Joint Traffic Engineering for IP over WDM Networks
Jinhan Song and Saewoong Bahk
School of Electrical Engineering, Seoul National University, Seoul, Korea
sbahk@netlab.snu.ac.kr
The architecture of next generation high speed networks is expected to consist of IP and WDM layers. IP layer
runs packet switching while WDM layer uses circuit switching. While they have the similar goal of efficient use of
network resources, they lack harmony due to different traffic engineering capabilities. To this end, we introduce a
combined cost to quantitatively analyze the effectiveness of traffic engineering for the two layers. First, we model
an edge router and devise a traffic engineering mechanism to minimize the combined cost over a single path. We
show that both the explicit price which characterizes the traffic engineering cost, and the implicit price which
determines the network congestion cost, play a key role in wavelength allocation. We consider wavelength
granularity that significantly affects overall performance when the number of wavelengths in a link is small. To
solve this granularity problem we propose an approximation algorithm that finds an integer solution from the real
solution. Through simulations, we verify that our proposed algorithm achieves near-optimal performance with
greatly reduced complexity and guarantees the feasibility of solutions.
OCIS codes: 060.4250
1.
Introduction
The universality of IP (Internet Protocol) places the Internet at the core of network convergence. As Internet traffic
volume increases explosively, WDM (Wavelength Division Multiplexing) is spotlighted as the first and foremost
candidate to meet the bandwidth demand since it increases the network capacity by orders of magnitude. As a
consequence, IP over WDM is expected to be the dominant architecture in future Internet backbone networks. Fig. 1
shows the IP over WDM network architecture considered in the IPO WG (IP over Optical Working Group) of the IETF
(Internet Engineering Task Force) [1], [2]. Router networks use optical subnetworks as a means for transporting data. So
the router networks set up lightpaths for communications across optical networks. A lightpath works like a circuit formed
by cascading multiple wavelength channels.
Traffic engineering is concerned with “performance optimization of operational networks” [5]. In the IP layer, it is
necessary to overcome the limitations of current IP routing protocols, which typically select a shortest path, i.e.,
minimum-hop path and often produce hot spots [5]. To avoid these limitations, traffic engineering tries to use diverse
routes based on metrics other than hop count and distributes the traffic load throughout the network. Explicit routing
implemented in MPLS (Multi Protocol Label Switching) seems a viable solution to this end and MP  S (Multi Protocol
Lambda Switching) is an approach to introduce MPLS into the optical domain [6]. In the WDM layer, the need for traffic
engineering capability grows as the number of wavelengths available increases. By using the greatly increased
wavelengths in number, we can allocate multiple wavelengths for a lightpath or form a virtual topology with denser
connectivity. Typical traffic engineering methods include virtual topology construction, traffic grooming, and dynamic
RWA (Routing and Wavelength Assignment) [3], [4].
A natural question arises as to whether existing traffic engineering techniques for IP over WDM networks accomplish
their goals as desired. The use of contradictory assumptions by the various methods presents a problem. In IP traffic
engineering, it is usually assumed that the topology is predetermined and given. In WDM traffic engineering such as
RWA, lightpaths are given as a part of the input. Hence, before IP and WDM traffic engineering techniques are applied,
the set of lightpaths is determined in advance. The lightpath determination is the core of the joint traffic engineering and
deals with the problems of 1) deciding the number of allocated wavelengths for each lightpath and 2) selecting the set of
lightpaths among the feasible ones. Specifically, we focus on the former one, computing the optimal number of
wavelengths for each lightpath by considering the combined costs of the two layers. The latter can be classified as a
problem of virtual topology construction which will be addressed in a separate paper.
Our joint traffic engineering framework for IP over WDM networks starts with characterizing and modeling an IP over
WDM network. Then we define an objective function considering both IP and WDM layers, i.e., their combined cost.
Our motivation of using the combined cost comes from the contradictory objectives of IP and WDM layers though they
have the same design goals of efficient use of network resource such as minimizing the network configuration cost and
maximizing profit. The combined cost is represented as the sum of the costs incurred by the two layers to transport a
given traffic. We present a mathematical formulation for a general case and provide its solution. In our previous work,
we obtained some basic results on this topic [7], [8]. However, the analysis was for a simple system model and the
formulation was confined to a single hop network. In this paper, we complete our work to deal with general multi hop
networks, and explore previously uncovered aspects of traffic engineering for IP over WDM networks.
The rest of the paper is organized as follows. We present the basis for the quantitative analysis in Section 2, and
provide the formulation and its solution. We describe and explicate the properties of the multi-hop model defined in
Section 3. We discuss granularity issues in Section 4, followed by concluding remarks in Section 5.
2.
Combined Cost Concept
In an IP over WDM network, the IP layer runs on top of the WDM layer. To model this network, we start by
characterizing both layers. We present a model of an edge router in router networks that interfaces with optical networks.
Then, we introduce the combined cost concept.
2.A.
IP and WDM layers
The IP network uses packet switching and generates elastic traffic, which adapts to the network state and often results in
the burst of IP packets. In contrast, the WDM network employs circuit switching to form lightpaths that consist of one or
more wavelengths. As wavelengths are allocated in integer units, there arises a granularity problem. In this paper, we
assume that wavelength conversion is possible at each node.
2.B.
Edge routers
An edge router is placed in router networks and interfaces with optical networks. It is equipped with signaling capability
and an IP and WDM protocol stack. Through the signaling protocol, it requests the setup and release of a lightpath.
Before doing this, it calculates the number of wavelengths that are needed. We will deal with the decision process in the
next section.
Fig. 2 shows the structure of the edge router where

is the packet arrival rate and
 is the average packet service
rate which is the inverse of unit transmission speed of a wavelength. When a packet arrives at the edge router, the
Classifier examines the packet header and determines the session it belongs to. We will use the term “session” to refer to
a lightpath between a source-destination edge router pair. As a lightpath can require multiple wavelengths, the Distributor
determines which wavelength to use for forwarding the incoming packet. Depending on the distribution algorithm, the
wavelength selection may be different.
In Fig. 2, session 1 identified by forwarding equivalence class (FEC) 1 uses two wavelengths (k=2). The distribution
algorithm chooses one of these two for forwarding each packet for FEC 1. Session N does not need a distribution
algorithm for wavelength selection because it has only one wavelength (k=1). We consider two distribution algorithms:
randomization and metering [9]. While the randomization algorithm distributes an incoming packet over the allocated
wavelengths for the corresponding session in a random manner, the metering algorithm directs an incoming packet to the
wavelength with the least queue length. The queues shown in Fig. 2 store incoming packets to convert their format for
the optical network. The packets are then transmitted over the allocated wavelengths. Our considered distribution
algorithms cannot guarantee in-order delivery, so end-hosts need to execute the reordering process. If in-order packet
delivery is in need, we have to adopt some other distribution algorithms such as 802.3ad link aggregation algorithm. The
purpose of our work is in providing a framework for traffic engineering which can be modeled simply, instead of
designing a detailed traffic engineering algorithm.
We can view IP traffic engineering as a modulation mechanism of incoming packets that selects an appropriate
delivery route, and WDM traffic engineering as a control scheme of the total service rate that adjusts the number of
allocated wavelengths. For ease of analysis, we fix the total packet arrival rate, and manipulate only the number of
wavelengths allocated by WDM traffic engineering. To determine the number of allocated wavelengths, we need to
consider a couple of factors. First, more wavelengths are required for bursty traffic than for smooth traffic since bursty
traffic experiences higher packet loss. Second, the resource allocation process should be fair and efficient. To
quantitatively analyze the costs involved in the decision process, we propose a combined cost next.
2.C.
Combined cost
Both the IP and WDM layers have their own traffic engineering algorithms and corresponding goals. A natural question
is how to harmonize them. We can achieve our goal by minimizing the detrimental interference between the independent
but functionally overlapping capabilities of the two layers. Therefore our previous work proposed the notion of a
combined cost that quantifies the costs at both layers for transporting user traffic [7], [8].
The cost in the IP layer increases as the queue builds up. This is because delay and loss probability of packets increase
as the queue grows. The cost in the WDM layer can be characterized by the number of allocated wavelengths since the
wavelength is a premium commodity. So we define the combined cost as the sum of the costs in the IP layer and in the
WDM layer laid out to handle the given traffic. We represent the combined cost C as
C  pb N IP  p NWDM ,
(1)
N IP is the number of buffered packets, NWDM is the number of allocated wavelengths , pb is the unit price per
buffered packet and p is the unit price per allocated wavelength. N IP and NWDM have different scales and they are
consolidated by using pb and p that are referred as the explicit prices. This motivated us to introduce the cost
where
concept.
The price in the IP layer is proportional to the queue length and the price in the WDM layer is proportional to the
number of allocated wavelengths. These quantities influence each other since the queue length closely depends on the
number of allocated wavelengths. In particular, they exhibit a tradeoff relationship. That is, given the explicit price,
decreasing
N IP results in an increase of NWDM , and increasing (i.e., allowing larger) N IP in smaller NWDM . This
is an intuitive relationship which will be explained next.
2.D.
Analytic modeling
Different distribution algorithms result in different queue behaviors. To model the bursty nature of IP traffic, we use a
bulk arrival model, M [ B ] / M /1 [10]. B is the bulk size and characterizes the burstiness of traffic. The larger B implies
the burstier traffic.
The system that employs randomization is equivalent to k independent queues [9]. When k wavelengths are allocated
for a session with the offered load r, each wavelength is responsible for serving the offered load of
r
k
 1k   . Assuming
an exponentially distributed service time, the average queue length is given by B  1 r . The total number of buffered
2 k r
packets for a session is N IP  k B  1 r .
2 k r
With NWDM  k , we get
the combined cost C  pb B  1 kr  p k . It
2
k r
2
can be easily shown that the combined cost has the global minimum
Cmin  pb
B 1 
2p 
1 
 r
2 
B  1 
at

B  1  where p  p .
k *  NWDM  1 
r
pb
2 p 

p determines which layer plays the key role in handling the traffic. By varying p, we can check the tradeoff relation
between IP and WDM layers. The system with metering is equivalent to a queue with k servers. To simplify the analysis,
we approximate this to a queue with a server of k times the unit speed. The approximation is valid since we can observe
that the number of wavelengths minimizing the combined cost in the approximated and the original models are about the
same [9]. Table 1 summarizes the analytic results. As B increases, the minimum combined cost and the number of
required wavelengths also increase as expected. Engineering the bursty traffic requires more resources and higher cost.
2.E.
Total combined cost
Now we extend the above definition for multiple sessions to represent the sum of individual combined costs of all the
sessions in the network. Assuming that the individual combined costs are summable, we write the total cost as follows.
Ctotal 
C
f F
f
.
(2)
C f is the combined cost for session f and F is the set of all the sessions under consideration.
3.
Minimization of the total combined cost
Minimizing the total combined cost is not so simple because some session requests may be blocked due to limited
network capacity. Some other sessions may not receive as many wavelengths as they request. In this case, wavelengths
should be shared among the sessions. Here we solve the problem of determining the number of wavelengths to minimize
the total combined cost. We assume that the wavelength conversion is possible at each node.
3.A.
Formulation
We formulate the problem of finding the optimal number of wavelengths by minimizing the total combined cost. It is
given as follows.
Objective
Minimize Ctotal 
C
f F
f
(3)
Subject to

f F ( l )
k f   l for all links l  L ,
k f  rf for all sessions f  F ,
k f : integer,
(4)
(5)
(6)
pb , p  0 : constant.
L is the set of optical links, F (l ) is the set of sessions that pass link l and
(7)
 l is the number of wavelengths on link l.
We assume that the load offered r f to session f is given and normalized by the transmission rate of a wavelength. (4)
states that the sum of allocated wavelengths for each session, k f , should not exceed the number of wavelengths of each
link. (5) means that k f should exceed r f for stability. This constraint prevents the cost of IP layer from diverging to
infinity. (6) and (7) state that k f is an integer number and that the explicit prices are constant .
As the objective function is nonlinear, we need to apply nonlinear programming techniques. Since nonlinear
programming with integer constraints is computationally hard, we ignore the integer constraints for the time being.
Without the integer constraints, the number of allocated wavelengths k f becomes a positive real variable. C and

are the functions F : R  R

N IP
depending on k f . To find the solution, we first derive the dual version of the
problem, and then apply the Kuhn Tucker theorem and the gradient projection technique [9], [11]. The resulting iterative
equations are as follows.
<Gradient Projection >
for every
l L,
 l (t  1)  [ l (t )  
D( )

 l
<Kuhn Tucker condition>

f 'F ( l )
D( ) 
]
 l
k f '  l .
(8)
(9)
for every f  F ,
f 

l 'L ( f )
(10)
l'
1
k f  C 'f ( f ) .
(11)
GP (Gradient Projection) decides the feedback information from the optical links and the KT (Kuhn Tucker) condition
determines the behavior at edge routers [11]. (8) and (9) are obtained by applying the gradient projection method to the
dual form of the original formulation. In (8) and (9),
l
is the Lagrange multiplier,
 is the step size, and D( ) is
the objective function of the dual formulation. (8) and (9) describe the procedures to update the Lagrange multiplier from
the given wavelength allocations, k f ’s. In (10), L( f ) is the set of links that a given session f passes through, and
f
is the sum of the Lagrange multipliers for those links. The Lagrange multiplier is positive if the constraint is active,
i.e., equality holds in (4). If the constraint is inactive, or the strong inequality holds, the Lagrange multiplier is zero. This
means that as the link is congested and becomes a bottleneck, the Lagrange multiplier value increases. This property of
the Lagrange multiplier makes it translatable as price [12][13]. (11) shows that
for bottleneck links, determines the number of allocated wavelengths. We call
f
f
, the sum of the Lagrange multipliers
the implicit price. The implicit price
and the explicit price are closely related. To investigate this relation, we consider a session f and apply partial derivation
to (2),
C f (k f )
k f
 pb
N IP (k f )
k f
 p
. (12)
From the KT condition (11)
C f (k f )
k f
  f .
(13)
Combining the above yields,
1
kf 
p  f
N IP
( 
).
k f
pb
(14)
(14) states that the number of allocated wavelengths is determined by the sum of the explicit price of the WDM layer and
the implicit price scaled by the explicit price of the IP layer. The implicit price as well as the explicit price significantly
influences wavelength allocation.
Now we discuss the implications and properties of these prices. First, the implicit price is compatible with the explicit
price and they have the same unit. An increase in the implicit price is equivalent to an increase in
p or the wavelength
price. Thus increase in the implicit price discourages wavelength allocation.
Second, increase in the implicit price causes increase of the IP layer cost and less wavelength allocation, resulting in
more buffer usage. The implicit price is not the actual wavelength price but reflects the relative wavelength value, so it is
not directly included in the calculation of WDM layer cost.
Third, the implicit price reflects network congestion. As stated before, the Lagrange multiplier of a link is positive
when wavelength requests to the link exceed the link capacity. Thus the positive Lagrange multiplier indicates that the
link is congested. The session with the positive implicit price goes through some congested links.
Fourth, the implicit price represents social penalty. As a session that passes through more congested links causes even
more congestion of the network, it is given a higher implicit price which discourages the wavelength allocation. The
session passing through congested links will get the penalty of fewer wavelengths.
Fifth, the implicit price conforms to the economic law of demand, stating that an increase in price results in a decrease
of demand. This applies to the case of the implicit price.
Sixth, the implicit price enforces fairness. As wavelengths are still a scarce resource, they need to be shared among
sessions. From (14), we see that the implicit price encourages link sharing. Sessions that pass through a certain congested
link have the same Lagrange multiplier. We call this “the price fairness.”
Last, the implicit price can be used for billing. Kelly called the sum of Lagrange multipliers as the shadow price and
proposed to use it as the basis of pricing [12], [13]. Its objective is represented as a nondecreasing utility function and
tends to make links more congested [14]. However, if a user can modify the utility function not to fill up the links, he or
she can use them free of charge. This is because the shadow price is positive only at the bottleneck link. The proposed
combined cost model suggests a more realistic billing model. Users know the entrance fee or the explicit price before
they access the network. Then they optimize the allocation by minimizing the combined cost. Once they are admitted
into the network, they need to pay additional money amounting to the implicit price during the busy hours. The behaviors
of users and the network can be explained by (8) and (11), respectively.
Let us point out a couple of items to refine our formulation. First, we have ignored the integer constraints. To enforce
the constraints, we need to take some additional steps. We will discuss this topic in the next section. Second, we assumed
that the route is predetermined. If the network uses dynamic routing, it can obtain additional performance gains [15].
Last, the formulation does not consider the wavelength continuity constraints since we assumed that the wavelength
conversion is possible at every node [4]. If we count the wavelength constraints, some allocations obtained from the
formulation can become infeasible and the utilization of wavelengths can be decreased [16]. The process of selecting
routes and wavelengths is called RWA (Routing and Wavelength Assignment). RWA algorithm affects the overall
performance, and devising an appropriate one that considers the combined cost is beyond the scope of this paper.
3.B.
Numerical analysis
Now we perform the numerical analysis for the topology in Fig. 3.
3.B.1 Single bottleneck case
There are 3 sessions and their ratio of the offered loads is 4:2:1. Each link has 100 wavelengths. We assume that each
session is subject to use the traffic distribution of randomization. We set B=1 and increase the traffic load of each session
until Link 2 begins to be congested, while maintaining the same load ratio. Fig. 4 plots the offered load of session 1
versus the number of allocated wavelengths and the implicit prices, respectively. When the offered load of session 1 is
less than 45, the links are not congested and the implicit prices of the two links are zero. If the implicit prices are zero,
each session occupies the wavelengths according to the explicit price which is proportional to the offered load.
Once Link 1 is congested, its implicit price begins to increase. If no more wavelengths are left for allocation at the
bottleneck link, the number of allocated wavelengths will be fixed even if the offered load increases. The increased
implicit price indicates that more allocation is not possible.
We can find the closed form solution for this simple topology. We write the solution as follows if Link 2 is not congested.
ki  min[(

1
 1), i ]  ri , i=1,2,
p
 ri
(15)
1
 1)r3 ,  2  k1 ] .
p
(16)
k3  min[(
Fig. 5 shows the numerical result when there is a single bottleneck and the sessions use different traffic distribution
algorithms. Session 1 employs the metering algorithm to lower the cost at the expense of increased complexity while the
others use the randomization algorithm. Link 1 becomes full when the offered load for session 1 is about 56 while it was
45 in Fig. 4. This is because metering requires fewer wavelengths than randomization for the same load. Implicit prices
are also getting lower for the same reason. The number of wavelengths allocated for session 2 begins to decrease as Link
1 is congested. As the link becomes saturated, session 2 returns the allocated capacity to session 1.
3.B.2 Multiple bottleneck case
Let us consider the case of multiple bottleneck links. We set the traffic ratio to
r1 : r2 : r3  1:1:1 to make both links
congested and employ randomization for every session. Fig. 6 plots the results.
When the links are not saturated, we observed the same behaviors as before. However, if the two links are congested,
the number of wavelengths allocated for session 1 decreases while those for sessions 2 and 3 increase. This is because
the implicit price is calculated as the sum of the Lagrange multipliers of the bypassing links. Session 1 passes the two
bottleneck links while the other sessions do only one bottleneck link. Compared to the single bottleneck case, session 1
should pay a higher implicit price and get allocated fewer wavelengths. If a session traverses more bottlenecks, it ends up
paying a higher price. Then the allocated wavelengths converge to the same value as the implicit prices go to infinity.
4.
Granularity Concern
In the previous section, we ignored the wavelength granularity constraints in solving the optimal wavelength allocation
problem. In order to apply the obtained analytic results to a real WDM network, we need to take the granularity problem
into account. Therefore we propose an algorithm to find an approximate integer solution from the real solution. To this
end we investigate the effect of the granularity constraints on the combined cost and the wavelength allocation through
simulations.
With the granularity constraints, the formulation becomes a nonlinear integer problem. For integer programming,
exhaustive search is a typical but costly approach. It checks every feasible point within the constraint set to find the
optimal solution. Though it guarantees optimality, its complexity is intolerable. For the considered formulation, the
complexity is
O( N ) where  is the maximal number of wavelength in a link and N is the number of links in the
network.
In contrast, rounding is the simplest approach with the complexity of O( N ) . There are three rounding techniques:
floor, ceiling, and rounding. Applying them to the real solution, we get the integer value close to the real one. However,
they often produce infeasible solutions which do not satisfy the constraints of (4) and (5). We will discuss the case of
infeasible solutions next.
4.A.
Proposed approximation algorithm
We propose a simple heuristic search algorithm that approximates the optimal solution and guarantees feasibility. Before
going into the details, we define some notations first.
K real is the allocated wavelength in real number without using
the granularity constraints. It is obtained by the gradient projection and the Kuhn-Tucker method described in the
K int is an integer and represents the allocated wavelength obtained by the exhaustive search.
K round and K prop are obtained by rounding and our algorithms, respectively. K min is the minimum number of
wavelengths that each session can get without violating the feasibility conditions. If K int is feasible, we say the
problem is “solvable.” In this case, K min exists, which is a key concern.
Our proposed algorithm aims to emulate the properties of K real , i.e., optimality and fairness. K real is optimal in the
previous section.
sense of minimizing the combined cost. It is also fair since the implicit prices are fairly imposed on sessions that pass a
common congested link.
K min and allocates wavelengths to approximate K real .
*
is the allocated wavelengths for session f and k f is the allocated value from K real . A
Our algorithm is presented in Fig. 7. It starts with
p f  k  k f where k f
*
f
wavelength is allocated to the session with the largest p f as long as the allocation is feasible and decreases the cost.
Our algorithm checks the feasibility at each step unlike in rounding.
4.B.
Performance evaluation
We evaluate the performance of our proposed algorithm over the ring and linear topologies shown in Fig. 8 and compare
it with those of rounding and exhaustive search methods. The active sessions are presented, and the offered load for each
session and the price of each wavelength are randomly selected. We consider two cases with of 10 and 100 wavelengths
on a link and conducted more than 10,000 simulation runs for each case.
Table 2 shows the ratio of infeasible solutions produced by the rounding techniques. It shows that the rounding
techniques produce infeasible solutions with high probability. Floor may violate (4), ceiling may violate (5), and
rounding may violate both (4) and (5). Therefore the results verify that the rounding techniques are not practical. To
make solutions feasible, we need to take some post procedures. Randomized rounding can be one of such procedures but
is known to be very complex [3].
Table 3 shows the combined cost of each algorithm.
K real is the lowest. However this is not realistic since the
wavelength should be allocated in integer units. For  =10, the costs of the real solution and our algorithm differ by
20~60 %. For  =100, they differ by less than 1%. This is because the fractional part is relatively small with the
increase of the number of wavelengths. It is also the same for exhaustive search.
Table 4 compares the wavelength allocation patterns of K prop and
For  =10, the wavelength allocations of K prop and
K int with that of K real to investigate fairness.
K int are exactly same, so they have the same fairness. For
 =100, the proposed algorithm tends to allocate slightly more wavelengths than exhaustive search. This means that
K int is a little bit fairer than K prop but the gap is quite small. So we can see that the proposed algorithm keeps the
combined cost near the optimal values and produces near optimal integer solutions.
In summary, the exhaustive search of K int produces optimal and fair solutions, but its complexity makes it, in general,
infeasible. On the other hand, our algorithm shows near-optimal performance with the complexity of O ( N ) .
Therefore we conclude that our scheme produces feasible solutions with desirable optimality and fairness properties.
Table 5 summarizes the comparison results.
5.
Conclusion
In this paper, we presented a framework for the joint traffic engineering in IP over WDM networks. A generic edge router
model and the notion of combined cost were presented and analyzed for various cases of traffic loads and distribution
algorithms. We formulated the wavelength allocation problem by employing the total combined cost, and solved it by
using the gradient projection method. The solution verifies that both the explicit price incurred by the loading and the
implicit price created by the link congestion play a key role in wavelength allocation. This result can be used as the basis
of billing.
We also considered wavelength granularity which is an important issue when the number of the wavelength in a link is
small. We proposed an approximation algorithm to find an integer solution from the real solution. Through simulations,
we showed that our algorithm achieves near-optimal performance with significantly reduced complexity and guarantees
the feasibility of solution.
Our proposed formulation and solution are useful for the efficient operation of an IP over WDM network, especially
for traffic engineering and billing. The modeling of IP traffic engineering and the design of realistic RWA algorithms are
left for future study.
Acknowledgements
This work was supported by NRL Program of KISTEP, KOREA.
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16. E. Karasan and E. Ayanoglu, “Performance of WDM transport networks,” IEEE J. Select. Areas Commun., vol. 16, Sept. 1998.
17. S. H. Low, “Multipath optimization flow control,” in Proc. of ICON2000, Singapore, September 2000.
18. L. Masooluie and J. Roberts, “Bandwidth Sharing: objectives and algorithms,” in Proc. IEEE Infocom 99, 1999.
Table 1. Analytic result according to traffic types and distribution algorithms (
Randomization

Combined cost
at the minimum


at the minimum
Number of
wavelengths
at the minimum
Table 2.
 ( B  1)r

 pb 
 pk 
 2(k  r )

2
Queue length

C  pb N M [ B ] / M /1( k )  p NWDM
 ( B  1)r

 pb k
 pk 
 2(k  r )

Cmin
)
Metering
C  pb kN M [ B ] / M /1  p NWDM
Combined cost
p
pb
p
2
B 1 
2p 
 pb
1 
 r
2 
B  1 
Cmin
B 1 
2p 
 pb
1 
 r
2 
( B  1) p 

2p 
N IP  1 
r
B  1 


2p 
N IP  1 
r r
B  1 


B 1 
NWDM   1 
 r
2
p



B 1 
NWDM   1 
 r
2
pr


Ratio of infeasible solutions.
Topology 1
Topology 2
Rounding
16.18 %
37.56 %
Floor
41.53 %
72.33 %
Ceiling
15.53 %
14.19 %
Table 3. Combined cost comparison.
Topology1
Table 4. Comparison of
 =10
 =100
 =10
 =100
K real
Kint
286.72
5120.2
272.55
5138.7
344.06
5140.7
437.23
5183.8
K prop
344.06
5141.5
437.23
5190.7
Kint and K prop .
Topology 1
Topology 2
 =10  =100
 =10  =100
| Kint  Kreal |
0.2356
1.0950 0.6863 2.0444
Mean of | K prop  K real |
0.2356
1.1167
Mean of
Topology2
0.6863 2.0512
Table 5.
Summary of algorithm comparison.
Procedure
Pros
Cons
K real
K round
Apply KT-GP methods
Optimal and fair real solution
Ignored granularity
Simplicity
Frequent infeasible solution
Kint
Exhaustive search
Round
K real
Optimal and fair integer
Intolerable complexity
solution
K prop
Approximate
K real
Low complexity and feasibility
Near-optimality

Router Network
Optical Network
Optical
subnet
Router Network
FEC1
(k=2)
Classifier
&
Distributor

NNI

Router Network
UNI
Optical
subnet

Optical
subnet
Fig. 1. An example of IP over WDM networks.
Fig. 2. Model of an edge router.
FEC 1
Link 1
FEC 2
Fig. 3. Simulation topology. ( 1
Link 2
FEC 3
 2  100 ,
pb  1 , p  4 )
FECN
(k=1)
70
Link 1
LInk 2
500
50
400
40
Implicit Price
Number of Allocated Wavelengths
600
60
30
20
FEC 1
FEC 2
FEC 3
10
0
0
10
20
30
40
50
60
300
200
100
0
70
0
10
20
Offered Load of FEC 1
30
40
50
60
70
Offered Load of FEC 1
(a) Number of allocated wavelengths
(b) Implicit prices
Fig 4. Single bottleneck – homogeneous case. ( r1 : r2
: r3  4 : 2 :1 )
70
FEC 1
FEC 2
FEC 3
Link 1
Link 2
100
50
80
40
Implicit Price
Number of Allocated Wavelengths
60
30
20
60
40
10
20
0
0
0
10
20
30
40
50
60
70
0
10
Offered Load of FEC 1
(a) Number of allocated wavelengths
Fig 5. Single bottleneck – heterogeneous case. ( r1 : r2
20
30
40
50
Rate of FEC 1
(b) Implicit prices
: r3  4 : 2 :1 )
60
70
60
Link 1,2
400
300
40
Implcit Price
Number of Allocated Wavelengths
50
30
20
200
100
FEC 1
FEC 2,3
10
0
0
0
10
20
30
40
50
0
10
Rate of FEC 1,2,3
20
(a) Number of allocated wavelengths
40
50
(b) Implicit prices
Fig. 6. Double bottleneck – homogeneous case. ( r1 : r2
Initialization:
30
Offered Loads of FEC 1,2,3
: r3  1:1:1 )
K : K min , Finfeasible : 
Step 1)
f  arg max p f
Step 2)
= k f +1
k new
f
Step 3)
if
f Finfeasible
where
p f  k *f  k f
new
makes K infeasible and C f (k f )  C f (k f )
k new
f
then Finfeasible : Finfeasible  { f }
else allocate a wavelength to session f
Step 4)
if
( Finfeasible )c   , terminate else go to Step 1
Fig. 7. Proposed approximation algorithm.
FEC 1
FEC 1
FEC 4
FEC 2
1
FEC 3
2
1
2
3
4
5
FEC 4
3
4
FEC 5
FEC 6
FEC 2
FEC 7
FEC 3
(a) Ring topology
Fig. 8.
(b) Linear topology
Topologies for the performance evaluation.