Table Of ContentA Highly Tunable Virtual Topology Controller
Y.Sinan Hanay∗, Shin’ichi Arakawa†∗ and Masayuki Murata†∗
∗ National Institute of Information and Communications Technology, Japan
Email: [email protected]
† Graduate School of Information Science and Technology, Osaka University, Japan
Email: {arakawa, murata}@ist.osaka-u.ac.jp
Abstract—Much research in the last two decades has focused methods fail to provide a “control” on the quality of finding
5 on Virtual Topology Reconfiguration (VTR) problem. However, a solution [3].
1 most of the proposed methods either has low controllability, In this work, we transformed two previously proposed
0 or the analysis of a control parameter is limited to empirical
VTR algorithms,Heuristic Logic TopologyDesign Algorithm
2 analysis. In this paper, we present a highly tunable Virtual
Topology (VT) controller. First, we analyze the controllability (HLDA) [2] and Attractor Selection Based (ASB) method
n
of two previouslyproposed VTR algorithms: a heuristicmethod [6], to their tunable versions. In the first part of this paper,
a
andaneuralnetworksbasedmethod.Thenwepresentinsightson we present our simulations on tunability of these methods.
J
how to transform these VTR methods to their tunable versions. We first start by showing that without a control parameter,
3 To benefit from the controllability, an optimality analysis of the
2 control parameter is needed. In the second part of the paper, these methods waste resources by excessively establishing
through a probabilistic analysis we find an optimal parameter lightpaths.
] fortheneural networkbased method.Wevalidated ouranalysis In the second part of the paper, we present a probabilistic
I
N through simulations. We propose this highly tunable method as analysis of ASB to find an optimal parameter for adaptability
a new VTR algorithm.
. to address the highly dynamic traffic. In order to see if our
s Index Terms—optical networks; virtual topology reconfigura-
c tion; tunable network topology. assumptions regarding the optimal parameter holds, we did
[ simulations and the simulations validated our approach.
Theremainderof thispaperis organizedasfollows.Section
1 I. INTRODUCTION
v II presents the problem setting, Section III presents prelimi-
0 Optical fiber has been the main choice of communication naries. Section IV presents tunability and optimality analysis
1 medium for long-haul networks because of its low transmis- of the methods. In Section V simulation results are presented
7 sionloss.Inaddition,afibercablecancarrymanychannelssi- and finally Section VI concludes the paper.
5
multaneouslyusingwavelength-divisionmultiplexing(WDM).
0
which makes it possible to establish many different virtual II. PROBLEM DEFINITION
.
1 topologies on top of the physical topology. This paper focuses on VTR problem. Figure 1 illustrates
0
Virtual topologies consist of lightpaths, which can be as the problem setting. A physical network consisting of four
5
1 short as a segment of a fiber between two hops, or as routers is given and the routers are connected through optical
: long as the span of sevaral fibers. A virtual topology where fiber links. In the illustration, it is assumed that each optical
v
each node pair is connected to each other (i.e. a complete fiber can carry three wavelengths.The virtualtopologieshave
i
X graph) is ideal. However,setting up such a high degree graph four light paths.
r may be unattainable as the number of transceivers per node Wavelength assignment, lightpath and traffic routing are
a
is inadequate. Instead, virtual topologies are constructed to other aspects of virtual topology design problem [7]. In
target a performance goal such as minimizing the maximal this work, we only focus on VTR aspect. In other words,
load on any link, minimizing average hop or minimizing the we are interested to find out virtual topologies that meet
latencybetweenthepairs.Thevirtualtopologyreconfiguration a performance requirement. We assume that the routers are
(VTR) problem is to find a suitable topology satisfying the equipped with wavelength converters, and we use Dijsktra’s
performance metric for the given traffic and resources (i.e. shortest path algorithm for lightpath and traffic routing.
transceivers, number of wavelengths).
A. NP-complete Problems
In the last decade, a lot of effort has been devoted to VTR
problem in fixed WDM networks, where a fixed bandwidth VTR problem is known to be NP-complete [8]. Like many
is allocated between nodes [1]–[3]. Elastic optical networks otherNP-completeproblems,exactsolutionsto VTRproblem
(EON) is a recently emerging paradigm. As opposed to can be obtained using mixed integer linear programming
fixed WDM networks, EON proposes fine-grain bandwidth (MILP).Figure2summarizestheapproachestotheVTRprob-
allocation that depends on traffic demand [4]. Such a flexible lem. In this paper we use terms controllability and tunability
physical layer requires the logical layer to be tunable as well toreferthatanalgorithmcanrunbasedonaspecification.Itis
sincetrafficpatternswillchangemorefrequentlyinnearfuture possible to fully specify the constraints using a MILP formu-
[5].AnothermajorchallengeinVTRproblemisthatheuristic lation, however MILP methods are intractable for more than
Fig.1. Avirtualnetworktopologyexample.Inthephysicaltopology,eachphysicallinkisafiber,andeachfibercancarrythreewavelengths.Bothvirtual
topologies have4lightpaths. Twopossiblevirtualtopologies areshown.VTRproblem istofindthetopology thatperformsbetter.
compares efficiencies of the three methods.
Network operators are reluctant to make drastic changes in
their topologies, even if that means only changing the link
weights [11]. Thus, a VTR algorithm that can control the
number of lightpath change is desirable. We aim for devising
such an algorithm, and present the underlying probabilistic
analysis.
C. Comparison Method
WechoseHLDA,becauseitisanefficientheuristicanditis
designed to minimize congestion, like ASB. Note that, MILP
basedmethodsareunabletosimulatelargetopologies(i.e.100
Fig.2. MILPbasedmethodsarehighlytunable,howevertheyareintractable
more than about 10nodes. Heuristics methods are efficient, yet they cannot nodes), thus we are bounded to use heuristic methods.
provideanycontrollability onthesolution. Some other efficient methods aim to minimize single-hop
traffic, end-to-end delay. ASB is selected because its analysis
is straightforward as we show in Section IV-B. Although
tennodes,andearlierworkconsideredonlytopologieshaving
there are more efficient methodsthan ASB such as Multistate
less than 14 nodes [1], [2]. However, real world topologies
AttractorSelectionwithDependentNoise(MADN)[12],[13];
consist of more than a few dozen nodes, for example AT&T
such methods are generally intractable. Our goal is not to
consists of 154 nodes and DFN network topology consists of
evaluatetheperformanceofdifferentVTRalgorithms.Instead,
30nodes[9].EvenaveryrecentworkthatpartiallyusesMILP
we are interested in exploring the controllability of VTR
considers topologies of 6, 11 and 23 nodes [10].
algorithms. We explored the existence of a optimal control
Mostoftheheuristicmethodsassignlightpathstoremaining
parameter for ASB.
resources after the algorithm finishes. In order to evaluate
different VTR algorithms, the algorithms must be evaluated III. PRELIMINARIES
also based on their overhead. This work evaluate methods
ASB uses traffic loads of the links and HLDA uses the
not only by their performance, but also the overhead they
traffic matrix. We assume that traffic loads are continuously
introduce.
monitored by a central controller for fixed intervals, and
this is easy to implement in practice using Simple Network
B. Motivation
Management Protocol (SNMP) [14].
Figure 3 presents the motivating example for this work. The following sections review the relevant part of ASB
Three different VTR algorithms were compared by running briefly,foramorethoroughexplanationofthesetwomethods,
each method 30 times for each traffic load. HLDA clearly reader can refer to the corresponding paper [6], [2]. Yet,
performsbetter than ASB and MADN for traffic loads higher the following section should be self-sufficient to follow the
than 0.2 as shown in Figure 3a. On the other hand, Figure discussion. After the overviewof those two methods,we then
3a reveals that average number of lightpath change per round explain how these two methods can be made tunable.
with HLDA is drastically higher than ASB and MADN. All
A. Attractor Selection Based VT Control
three methods established similar number of lightpaths, close
to 1600 (the physical upper limit for 100 nodes carrying ASB method searchs for topologies randomly, and if a
16 transmitter/receivers). The quality of the solutions can be satisfactory topology is found, it is saved in a memory.
determined as the ratio of performance to the number of This saved topology is called an “attractor”. ASB method is
lightpath changes, which we define as efficiency. Figure 3c described in Algorithm 1.
1600
1 es 1400 1 HALDSBA
0.8 hang 1200 HALDSBA 0.8MADN
performance 00..46 er of lightpath c 1 468000000000 MADN efficiency 00..46
0.2 HLDA mb 0.2
ASB u 200
MADN n
0 0 0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
traffic load traffic load traffic load
(a)Underlowtrafficallmethodsperformsimil- (b)HLDAchangeslighthpathsmuchmorethan (c)MADNisthemostefficient ofallthree.
iar,whileforheavytrafficHLDAperformsbest. ASBandMADNperround.
Fig.3. Theperformancecomparison ofvarious VTRalgorithms.
1: procedure ASB(time t) meaning of the variables. Assume that there are 100 nodes,
2: VG ←ComputeVg() ⊲ using umax by Eq. 1 n=100,thenthepossiblenumberofpairsis9900,n×(n−1).
3: ComputeWeightMatrix(VG,t) Thusatopologycanbedescribedbyabitvectorofsize9900.
4: ComputeExpression() For example, the first virtual topology in Figure 1 can be
5: UpdateLightPath() described by the following bit vector:
6:
7: procedure COMPUTEWEIGHTMATRIX(VG, time t) [010 011 111 011] (2)
8: if (VG(t−1)<Tmax&VG(t)>Tmax) then node1 node2 node3 node4
9: for i←1,n do |{z} |{z} |{z} |{z}
wherea 1bitindicatesthatthecorrespondingpairhasalight-
10: for j ←1,n do
path between them. For example, since node 3 is connected
11: weightMatrix[i,j]−=Hebb(i,j,Ak) to all nodes, the bits belonging to node 3 are set to 1’s.
12: for i←1,n×(n−1) do The attractors are stored in an attractor matrix A. For
13: Ak[i]=LightPath[i] ⊲ Update attractors example,to store 5 topologies,the size of the attractor matrix
14: for i←1,n do A must be 5 by 9900. A denotes the ith attractor and the
i
15: for j ←1,n do attractors are added in a FIFO sense (line 17).
16: weightMatrix[i,j]+=Hebb(i,j,Ak) After calculating VG, the system compares whether the
17: k =(k+1) mod numberOfAttractors performance improved with respect to previous round sig-
nificantly by checking against a threshold parameter T
18: max
19: function COMPUTEEXPRESSION ⊲ by Eq. 3 as shown in line 8. If so, the topology is added as a new
20: for i←1,n×(n−1) do attractor by the means of changing the weight matrix as
21: for j ←1,n×(n−1) do shown in line 16. Hebb function calculates the weights based
22: x[i]+=ComputeDeltaExp(i,j) on Hebb learning, which is given by Equation 4. This new
weight matrix generates new expression levels, x, which can
23:
be thought as a measure of how likely a lightpath has to be
24: procedure UPDATELIGHTPATH
established. For each lightpath l , there is a corresponding
25: for i←1,n×(n−1) do i
expression level x .
26: if (x[i]>0.5&CanEstablish(i)) then i
Auto-associative Memories: Auto-associative memories
27: EstablishLighpath(i) ⊲ LightPath[i]=1
are neural memories, which are used to store patterns based
28: else if (IsEstablished(i)&x[i]<0.5) then
on a correlation matrix [15]. Neural memories are different
29: RemoveLighpath(i) ⊲ LightPath[i]=0
than the computer memories; the values are not read in
Algorithm 1. ASBmethod neuralmemories,theyarecalculated.Auto-associativememo-
ries are conceptually similar to content-addressablememories
(CAMs). To query the memory, user provides a data word
At the beginning of a round, the ASB controller gets
instead of an address. In addition,unlike computermemories,
the utilization of the maximally loaded link and computes a
neural memories are not physical devices. The values are not
performance metric V , which is given below.
G readfrombitcells,butrathertheyare“calculated”byamatrix
1 multiplication. The concept of auto-associative is illustrated
V = (1)
G 1+e50(umax−0.5) in Figure 2. The attractors are stored in an auto-associative
In this equation, u represents the utilization of the max- memory.
max
imally loaded link. Before getting into further details of the In line 23, ComputeDeltaExp function calculatesthe dxi
dt
algorithm, we give some numerical examples to clarify the according to Equation 3. The following equation shows how
Here, α is learning rate, which was set to 1. The weight
matrix is updated when an attractor is found.To speed up the
calculation, instead of a 9900 by 9900 matrix multiplication,
we first subtract the contribution of the oldest attractor(line
11), and add the new one (line 16).
IV. CONTROLLABILITY AND OPTIMALITY
In the following sections we discuss tuning of the two
algorithms, and present the analysis on optimizing search
space for ASB.
A. Controllability
Fig. 2. Auto-associative memories can recognize andcorrect noisy inputs
Mostofthemethodspresentedpreviousarenottunable.The
tosomeextent.AnoisyinputFalifornibissupplied,andthememoryreturns
the closest element California. On the other hand, if a non-existent entry most distinctive example is TILDA [2], which assigns light-
(i.e.Texas)ispresented, thenapermutation ofsomestoredinputpatterns is pathsbasedonthehopdistance.TILDAistraffic-agnostic,that
returned astheoutput.
is, regardlessof the traffic it generates the same virtual topol-
ogy as long as the physical topology stays same. TILDA can
bemadetunableinseveralways.Forexample,anoperatorcan
the expression level is updated [6]:
modify TILDA so that the lightpaths assigned only between
n nodesthathaveahopdistancelessthansomespecificnumber
dx
i
=f wijxj−xiVG + N(0,1) (3) h. It is possible to find a sub-optimalh by empiricalanalysis,
dt
Xj=1 but it is rather intractable to analyze mathematically. Thus,
althoughit is possible to make any given methodtunable,the
auto-associativememory randomwalk
analysis of the control parameter is not straightforward.
| {z } | {z }
The VTR method Adaptive presented in [1] can be consid-
In this equation, N(0,1) is the standard normal random
ered as one of the first tunable method. Adaptive uses two
variable, and x captures the importance of a lightpath. If x
i i
watermarkstoestablishalightpath:W andW .Itiseasyto
is greater than 0.5, a lightpath is established provided that H L
minimizethesearchspaceforAdaptive,bysettingW =100
there are enough resources. A higher V means the system is H
G
and W = 0, but it is not possible to analytically calculate
in better condition. f(.) is the sigmoid function. L
howtomaximizethesearchspace(i.e.forwhichW andW
ThesystemdynamicsshowninEq.(3) consistoftwo com- H L
values search space is maximized). We can only say the that
ponents:auto-associativememoryandrandomwalk.Whenthe
search space is maximized when W = W , but at which
system is in good conditions, that is when V is high, then H L
G
value this will be minimized depends on the traffic.
the x is mostly determined by the auto-associative memory,
i
1) Tunable HLDA (tHLDA): We tried two different ap-
which inclines towards to the stored memory elements. On
proaches to make HLDA tunable. In the first approach, the
the contrary, when the network conditions get worse and the
maximum number of lightpaths HLDA is fixed. In the other
controller needs to find a new attractor suitable for the new
approach, the minimum amount of traffic that is able to
conditions, it randomly searches for a new attractor.
establish a lightpath is changed. In terms of efficiency and
ASB does not make any assumption about network re-
performance, the first method performed better, and we use
sources availability. If there are not enough resources for
this version of HLDA in this work.
a lightpath to be established, ASB does not establish the
2) TunableASB(tASB): InASB,thenoisetermisanormal
lightpath, and skips to the next lightpath. An alternative
random variable with zero mean and unit variance. We make
approach is to continue looking for a topology in which all
ASB by tunable by changing the noise term as follows:
the lightpaths can be established. There are several ways to
build the weight matrix using different learning algorithms, N(0,1)→ N(µ,1) (5)
such as Hebbian, Oja and APEX learning; and the effects of
ASB tunableASB
differentlearningalgorithmshasbeenstudiedbefore[16].We
use Hebbian learning for constructing weight matrix. Thus,µbecomesapa|ram{zete}rfor|AS{Bz.B}ylookingEquation3,
Hebbian Learning: ASB’s auto-associative memory uses we can see that µ has an effect on xi. Statistically, a positive
Hebbian learning to store and read the elements. In our case, µ increases the value of xi and, a negative µ decreases the
virtual topologies are stored in a auto-associative memory of value of xi with respect to µ = 0 (original ASB). Since
which weight matrix is constructed using Hebbian learning. VTR is a NP-complete problem, an optimal µ minimizing
the congestioncannotbe foundin polynomialtime. However,
In the generalcase, the weightmatrix W is constructedfor
a topology vector X, as W = XTX. Weight matrix can be when a good topology is found for some µ value, it can
be changed incrementally in either direction by removing or
constructed,usingHebbianlearningweightupdaterulebelow:
adding multiple lightpaths at once. Note that changing µ has
∆w =αl l (4) overall effect on all lightpath establishments and deletions.
i,j i j
This approach should not be confused with a pseudo-tuning considered also. We proceed by calculating probabilities for
strategyofotherheuristicbasedmethods,wherethecontroller V >0 by considering each transition separately.
G
triestoincreasethelightpathonebyone.Here,inourmethod,
P(0→0)=P(η <0/5) (12)
µ can have differenteffects based on traffic and resources. In
order to find an optimal µ value, we present our analytical P(0→1)=P(Vg+η >0.5) (13)
calculation in the next section. P(1→0)=P(−Vg+η <0.5) (14)
P(1→1)=P(η >0/5) (15)
B. Optimality
In this section, we present our analytical approach to In order to maximize the topology search space, probabilities
calculatean optimumµ . Somedefinitionsthatwillbe used in Equation12 mustbe equal to 0.5. In other words, we need
opt
in this section are presented in Table I. tofindtheη whichwouldmakealltheseprobabilitiesequalto
0.5whenV =0. Sucha requirementis satisfied with setting
G
TABLEI µ as below in each case:
DEFINITIONSFORVARIABLESANDEXPRESSIONS.
µ0→0 =µ1→1 = 0.5 (16)
Pi(j→k,t) Probability oflightpath ichanges fromj tokattimet.
PPii((01→→10,,tt)) PPrroobbaabbiilliittyy ooffetesrtmabilnisahtimonenotfolfiglhigtphatpthathiaitattimtiemte.t. µ0→1 =(cid:26) 00.,5, iiff VVGG =>00.5 (17)
Xli resourceavailability Indicator randomvariable forli.
0.5, if V =0
One problem with ASB is that when the network condi- µ1→0 =(cid:26) 1, if VGG >0.5 (18)
tions are poor, its new topology finding capability is limited. Combining all these equations together, we can write piece-
Specifically, the network is performing poorly when VG = ǫ wise linear functions between boundaries for each µ. The
for some small ǫ≈0. When the network conditionsare poor, optimum mean, µ (t) is found by
opt
the probabilities of lightpath changes can be calculated by
µ (t)= 0.5(1−x (t−1))(1−x (t)) + (19)
opt i i
P(f(xi):0→1,t) =1−P(η<0.5|xi(t−1)<0.5) (6) h i
P(f(xi):1→0,t) =P(η<0.5|xi(t−1)≥0.5) (7) 0.5xi(t−1)xi(t) +
h i
Pi(0→1,t) =P(xi:1→0,t|Xpi) (8) (0.5+V )x (t−1)(1−x (t))f(0.5−V ) +
G i i G
PC =Pi(0→1,t)∪Pi(1→0,t) (9) h i
(0.5−V )(1−x (t−1))x (t)f(0.5−V )
Above, P corresponds to probability of a lightpath addition G i i G
C h i
ordeletion.Noticethatintheequationweusedtheexpression
InEquation19,thefirsttermisfor(0→0),themiddleterms
level, xi, instead of the path indicator variable, pi. We can are for (1 → 1) and (1 → 0), and the last term is for the
analyzetheprobabilityoflightpathestablishmentanddeletion
(0→1) transition.
using probabilitytransitions. The transition probabilitymatrix
The discussion above assumes that network resources can
T is defined as:
over-provisionthenetwork.Onthecontrary,whenthenetwork
P(0→0) P(0→1) resources cannot over-provision the traffic, the network re-
T = (10)
(cid:20)P(1→0) P(1→1)(cid:21) sourcesneed to be considered.Instead of setting the lightpath
establishmentprobabilityto0.5,theavailabilityofthenetwork
Note that we dropped the subscript i and input t from P for
resources should be taken into account as below.
abrevity,sincetheprobabilitytransitionsaresameforallpaths
min(num ,num )
and rounds when VG = 0. For ASB, we can calculate the Pi(0→1,t)= ports wavelength (20)
transition probabilities for V =0, for which T becomes: n−1
G
Here num denotes the number of resources. µ (t) can be
0.69 0.31 opt
TASB =(cid:20)0.69 0.31(cid:21) (11) calculated similarly for this new P.
When the V is low, the VT controller has to find a new V. SIMULATIONS
G
topology that satisfies the new traffic. If the traffic variance In this section we present our simulations results. The net-
is high enough, then a topology that is much different than workconsistedof100nodes,andthelog-normaltrafficmodel
the last satisfactory topology must be found. In order to find was used [17]. For ASB method, an initial list of attractors
such a diverse topology, the topology search space has to be was generated randomly. Dijsktra’s shortest path algorithm
increased. In the extreme case, the search space has to be was used for routing, and the lightpaths were assumed to be
maximized. The search space can be maximized by making unidirectional. We assumed that the nodes are equipped with
the transition probability matrix T = [0.5,0.5;0.5,0.5], so wavelength converters. The physical topology has 100 nodes,
that at anytime the probabilityof lightpath establishmentand and its graph characteristics are given in Table II.
deletionisequal.thenoisetermbecomesN(0.5,1).However, First we compared tHLDA and tASB for different control
if V > ǫ, the deterministic term in Equation 3 has to be parameters.ThecontrolparameterintASBisµwhichchanges
G
3000 1
1 tHtALDSBA es tHtALDSAB 0.9 tHtALDSBA
ng 2500 0.8
performance 000...468 er of lightpath cha 112050000000 efficiency 00000.....34567
0.2 mb 500 0.2
nu 0.1
0 0 0
0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12
control parameter control parameter control parameter
(a) tHLDA performs better than tASB as the (b)Theaveragenumberoflightpath changes per (c)tHLDAhasalmostaconstantefficiencywhile
controlparameter increases. round increases for tHLDA, but it reduces for theefficiency oftASBcan beincreased with the
tASB. controlparameter.
Fig.3. ThecomparisonoftASBandtHLDA.
TABLEII
When the traffic load is low, higher µ values increases
PHYSICALTOPOLOGYCHARACTERISTICS
the value, as more lightpaths start to establish. However,
Degree Avg.PathLength Clus.Coeff. Diameter for medium and high loads, increasing µ beyond 0.5 results
4 3.41 0.05 6
in poor performing topologies. This is due to the sigmoid
function,whereµ=0.5is a saddlepoint.µ=0.5meansthat
mostnodespairsstarttohavealightpathassigned.Sincelight-
from 0 to 0.5, and the control parameter in tHLDA is the
path assignment is random, less important paths deplete the
number of lightpaths which was set betwen 800 to 1600,
resources and results in resource scarcity for more important
with an increment of 80. The lower bound 800 was chosen
lightpathsthatareassignedlaterinlightpathsassignment.This
as this was the point where tHLDA and tASB performance
situation, results in use of all available lightpaths to be used
was equal. For each control parameter, each method was run
as it can be seen in Figure 5c. Thus, every pair experiences
30 times, and the averageof those runswere taken. Figure 3a
the same amount of increase, and a µ = 0.5 or higher is not
and3bshowtheperformanceandoverheadcomparisons.sthe
meaningful.
controlnumberincreasestHLDAoutperformstASB.Hovewer,
Figure 5c emphasizes the linear relation between µ and
Figure 3b reveals that as the control parameter increases the
the number of lightpaths establishment. As the µ increases,
number of lightpath changes is drastically higher than tASB
P(0 → 1) increases, and the more lightpaths establish. This
again. More importantly, Figure 3c reveals that for HLDA,
explains the behavior for µ > 0.35. However, for µ < 0.35
the tunability is quite low. The figure shows that the number
configurations, the network initially starts with lower number
of lightpath change is also a fraction of total number of
of lightpaths. Since its search space is small, it cannot find
lightpaths. The efficiency stays constant across all the control
a good topology; and the controller incrementally increases
parameters.
the number of lightpaths. Throughout the simulation, those
In the second part, µ parameter was swept from 0.2 to 0.6.
configurations failed to find a good topology even when the
Eachconfiguration(foreachµvalue)wasrunwith10random
number of lightpaths reached the maximum. Because the
traffic patterns, and the mean of 10 runs was taken. µ was
opt network fails to find a good topology for µ < 0.35 as can
sampled in each of these runs, for observation. We observed
be seen in Figure 5b. For example,setting µ=0.4 costs 40%
that µ has a mean of 0.46, with a standard deviation of
opt fewer lightpath establishments. The figure also shows that the
0.13. The histogram of µ was given in Figure 4. It shows
opt
that µ=0.5 appears 100 times more frequently than µ=0.
Figure 5a shows the comparison of ASB vs our extended 106
analytical model with µ . The figure indicates tASB with
opt
µopt performsbetter thanASB. Thusit is safe to say, ouran-
alytical findingsabout optimal µ agrees with our simulations. 104
The simulations were run for 400 steps. We conjecture that
increasing simulation time would increase the performance
further. The main drawbackof the µ approachis its longer 102
opt
runningtime, whichis about10XslowerthantASB. Thuswe
madeanothersetofsimulationstoanalyzehowtASBperforms
100
under constant µ values. Figure 5b shows the performanceof −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
optimal mean
thecontrollerwithvariousµvalues,underthreedifferenttraf-
fic patterns. As our analytical calculationssuggested, µ<0.5 Fig. 4. The distribution of uopt values are concentrated around 0.5. Note
gives the best results. thatthey-axis islog-scale.
1600
1 1
1400
performance 0.5 performance 000...468 mber of lightpaths 11020000
nu
AmSopBt 0.2 MedHLiuiogmwh LLLoooaaaddd 800 MedHLiuiogmwh LLLoooaaaddd
0 0 600
0.25 0.5 0.75 1 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
traffic load m m
(a) Optimal mean value vs ASB performance. (b)tASBperformsbestforµ=0.4foranytraffic (c) The minimum number of lighpaths estab-
Asthetrafficloadincreases, VTcontroller with load. lished when µ = 0.32,µ = 0.36,µ = 0.38
µopt outperforms ASBasexpected. forlow,mediumandhightraffic loads.
Fig.5. Optimality ofµforvarious traffic loads.
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