larger areas. Instead of broadcasting data to the stations all over the network, a connection
between two stations is established by setting up lightpaths from the source to the
destination. A lightpath is a route between two stations in the network. It supports
a connection between the two stations with a certain bandwidth. It is set up by allo10
cating a dedicated wavelength channel along all the links in the route. The station at
one end of the lightpath transmits signal, which propagates along the path, optically
bypasses all the intermediate nodes and is eventually dropped at the other end. A
lightpath usually provides very high bandwidth (a few Gb/s).
Wavelength routed networks using lightpaths allow spatial reuse. This means
stations can set up lightpaths using the same wavelength as long as the lightpaths
do not share a common physical link. Therefore, a wavelength routed network can
support simultaneous transmissions over the same wavelength. However, this requires
a more complicated Media Access Control mechanism, as it is important to make sure
no lightpaths using the same wavelength share a common link.
From the perspective of upper layers, a lightpath behaves like a physical link. In
fact, the lightpaths in a WDM wavelength routed network present a virtual topology
of point-to-point connections to the upper layers. If the virtual topology is static,
i.e., all the lightpaths are set up in advance and do not change over time, the upper
layer protocols can simply consider the lightpath connections as a physical network.
Figure 1-1 shows the physical topology and logical topology of a 2-wavelength
WDM network. In 1-1(a), the physical connections among the nodes are shown. In
this figure, lightpaths are set up among the nodes on the two wavelengths, A, and A2 .
Note that no two lightpaths with the same wavelength share a common physical link.
The nodes are logically connected by the lightpaths. This gives the logical topology
of the network, as shown in 1-1(b).
At the optical layer, an alternative to the lightpath service is the virtual circuit
service. Instead of allocating an entire lightpath for each request, the virtual circuit
service network can accept connection requests with lower rates. It provides circuitswitched connection using only part of the bandwidth in a wavelength channel. As a
result, several virtual circuits can share a wavelength channel over the same physical
link.
Compared to lightpath service, virtual circuits allow requests to give a more finegrained specification of their bandwidth requirements. This is especially useful for
access networks where the bandwidth requirement for each connection is small com11
-2
(b)
4
4
,
3
3
(b)
(a)
Figure 1-1: (a) The physical topology of a WDM network and the lightpaths set up.
(b) The logic topology of the network.
pared to the capacity of a lightpath. The multiplexing of a lightpath by many users
helps to reduce the cost of each user.
1.2
Previous Work
Previous research on WDM networks can broadly be divided into two categories based
on the type of networks under study. The two types of networks, as discussed above,
are broadcast and select networks and wavelength routed networks.
For broadcast and select networks, lots of prior research has focused on dynamic
reconfiguration policies. In those network models, the stations are equipped with
tunable transmitters and fixed receivers (TT-FR), or fixed transmitters and tunable
receivers (FT-TR). The tunable transmitters (or receivers) of a station can be dynamically configured to different wavelengths. Therefore, a station can make logical
connections with different stations at different time by tuning their transmitters (or
receivers).
Scheduling polices of the tunable components have been proposed for
providing high network connectivity [9], achieving load balancing [2], maximizing
throughput [15, 12], or providing Quality of Service guarantees [10].
For wavelength routed networks, previous work has focused on the routing and
wavelength assignment (RWA) problem [18, 1, 14]. In a standard RWA problem, the
physical topology of the WDM network is provided as input. The goal is to design a
virtual topology, given the traffic demand, in order to optimize certain performance
12
metrics (such as congestion and throughput) or to satisfy certain constraints (such as
delay).
There are also studies of the RWA problem on ring topology [13, 7, 17]. The goal of
these work is to maximize the number of request accepted or to minimize the number
of wavelengths required for certain class of traffic. There are also studies [6, 21, 3] that
focus on effective traffic grooming to reduce the number of WADMs required at the
nodes. The type of traffic in their models usually consists of fixed rate connections.
1.3
Contribution of this work
In this thesis, we propose a model on WDM ring networks where the number of
WADMs is fixed. We give an algorithm Least Load LL for multiplexing traffic to maximize the network throughput. While most of the previous research on WDM rings
focused on fix rate connections, we allow traffic to have variable rates. We compare
the performance of LL with the best performance achievable by any algorithm, and
prove that LL always performs at least 1 as good as any existing algorithm.
13
Chapter 2
The ONRAMP Architecture
The ONRAMP architecture (figure 2-1) is a hierarchical optical network architecture.
It consists of a feeder ring network, which is a wavelength routed ring network with a
few hub nodes and tens of access nodes physically connected by optical fibers. The hub
nodes connect the feeder network to the backbone networks, whereas the the access
nodes connect the feeder network to the local distribution networks. The access nodes
route traffic from the distribution networks to the backbone via the hub nodes.
Hub Node
Backbone
Access Node
End User
Network
Distribution
Networks
Figure 2-1: The ONRAMP Architecture
Each distribution network is a smaller local network, which can be a broadcast
14
and select network or a wavelength routed network. It carries data among the local
users. A user in a distribution network can communicate with other users inside the
same distribution network or outside the network. Transportation of traffic within
the same distribution network is handled by its own local routing mechanism. On
the other hand, traffic destined outside the distribution network has to be sent to the
feeder ring via the access node that connects the distribution network to the feeder
ring.
2.1
The Access Node Architecture
An access node connects a distribution network to the feeder ring network.
The
access node handles two types of traffic. First, it routes traffic entering or leaving the
distribution network. Second, it forwards bypassing traffic coming from the upstream
access node to the downstream access node in the feeder network.
In order to achieve the above functionalities, an ONRAMP access node employs
the wavelength partitioningtechnique. Each access node partitions the wavelength
channels into two sets, namely the distribution band and feeder band. The distribution
band contains the set of wavelengths dropped at the access node. These are the
wavelengths designated to add (or drop) traffic to the distribution network. On the
other hand, the wavelengths in the feeder band are switched optically at the access
node. No traffic can be added to (or dropped from) the feeder band channels. They
are used to support optical bypass of the feeder ring traffic.
Figure 2-2 illustrates a simple access node architecture. The WDM system supports two wavelength channels, A, and A2 . In the access node, one WADM is installed
to add (or drop) wavelength A2 . The other wavelength, A,, optically bypasses the
switch. Therefore, the distribution band of the access node is {A 2 } and the feeder
band is {A}.
Traffic carried by A2 is converted into electronic form and buffered.
The data will be routed to the distribution network if it is destined to the distribution network. Otherwise, the data will be added back to A2 , multiplexed with the
incoming distribution traffic.
15
traffic from
distribution network
traffic to
distribution network
Electronic Buffer
electronic data
electronic domain
...... .....
optical signal
W ADM -
X,
.
optical domain
XA
Figure 2-2: A Simple ONRAMP Access Node
Different designs for access nodes employing wavelength partitioning are possible.
Usually, such alternatives reflect a tradeoff between performance and complexity. For
instance, one design is to install a fixed set of non-reconfigurable WADMs at every
access node . With this implementation, the distribution band and feeder band are
fixed. On the other hand, if WADMs are reconfigurable, the access node can adapt
to the changing traffic demand by dynamically changing the distribution band and
feeder band. While the system with reconfigurable WADMs are more costly and
complicated, they provide the network with the flexibility to make adaptive changes.
Our work focuses on the wavelength partitioning problem on the feeder ring network. We will introduce a simple algorithm to determine the distribution band and
feeder band of each access node based on its traffic demand. For simplicity, we assume
all the feeder traffic is either sourced at or destined to the hub node. A connection
between two access nodes can be established by joining a source-hub connection and
a hub-destination connection.
16
Chapter 3
Model and Formulation of the
Wavelength Partitioning Problem
In the wavelength partitioning problem, we are given the traffic demand of the access
nodes in the WDM network. We allocate channel bandwidth for the access nodes so
that the traffic demand is satisfied. In effect, the distribution band at each access
node needs to be determined. The goal is to ensure high utilization of the network.
In the following section, we give a simplified model on the ONRAMP architecture.
Then we formulate the wavelength partitioning problem as an integer programming
problem, in which the objective is to maximize the throughput.
3.1
Architecture and Traffic Model
The feeder ring network consists of 1 hub node and N access nodes. The hub node
connects the ring network to the backbone.
Each access node is connected to a
distribution network, which requests to send certain aggregated traffic into the feeder
network. As discussed at the end of chapter 2, we assume all traffic is either sourced at
or destined to the hub node. In the bidirectional network, one direction is dedicated
to traffic to the hub and the other direction is dedicated to traffic from the hub. In
this thesis, we focus on one direction, the direction with the hub as the destination.
The traffic in the other direction can be handled similarly.
17
We model the traffic from the distribution networks by a traffic vector v =
[vi, v 2 , ... , vN]. The traffic demand vi specifies the amount of bandwidth to the hub
node requested by the distribution network of access node i.
.
The ring network supports W wavelength channels, 1, 2,.. . , W. Each channel has
a maximum capacity of B bits per second (bps). Different nodes can multiplex their
traffic in the same channel, as long as their aggregated traffic does not exceed the
channel capacity.
Each access node is equipped with a fixed number k E {1, .. ., W} of WADMs at its
output port. We call this number the termination degree of the node. The k WADMs
are configured to drop k different wavelengths. Therefore, their configurations specify
the distribution band at the access node. The access node can add its own distribution
traffic to the distribution channels. In effect, the nodes adding their traffic to the same
channel are sharing the channel. On the other hand, the traffic carried by the feeder
band channels will optically bypass the access node. No distribution traffic from the
access node can be added to these channels.
With the traffic model, our problem is to determine the distribution band for
each access node, (i.e., the set of k wavelength channels where the access nodes
adds its distribution traffic to,) and the amount of channel bandwidth allocated for
each access node's traffic, subject to the channel bandwidth constraint. Our goal
is to maximize the throughput, which is the total amount of traffic assigned to the
wavelength channels.
Formulation of the problem
3.2
The throughput maximization problem can be formulated as an integer programming
problem.
let v
=
For a WDM ring network with N nodes and W wavelength channels,
[vi, v2 ,.
.. , VN]
be the traffic vector of the N access nodes. Let k be the
termination degree of each access node. The problem can be formulated as follows:
e Input: The number of nodes N, the number of wavelength channels W, the
channel bandwidth capacity B, the termination degree k
18
C {1,
...
, W},
and the
traffic vector v = [v 1 , v 2 ,.
"
. . , VN]-
Output: A N x W allocation matrix C and a N x W distribution band matrix P. The allocation matrix specifies the amount of distribution traffic each
access node adds to each channel. The distribution band matrix specifies the
distribution band of each node.
N
"
W
Objective Function: Maximize E E cij.
i=1 j=1
" Constraints:
w
Epi_< k
j=1
Vi
(3.1)
(3.2)
pij E {O, 1}
N
Zci < B
Vj
(3.3)
i:=1
c =0
if pij =
(3.4)
W
j=1
cij < vi
Vi
(3.5)
In the above formulation, the matrix P defines the distribution band for each
node. For any node i, a channel j is in its distribution band if and only if pij
=
1.
Constraint 3.1 restricts the size of each node's distribution band to at most k.
The matrix C gives the amount of bandwidth of each channel allocated for each
access node. Each element cij is the amount of traffic access node i adds to channel
j. Constraint 3.3 specifies the total amount of bandwidth allocated on each channel
does not exceed the capacity of the channel.
Clearly, the matrices P and C are closely related. In particular, if a channel j is
not in the distribution band of node i, the node cannot add any distribution traffic
to the channel. Constraint 3.4 specifies this restriction.
Finally, constraint 3.5 specifies that the total bandwidth allocated to a node does
not exceed the traffic demand of the node.
19
Given the meaning of the allocation matrix, our objective is to maximize the total
amount of channel bandwidth allocated to all the nodes. This is the amount of traffic
carried to the hub node from all access nodes. We call this quantity throughput of the
network.
In the remaining of this chapter, we show that the maximization problem is NPhard.
Therefore, it is computationally hard to find the optimal configuration for
maximum throughput. In the next chapter, we propose an approximation algorithm
Least Load (LL) that guarantees to achieve at least 1 of the optimal throughput
performance.
3.3
Complexity
The maximization problem can be formulated as a decision problem. The question
to be asked will become: "Given an instance of the wavelength partitioning problem,
is it possible to find a configuration achieving throughput T?"
We argue that this decision problem is NP-hard. In fact, it is a generalized version
of the NP-complete MultiprocessorScheduling Problem (MSP) [5], which is formulated
as follows:
Multiprocessor Scheduling Problem
Instance: A finite set of tasks T, with size s(t) for each task t, the number of
processors m, and a deadline D.
Problem: Can T be partitioned into m subsets T 1 ,. . . , Tm, such that
E s(t) < D
Vi,
tETi
that is, each processor handles a subset of tasks and all the processors meet the
deadline?
In the context of our throughput maximization problem, the access nodes are the
jobs. The traffic vector v defines the job sizes. The number of wavelengths W is the
20
number of processors. The bandwidth capacity B is the deadline D. The termination
degree is simply 1, because the jobs cannot be shared by more than one processor.
We can now translate the multiprocessor scheduling problem into the wavelength
partitioning problem: "Given W wavelength channels and N nodes with traffic vector
v, is it possible to assign the traffic at each node to a wavelength channel such that
all the nodes' traffic demand is satisfied with no bandwidth capacity violated?" If
we are able to solve this problem in polynomial time, we would be able to solve the
MSP in polynomial time as well. Therefore, the wavelength partitioning problem is
NP-hard.
Since it is hard to find an optimal solution for the problem, we focus on efficient
approximation algorithms instead. In the next chapter, we give a polynomial time
approximation algorithm Least Load (LL) which guarantees to achieve at least
the optimal throughput performance.
21
34
of
Chapter 4
The Least Load Algorithm
We propose a simple algorithm Least Load (LL) to control the wavelength partitioning
at each node. In section 4.1, we first define some terminology to help describe the
algorithm LL. Then in section 4.2, we describe LL and analyze its complexity. In
section 4.3, we prove the performance guarantee of the algorithm.
4.1
Terminology
To make our description of the algorithm easier, we first define some terminology:
1. The utilization of a wavelength channel
j
is the amount of traffic assigned to
N
the channel, i.e., Zcij.
i=1
2. A wavelength channel
j
is saturated if its utilization reaches its maximum ca-
N
pacity, i.e., Zcij = B.
i=1
3. The residual traffic of an access node i is the amount of its distribution traffic
w
cij.
not assigned to the channels, i.e., vi -
Z
j=1
W
4. An access node i is satisfied if it has no residual traffic, i.e., E cij = vi.
j=1
5. The distribution band of an access node i is complete if the distribution band
W
consists of k wavelength channels, i.e., E pij = k.
j=1
22
4.2
Description
In this section, we first give a high level description of the algorithm LL. Then we will
describe the algorithm in greater detail in subsection 4.2.2. After that, we analyze
its complexity in subsection 4.2.3.
High Level Description
4.2.1
For each access node, we need to select channels for the node's distribution band and
determines the amount of traffic assigned to each of these channels. LL repeatedly
picks a node, selects new a wavelength channel for its distribution band, and allocates
bandwidth of the channel to the node's traffic. It proceeds until either the distribution
band of each node is complete, all nodes are satisfied, or all channels are saturated.
For each iteration, LL arbitrarily picks a node whose distribution band is not
complete. Then it greedily selects a wavelength channel for the node's distribution
band. Specifically, LL selects the least utilized wavelength for the distribution band.
The algorithm allocates as much channel bandwidth as possible without violating the
constraints.
4.2.2
Detailed Description
The algorithm is based on a primitive operation PUSH, which we define as follows:
9 PUSH(i, j): Assign as much traffic at node i as possible to wavelength channel
j. Specifically, the amount of traffic assigned is either the residual traffic at
node i or the available bandwidth at channel
j,
whichever is less. In our matrix
N
W
notation, this operation increases cij until E
k=1
Cik
= vi or E
Ckj =
B. Besides,
k=1
it sets pij +- 1.
Throughout the computation, the algorithm maintains 3 sets of variables. For
each node i, the variable di counts the number of channels currently selected for the
distribution band. The variable ai keeps the amount of channel bandwidth currently
allocated for node i. Lastly, the variable uj keeps track of the utilization of channel
23
j. Initially, all variables (including cij and pij) are set to 0. The following gives a
pseudocode for the algorithm LL.
9 do until either all nodes are satisfied, all wavelength channels are saturated, or
all nodes' distribution bands are full.
- Pick any unsatisfied node i whose distribution band is not complete.
- Find
j
such that uj is the minimum, breaking ties arbitrarily.
- Do PUSH(i, j).
- Update di, ai and uj.
4.2.3
Complexity
In the algorithm, the loop iterates for at most Nk times. This is because in each
iteration, the algorithm adds a wavelength to the distribution band for a node. After
k wavelengths have been selected for a node, the node will never be considered again.
Therefore, each node is considered for at most k times. It follows that the total
number of iterations is O(Nk).
In each iteration, the algorithm can find an arbitrary unsatisfied node with incomplete distribution band in 0(1) time by keeping a simple list structure. The list
contains all the unsatisfied nodes with incomplete distribution band. Initially, all
the nodes are inserted into the list. At the beginning of each iteration, LL picks the
first element from the list as the selected node i. At the end of each iteration, the
algorithm updates the variables di and a2 for node i. It checks whether the node
is satisfied (i.e., is ai = vi?) and the node's distribution band is complete (i.e., is
di = k?). If both cases are not true, it puts the node at the end of the list for future
consideration. Otherwise, the node is deleted from the list and will not be considered
again. Therefore, in each iteration, selection and insertion/deletion of the node takes
0(1) steps. Updating and checking the variables di and a2 takes 0(1) steps as well.
Using the binary heap data structure [4] to maintain the channel utilization uj,
finding the least utilized channel and updating the variable uj in each iteration can be
24
done in O(log W) time. Therefore, the running time for each iteration takes O(log W)
time. The overall complexity of the algorithm is O(Nk log W).
Though not optimal, LL provides some performance guarantee on the output, as
specified by the following theorem. We will prove the theorem in the following section.
N
W
Theorem 1 For any allocation matrix C, let f(C) = E E cij be the throughput
i=1 j=1
given by the matrix. Given an instance I of the wavelength partitioningproblem, let
C* be the optimal allocation matrix and CpL be the allocation matrix given by LL.
Then for any instance I of the problem, f(CLL) ;> If(Cj ).
4.3
Proof of Performance Gaurantee
First note that a node can add at most kB units of traffic to the feeder network because
it only terminates the k channels in its distribution band. If the traffic demand from
an access node is greater than kB, the excess amount cannot be handled by any
algorithms, including the optimal algorithm. In this sense, the excess traffic demand
is not interesting to our wavelength partitioning problem. Therefore, we have the
following assumption on the traffic demand.
Assumption 1 The traffic demand vi at every access node i is at most kB.
In the following subsection, we first define some notations for the proof. In subsection 4.3.2, we explain intuitively why LL performs well. Then we present a formal
proof of the performance guarantee of LL in subsection 4.3.3.
4.3.1
Notation
Given an output from LL, we consider a pair of values ri and si for each node i. The
w
variable ri is the residual traffic of node i, i.e., ri = vi - E cij. The variable si is an
j=1
indicator variable specifying whether node i is unsatified. We define si = 1 if node i
is unsatisfied and si = 0 otherwise.
25
Let r = max{ri, r 2 , ...
,
rn} and s =
Zi si.
Intuitively, r is the maximum residual
traffic among all the nodes and s is the number of unsatisfied nodes.
Note that if si = 1, node i is unsatisfied. This happens only if all channels are
saturated or the distribution band of node i is complete. In the first case, all channels
are saturated in the output given by LL. Therefore, in this case LL has achieved the
maximum throughput possible.
Hereafter, whenever a node i is unsatisfied (i.e., si = 1), we always assume the
second case. In other words, node i has sent as much traffic as possible via its k
distribution channels. But the available bandwidth in those channels is not enough
to satisfy all the traffic at node i.
In this case, every wavelength in the distribution band of node i is saturated.
As a result, each unsatisfied node corresponds to a set of k saturated wavelengths.
Note that for any two unsatisfied nodes, their distribution bands are disjoint. This
is because when a wavelength is first selected for one of the nodes, the algorithm LL
will saturate the wavelength. Therefore, the wavelength will never be selected for the
other node. As a result, if there are s unsatisfied nodes in the output given by LL,
there are at least ks saturated wavelengths.
4.3.2
Intuitive Argument
We first explain intuitively how the values of r and s relate to the throughput of LL.
We argue that, for any input, the throughput given by LL's output will not be "very
low" compared with the optimal solution. We categorize LL's outputs according to
the values of r and s in the outputs. We argue that in any of the following 3 cases,
LL's output will not be "too bad".
s large : If s is large, ks is large. Therefore, there is a large number of saturated
channels. As a result, the throughput is high.
r large :Assume node i has residual traffic equals r. When the last channel is selected
for the distribution band of node i, this channel is the least utilized at that
moment. If r is large, from assumption (1) it implies only little traffic at the
26
node is pushed to the channel. Therefore, the utilization of that channel is high
at that moment. This implies utilization of all channels is high. As a result,
the throughput is high.
r and s small : If both r and s are small, the amount of residual traffic, which is at
most rs, is small. This means most of the traffic demand is satisfied. Therefore,
the throughput given by LL would not be much worse than the optimal solution.
Therefore, for any input, LL will not perform much worse than the optimal algorithm. Now we formalize the above argument mathematically.
4.3.3
Formal Proof
Suppose node i is a node with the maximum residual traffic r. Let {A,
,..
,
, Ak} be
the distribution band of node i, and assume the channels were selected in the above
order. (i.e., A, is first selected for the distribution band of node i, followed by A2
and so on.) For each distribution band channel Aj, the value of ci, is the amount of
traffic at node i assigned to channel Aj. For a cleaner notation, we let aj = ca Since LL always chooses the least utilized channel and tries to saturate the channel
by a PUSH operation, it follows that a,
a2
> ak. Also, since the total traffic
...
demand at node i is the sum of its residual traffic and its assigned traffic, we have
k
vi = ri + E aj. It follows that
j=1
k
Vi - ri
a
(4.1)
kak.
(4.2)
=E
j=1
>
Now we can give a lower bound of ak in terms of B, r and k:
ak
<
vi - r
k
kB- r
k
27
(4-3)
(4.4)
k
Therefore, when LL chooses Ak for node i, the utilization of Ak equals B -
(4.5)
ak >
Since it is the least utilized channel at that moment, we know that the utilization of
each channel is at least
.
Also, since there are s unsatisfied nodes, there are ks saturated wavelengths.
In conclusion, when the algorithm terminates, there are at least ks saturated wavelengths; and the utilization of other wavelengths is at least E. Let TLL be the throughput given by LL, we have:
TLL
>
=
=
where a and
#
are defined as
ksB+(W - ks).k
sr
r
ks
WB - (
+
W )
kB
WB
W
WB - (a + - a#),
(4.6)
(4.7)
(4.8)
and ' respectively.
This gives a lower bound on the throughput performance of LL. Next, we give an
upper bound on the optimal throughput.
The optimal throughput, denoted by T*, is bounded above by the total traffic
N
demand V =
vi. The total traffic demand is also the sum of throughput give by
LL, which is T"; and the total amount of residual traffic given by LL, denoted by R".
Note that the total amount of residual traffic is at most rb. The following equation
gives an upper bound of the total traffic demand.
N
V
(4.9)
=
Zvi
-
TLL + RLL
(4.10)
<
TLL + rb
(4.11)
Also, since all traffic has to pass through the link connecting the hub node (figure 428
1), the optimal throughput cannot exceed the link's bandwidth capacity, which is
WB. Therefore, the optimal throughput is at most min(WB, V).
Hub Node
Hub Link
Access Node
Figure 4-1: All unidirectional traffic destined to the hub must pass through the hub
link.
Now, let I := 7
be the performance ratio of LL to the optimal algorithm, it
follows that
1 >
TLL
V
m(
-min(W B,V )
>
=
TLL(4.13)
mmn(WB, TLL + rb)
TLL
TLL
max(B, TLL
).
WB' TLL + rb
(4.12)
(4.14)
First, from equation (4.8),
TLL
-- >
WB
=
For the second term,
a +/3 - Y#(41
1 - (1 -i)(1 -3).
(4.15)
(4.16)
TLL
TLL+rb
TLL
TLL + rb
rb
=
TLL + rb
29
(4.17)
> 1
-W
B(a +,3 - ceo) + rb
(4.18)
rb
=
1 -
WB
1 --
Q+/%-/
- -(a +
1
a/3
=a +0
1(4.19)
-a,3) + I
+ 1
(4.20)
(4.21)
=1
-
(4.22)
a +0
1 - (1 - a)(1-/#)
a+
(4.23)
Therefore, from (4.16) and (4.23),
1
=
Since
>
and
(4.24)
).
max(1- (1 - a)(1-3),
r , it follows that 0 < a,
<3 1. This is because ks is
the number of saturated wavelengths, which never exceeds W. The variable r is the
residual traffic at a node, which never exceeds the total traffic demand at the node.
By assumpution 1, this number is at most kB. As a result, both a and 3 lie within
0 and 1.
It follows that the two terms on the right hand side of equation (4.24), 1 - (1 a)(1 - 3) and
1
-l$~/
), are both continuous and differentiable [19] in the domain
of our interest. Since the two terms change in the opposite direction when a or
#
changes, the performance ratio 1 may reach its minimum only at the boundaries (i.e.,
a, /=0 or 1) or when the two terms are the same. We check the value of 1 at each of
these cases.
First, we check the values of 1 at the boundaries. When a or 3 is 1, the performance
ratio 1
=
1.
When a or / approaches 0, it also approaches 1.
Intuitively, the
throughput given by LL approaches the optimum when a or /3 approaches 0 or 1.
This is consistent with the meaning of a and
/.
In the case with a
=
0, the number
of unsatisfied nodes s is also 0, since a = g. That means all nodes have been satisfied;
30
so LL has achieved the optimum throughput. On the other hand, if a
of unsatisfied nodes s equals 1.
=
1, the number
In this case, all the wavelengths are saturated as the
number of saturated wavelengths is at least ks = W. Thus the throughput reaches
the maximum capacity of the network.
Similarly, if 0
=
satisfied. If 0 = 1, r
0, r
=
=
0, and there is no residual traffic. All traffic demand is
kB. That means there is a node which does not receive any
bandwidth allocation at all. This can happen only if all the wavelength channels are
saturated before the node is ever considered. In this case, the throughput reached
the maximum capacity of the network.
Next, assuming a, 3 = 0,1, we check the relationship between a and 3 in the
case where the first term, 1 - (1 - a)(1 - /) (denoted by ti), equals the second term,
1-(1-0a1-)
(denoted by t 2 ). First note that if a, # L 0, 1, the term 1-(1-a)(1-) #
a+,3
0. It follows that
a+0a+/3=+
=
a+#3
(1 - (1 - a)(1 - 0))
a
1 - (I - a)(1 - 0)
1
1_ ( _a)(-_ ) -(1 - (1 - a)(1 - 0))
t2
(4.25)
(4.26)
(4.27)
tl
(4.28)
=1.
As a result, when the two terms are equal, a +
(1
-
a)(1
-/3)
/
= 1.
In this case, 1 = 1 -
=1- (1 - a)a; and the function reaches its minimum when a
1
=
2.
Therefore,
>
1
1
2 2
=-(4.30)
4
U
31
(4.29)
This means the throughput performance of LL is at least 14 of the optimal algorithm.
32
Chapter 5
Conclusion
Wavelength partitioning is the technique used in ONRAMP architectures as a way
to reduce cost and enable multiplexing of user traffic. In theory, the wavelength
partitioning problem itself is interesting. Even with the simple model proposed in
this thesis, the problem is shown to be NP-hard.
It is conceivable that a more
realistic/complicated model may involve even more interesting algorithmic issues.
A plausible way to extend the model in this thesis is to allow dynamic traffic.
Since real traffic is changing from time to time, it would be useful to capture this
characteristic into our model. This would certainly bring up new questions such as:
How do we find the "best" configuration given a certain changing traffic pattern?
If the network is equipped with configurable WADMs, how should the WADMs be
configured in order to adapt to the changing traffic? The problem of minimizing
receiver tuning on dynamic traffic was studied for broadcast-and-select networks [2].
We believe a similar problem for the wavelength routed ring network would be an
interesting and challenging topic to work on.
33
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