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Node arrangement optimisation in linear multihop lightwave networks

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H.-J.Ho and S.-L.Lee

Abstract: Virtual topologies of WDM-based multihop networks can be arranged by assigning wavelengths to tunable transceivers of nodes. An optimal node arrangement problem in multihop lightwave networks with linearly virtual topologies is considered. For a given trallic matrix, this problem is to arrange node locations to adapt traffic pattems and optimise network performance. However, the problem of node arrangement for constructing optimised linear virtual topologies has been proved as NP-complete. Two lower bounds are proposed for this problem according to the trafic-weighted mean hop distance and the maximum link flow, respectively. An efficient node arrangement algorithm, called k-ENA, is also proposed. The performance of the k-ENA algorithm is compared with the best algorithms in previous work. Simulation results indicate that the k-ENA algorithm generally yields the hest solutions.

1 Introduction

Wavelength division multiplexing (WDM) tcchnology has high potcntial to upgrade the capacity of existing transmis- sion links in a cost-effective way. Using the WDM tech- nique, a number of wavelength channels can bc accessed concurrently by nodes through tunable or fixed-tuned transceivers. If network nodes are equipped with tunable transceivers, single-hop connections can bc accomplished by tuning their transceivers to the same wavelength. In the single-hop system [ I ] , information is transmitted directly from the source to the destination without passing through intermediate nodes. However, the main drawback is the need to use fast tunable transceivers.

Another attractive usage of the wavelength tuning is to configure a virtual topology in multihop WDM networks [2]. Here, only a small number of tunable or fued-tuned transceivers are required. The virtual structure can be constructed on any physical topology (such as stars, buses and rings) by exploiting the broadcast-and-select property of the WDM lightwave network as shown in Fig. 1. By properly assigning the wavelengths to the transceivers, ether an irregular or regular virtual topology can be embedded on the physical network.

Research on constructing optimal structure of multihop lightwave networks is reported for the irregular topology [3, 41, linear topology [S, 61, ring topology [7], and Shuf- tleNet [SI. The study on linear multihop topologies is important since it will allow DQDB (IEEE 802.6) networks to rearrange and scale up. The IEEE 802.6 metropolitan area network has a linear topology. Studies have shown that the DQDB protocol allows inmediate transmission if

the network is lightly loaded but suffers limitation from the reservation mechanism if the propagation delay is large [9]. By using a particular arrangement of nodes the perform- ance of the general DQDB system can possibly be made better. Nevertheless, the network capacity is still limited because only one channel is employed in the optical fibre. To overcome such problems, several WDM-based DQDB networks [IO, 111 and several heuristic algorithms [S, 61 were proposed to construct photonic implementations of adaptive and optimised linear multihop topologies.

broadcast-and-select network

shown in the Figure corresponds to a distinct optical wave- length. The specific problem can be stated as follows [61: Given that the network nodes must be connected linearly and that the node positions in the network can be adjusted by propcrly tuning their optical transceivers, find the best pattem for interconnecting them. This is known as a node arrangement problem that focuses on optimising initial placement of nodes into a linear multihop topology given average traffic requircmcnt between nodes.

Two commonly used objective functions are considered. For a given traffic demand matrix, the first objective is to minimise the mean hop distance weighted by the traftic. This is equivalent to minimising the mean packet delay of the network when it consists of equal length links. The sec- ond objective is to minimise the maximum offered traffic o f a logical link in the linear bus topology. Minimising the maximum offered link trdffic is equivalent to minimising the maximum link congestion and maximising the network throughput. By reducing the problem to a minimum-cut linear arrangement problem. it has been shown that the node arrangement problem for minimising the maximum link congestion on a linear multihop topology is NPcom- plete [6].

In previous studies [5, 61, network nodes were considered to he connected via direct point-to-point links to form a linear multihop network, as shown in Fig. 2. Such an ti-node linear multihop network requires 2(n - I) wave- length channels, and two transmitters and two receivers per node. The construction of a linear dual unidirectional bus topology is proposed in the photonic bus network (PBNet) (51. In the PBNet investigation, the min-max algorithm in building locally optimal configurations, for a given traffic matrix. is found to have the best performance among the various algorithms. It is a greedy algorithm with O(n3) running time. The optimality criterion of this algorithm is locally to minimise the maximum flow of any candidate link. On the other hand, the single bidirectional bus by assuming the special case of symmetric traftic flows is considered in [6]. Based on the assumption of symmetric flows. a number of heuristic algorithms to build near- optimal configurations ai-e proposed in [6]. Instead of using ordinary serial approaches, research was focused on parallel techniques as well. Among the various heuristic algorithms studied in [6], an iterative algorithm generally yields a better result than the others. Although different iterations in the algorithm can be done in parallel, it still takes O(d) time for each iteration.

We propose two lower bounds on the traffic weighted mean hop distance and the maximum link flow, respec- tively. An efficient node arrangement algorithm (k-ENA) with a time complexity of O(kri*) is also proposed, where n is the number of nodes and k is the number of refinements. This k-ENA algorithm is able to make better choice of can- didate nodes than the min-max algorithm in [5] by estimat- ing future traffic at the selective moment. That is, it makes a locally optimal choice according to more information in the hope that this choice will lead to a globally optimal solution. The performance of the k-ENA algorithm is con- pared with the best algorithm in previous works [5, 61. Numerical results indicate that the k-ENA algorithm gener- ally yields the k s t results when k 2 3. The k-ENA algo- rithm has a time complexity of O(n2) when k is a constant (practically 3 s k s 6). It is a significant improvement over the best-result algorithms proposed in [5, 61 that require 0(n3) and an4) time units respectively.

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2 Node arrangement problem

Let V = { I , 2, ..., P I } denote the set of nodes on the net- work. The location of a node I in a linear bus network can be represented by vi. Therefore a linear arrangement of nodes can be written as a(V) = [vi, v2, ..., .,J, where a is a permutation or V. We assume that the network nodes are logically interconnected via direct point-to-point links to fomi a linear multihop nctwork as in [5_ 61. There are 2(n - I ) logical links between I I nodes. Let E = ( ( v i , (vi+l, 13;)

1 vi E V, I s i s 11 - I } denote the set of unidirectional links (see Fig. 2).

The average traffic between nodes in an n-node network can be represented by a n x n matrix [Ag]. where Aq denotes the traffic arrival rate from node i to nodcj. Let Aii = 0 and let A = denote the total arrival rate of traffic to the network. Two commonly used objective functions are considered. The first objective is to minimise the mean hop distance weighted by the traffic. This is equivalent to mini- mising the mean packet delay of the network when it consists of equal length links. The second objective is to minimise the maximum offered traffc of a logical link, which is equivalent to minimising the maximum link congestion.

2. I Minimisation of traffic weight mean hop distance In multihop networks, reduced traffc hop distance will result in fewer electro-optic conversions and processing overheads at the intennediate nodes. Consider the overall traffic delay in the nctwork, the intemodal processing time (queueing delay) and signal propagation time are the major delay components. In the special case when the network consists of equal-length links, the minimisation of the network-wide average packet delay is equivalent to mini- mising the mean hop distance as well. Hence_ the first performance metric selected here is the mean hop distance weighted by the proportion of traffic intensity. Under the assumption of equal-length links, the traffic delay will also be minimised if the queueing delay at the intermediate nodes is considered insignificant.

Let /I( be the number of hops from node i to node j . Then mean hop distancc weighted by the traffic between two nodes is given by

For a specified node permutation a(V, = [vILv2, ..., !JJ where /I,,,, = 1 1 - j, the mean hop distance h, can be expressed as

Therefore the node arrangement problem on minimising the traffic-weighted mean hop distancc is to construct a node permutation U( V) t c the positions of the linear multi- hop network such that / I , is minimised among all node arrangements.

2.2 Minimisation of maximum link flow To consider the potential reduction in link congestion for a given traftic pattem, the measurement metric is to minimise the largest flow over the links in the network. This optimal- ity criterion is motivated by the fact that in a linear multi- hop networks a single heavily congested link is the bottleneck in the network.

IL'C P ~ ~ ~ ~ . - G ~ ~ ~ ~ ~ , ~ , mi. 148, A% 6. ncCenlhrr m i

For a specific node arrangement a(!') = [VI, 1'2, ..., v,,b any two adjacent nodes vi and v , + ~ are connected by two logically unidirectional links (vi, vi+,) and ( L ' ~ + ~ , v,). Thus, the total carried one-way traffic A(q, vi+J on the unidirectional link (I,{, v,+J can be defined as follows:

j=1 k = i + l

Similarly, the traftic flow through the link defined by

vi) can be

i n

.qv,+1 I .i) = A,,, ,vi 1 5 i i n - 1 (4) j=1 k=i+l

Our goal here is to find a node arrangement ci(M such that tlie objective function IV,, denoted as follows, is minimised:

tue = rnax{rn,tiz(.&(vi, ui+,)> .&(U,+, ) v i ) ) } ( 5 )

Note that mnx(A(L>, vnl), A(L';+~> 13;)) is the maximum unidi- rectional flow between node vi and node Then the overall maximum unidirectional link flow in the linear network corresponds to taking the maximum flow among the links ((L'~, vi+,), (v i+ , , vi) I I L i 5 17 - I}.

3 Lower bounds

A tight lower bound allows us to check and assess the qual- ity of the solution provided by the node arrangement algo- rithm. In this Section we derive a lower bound for the weighted mean hop distance and the maximum link flow.

n-1

2-1

3.1 Lower bounds on weighted mean hop distance In previous work [3, 121 the lower bound is usually dcrived by using the concept of minimum flow trees where a directed and balanced spanning tree is constructed for each source node. In gencral, the minimum flow trees are con- structed for each commodity independently to minimise the weighted distance from cach node. Thus it does not take into account the global traffic flow that is constrained by the network topology such as a linear bus topology.

We first obtain a lower bound on the weighted mean hop distance by observing that for a directed multihop bus topology, there arc exactly one source-destination (SA) pair 11 - I hops away, two s d pairs n - 2 hops away, etc. Now consider an idealised arrangement of nodes for which the s-d pair with the smallcst traffic is connected by n - I hop path, the next two s-d pairs in asccnding order of tratfic arc connected by n -2 hops paths, and so on. We show that the low bound A,,, for this idealised topology is a lower bound on the traffic-weighted mean hop distance for a linear multihop topology.

Let denote the bidirectional traffic requirement between node i and node .j, i.e. /$ = + Ai; for i < j 5 n. Note that there are a total of n(ii - 1)/2 bidirectional traffic requirements in an n-node network. Let PI, a, ..., &c,.ld be a permutation of bidirectional traffic requirements in the network, where we permute thc sequence of bidirectional traffic between nodes in an ascending order, such that

pi I i-33 if i < j Tileorrm I : For a directed linear multihop network with n nodes, let

n-1 k

then, for all node arrangement shcmcs, - h 2 ~ L B

Proof In the ideal arrangement the traffic requirements of nodes are sorted in increasing order, i.e. PI 5 a s ... li

/$Icn.ly2 Since there can be at most k pairs of nodes that are I 1 - I; hops away, the sum of weighted traffic requirements at k hops away is equal to (11 - k)/3(k2i;+2,,R. Let a(V) = [vi, t'2, ..., v,J be a feasible solution of an arrangement scheme. From eqn. 2. the weighted mean hop distance or the node arrangement n(M is

7L-1 k

= h1.B

3.2 Lower bounds on maximum link flow Once hLB is obtained the total amount of flow is distrib- uted equally on the feasible number of links to bound the maximum link flow. An immediate lower bound on the maximum offered load to a link can be found by dividing the lower bound on the total flow by the total number of unidirectional links. Let uLB denote the lower bound of the maximum link flow in the linear multihop topology, wc have

Another lower bound will be closer to the optimal solution of this problem. In a network with bus topology, the cen- trally-located links, i.e. (Ln/2], Ld21 + I ) , (Ln/2] + I , Ln/2]), ([n/21, [n/21 + 1) and ([id21 + 1, [n/2]), generally have higher traffic loading. Hence, the lower bound on the max- imum link flow in the bus network is at least equal to the minimum offered traffic on these links.

Consider the traffic flows in (Ln/2], Li7/2] + I ) and (Ln/2l + 1, Ln/2]), we found that traffic between nodes { 1. 2, ..., Ln/2]) and nodes (Lni21 + 1, ..., n - I , n ) will be transmitted via those links. Let [.x<(I), xi(2), ..., x,(n - I)] be a permuta- tion sequence of bidirectional traffic requirements between node i and nodes V - ( i } . Here we permute the sequence of traffic requirements between node i and other nodes in an ascending order such that

a i ( j ) 5 r , ( k ) if j < k (7) The sum of m smaller bidirectional traffic requirements of node i, denoted as T,(m), is

J=I

Also, let bl, y2. _.., y,] be a permutation of V such that

In eqn. 9 we permute T,,,([n/2]) in an ascending order, i.e q,,([n/21) L T,&[n/2]) 5 :.. i TJ[n/21), where I 5 i s n and TV,([n/21) represents the ideal (smallest) traffic flow between

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nodey, and other [id21 nodes. Next, consider the total traf- fic flow on the centrally located links (Lii/2], Ld2J + I ) and (Lid21 + I, Ln/2]) There are exactly Ln/2] nodes at the links’ left-hand side and [id21 nodes at the links’ right-hand side. Hence, the idealised bidirectional flow through these links is the summation of smaller traffic flows between Lnl2J nodes and [d21 nodes. Then we can get the possible mini- mum flow of links (LnI21, Ld2J + I ) and (Ln/2] + I , Ln/2]) as follows:

Following a similar approach, we permute Ty,(Ln/2J) in an ascending order, where I 5 i 5 n and T,,(Ln/2]) represents the ideal (smallest) traffic flow between node z, and other Ln/2] nodes. Then we can get the possible minimum flow of links ([n/21. [n/21 + 1) and ([n/21 + I , [n/21) as follows:

where [zl, z2, ..., ZJ is a permutation of V, such that

T/ieureni 2 For a directed Linear multihop network with n nodes, let

1

2 W L B = - - m a x ( w L B I ; w ~ H d

then, for any node arrangement pattern ci(M = [vI, v2, ..., v,J, we have we a wLB. P~uuf Since ivLBl and are obtained from the ideal construction of a minimum link flow bascd on the two-way traffic, fewer constraints are imposed on the construction as compared with the overall optimal pattem. Hence, for any node arrangement pattem c(V) = [ v , , v2, ..., v,,], the max- mum unidirectional link flow IY- is always equal or greater than a half of the idealised two-way traffic through the cen- trally located links, i.e. IV, t $ ” Y ( I Y ~ ~ ~ ; ivl.BL).

4 Efficient node arrangement algorithm

Several heuristic algorithms to build near-optimal configu- rations of linear multihop lightwave networks are proposed in [5, 61. Among the various heuristic algorithms studied in [SI, the min-max algorithm is found to perform the hest. This algorithm builds up an arrangement by adding nodes

to a partially formed photonic bus network (PBNet) one by one. An ‘un-added’ node is selected on the basis of the maximum traffic flow between this candidate node and the ’added’ nodes consisting of all nodes currently belonging to the partial PBNct, as shown in Fig. 3. The new node is added to the side of the partial PBNet which minimises the maximum link utilisation. However, the choice made by this algorithm depends only on the traffic flow between the candidate node and the partial PBNet so far, and it does not contemplate any future traffic on the other ‘un-added’ nodes. In this Section we propose an efficient node arrange- ment algorithm, called k-ENA, which is able to make a better choice of candidate nodes by estimating the future traffic at the selection moment.

Let G, and G, denote the sets of ‘added’ and ‘un-added’ nodes, respectively. For a candidate node vi E G , we denote Go = G , - { v i ) . Let &vc,,v,, denote the total traffic on link (vi-i, vi). Similarly, let &v8,a denote the total traflic from node vi to nodes in set G, and let &Gj,G/, denote the total traffic from nodes in set Gi to nodes in set G,. In Fig. 3, with respect to a candidatc node vi, nodes from v I through IJ;.~ are referred to as ‘added’ nodes and the other nodes are referrcd to as ‘un-added’ nodes. The total traffic flow through the link (vcI, vi) is given by

* [ t , ; - ~ , u i l = lylGn,t>rl + h(Go,Go) = A [ G ~ . G ~ ) (I3) A,j.,,v2, is known as the total traffic flow from ‘added’ nodes G, to ‘un-added’ nodes Go, and the choice of candi- date nodes on v, is independent of the total link traffic between nodes v . ~ and The optimality criterion of the min-max algorithm in [5] is locally to minimise the maxi- mum flow between the candidate nodes 11, € G , to the par- tial PBNet G, and without respect to the other nodes in Go Consider the total flow through link (vi, vi+J, we know that

.h(ui,t>z+l) = .hlGD.G0l +4”&) = A [ u - l , z ~ . ) +-%”*,Go) -.h(GD,W) (14)

The traffic flow through the link (v,, v , + ~ ) will be less than the flow through the link (v- ] , vi) when - &G8,,jl < 0. Additionally, since = &Ctr,Go,, only the term A,v,,Fol - &G,,,,,, in eqn. 14 which is dependent on the relative choice of v j , can be minimised. Hence, for minimising the weighled mean hop distance, thc optimalily criterion in the k-ENA algorithm is to select v, from the candidate nodes G , on the basis of

nlin {fi(t>z:ccll ~ A(Gn,t:,) + A(co ,v j ) - Aiu,,cD)} V,EGti

On the other hand, the optimality criterion for minimising the maximum link flow in the k-ENA algorithm is similar to eqn. 15. Here we focus only on the appended traffic ,+,,,Go, - hG,,,,;, as shown in eqn. 14. Since thcre are exactly lGol nodes in Go and iGai nodes in GR, we can amortise the appended traftic flow on the ICo\ + lGRl nodes into the current link (vi , vi+l) such that the flow in each link is mini- mised. During the selection process, a candidate node is preferred to the others if the amortised link flow is mini- mised, i.e.

(1G) The k-ENA algorithm is a greedy algorithm that builds up an arrangement by adding nodes to a partially formed bus network one by one. A formal description of the algorithm is given as follows. Algorirhni I : k-ENA algonthm Input: A n x n traffic demand matrix [Av] and an empty bus with n nodcs Output: A node arrangement pattern a(M = [ v , , v,, ..., U,,] I . Initialisation phase:

1.1 Let GR = 0, G, = [tq, v,, ..., v,], CUSI,,,~, = m and k = I (iteration). I .2 For each candidate node j E G , set up initial traffic flows.

CO = G,- U};

Computc h . ~ ~ ) and A(coJ$

~1 = Jl min {A(j ,cO) + *(c~,;)}; G, = G, U { v l l : Go= G,- {vI}; Sturring-Points = {vl};

2. Constructing phase:

Set it(,GB8) = 0 and hGB,,, = 0;

1.3 Choose vl to minimise the initial traflic flow.

;€Go

2.1 For all remaining nodes vi, i = 2 to n do 2.1.1 For all j t G , update internodal traftic flows.

Go = G, - U}; Update 141,~~) and A(cOd); Update and &G~:,);

2. I .2 Choose vi to minimise following optimisation measures.

if our objective is to minimise the mean hop distance then

- ( h ( 3 . G ~ J + A ( G ~ , 3 ) ) } ;

Otherwise, to minimise the maximum link flow, we set

2.2 After the construction, calculate the cost of the node anangement pattern.

IEE Proc.-Conm!un.. Vol. l4S, No. 6. Dccmther ZOO/

Cost, = Ohjective-Fu~fcti[)n([v,, vz. ..., d); if Cost,,,, > Costk then

Opt-Purtern = [vI , ‘ 2 . ..., v,,]; cos/,, = cost,;

end if 3. Check against the number of refinements to decide if one can terminate the iterating process. The expected number of refinements must not to exceed the value of n. Moreo- ver, numerical experiments indicate that the constant nuniber of refinements (practically 3 - 6) is performed quite well for any size of the problem.

if vn Starring-Points then if k < E.~pected-N~~he~-of~Reji~ien~enls then

For rapidly converging, we choice v, as the next starting point, i.e. let v1 = v,,;

Randomly assign a value to v , such that v I $! Starling-Poin ts

else

end if Srurting-Points = Startin.r-Poinrs U { 1’1 1; k = k + l ; Go to step 2;

Output the Cosr,,,,, and Opt-Prrtrern; else

end if The time complexity of this algorithm can be derived as follows. Each of the ‘compute’ and “in’ operations in step 1 of the algorithmic description can be done in O(n). Thus, step 1 takes O(n2). In step 2.1 the for-loop is repeated n - I times. In each repetition it takes O(n) time to update inter- node traffic flows for all candidates. Then, among the updated traffic flows, finding the minimum-cost element takes O(n) time. In step 2.2 the cost of objective functions can be found in O(n*). Therefore, it takes O(n2) time in step 2. Performance of this algorithm can be improved by running the algorithm for a number of refinements. If k starting points are chosen, where I 5 k a n, the time complexity of this k-ENA algorithm is O(kn2). Numerical results show that the performance of the 3-ENA algorithm is generally within 3% of the best possible results of n-ENA algorithm.

Finally, therc are three important properties of the k- ENA algorithm. Greedy: In each refining iteration, only one starting point is chosen and it might determine where the final local optimum is. Descent: I t is a descent method in the sense that once the initial solution is set up. perforn- ances of the algorithm can always improve for each itera- tion. Efficient: The initial arrangement can be rapidly set up in O(n2) time. Since the feasibility is maintained throughout the refining process in the algorithm, we may wish to stop the algorithm no matter how many refine- ments have been executed.

5 Simulation results

In this Section we evaluate the performance of the k-ENA algorithm and compare it with the lower hound and with the algorithms in [5, 61 for both symmetric and asymmetric traftic patterns. For the case of symmetric traffc, the offered flow from node i to j is the same as the flow from j to i. The offered traffic loads are almost invariably symmet- rical in several peer-to-peer applications. However, traftic may be asymmetric for other applications such as client- server applications. During the evaluation a multiplicative random number generator with period 2’2 is used for

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generating the random number. To reduce the bias of certain algorithms to certain traffic pattcms, a fixed set of 500 different traffic matrices are applied to each of algo- rithms. Various performance measures obtained from the experiments are then averaged over the set of 500 traffic matrices.

5.1 Symmetric traffic patterns Both uniformly and nonuniformly distributed random traf- fic are considered in the case of symmetric traffc pattems. The same numerical models used in [6] are adopted here. For uniformly distributed random traffic, the traffic rate between two nodes is a uniformly distributed random number between 0 and 1. For nonuniformly distributed random traffic, networks of 50 nodes with four clusters are considered. Specifically, cluster i consists of 5 , 8, 12 and I5 nodes for i = I, 2, 3 and 4 respectively. Let aCLF denote the cluster loading factor. Then traffc rate between two nodes, both belonging to the same cluster, is chosen to be high intracluster traffic with a uniform distributed random number between 0 and ScLF. The traffic rate between two nodes with different clusters is a uniformly distributed random number between 0 and I .

The first set of experiments was carried out under the symmetric matrix with uniformly distributed tranic. Fig. 4 shows the converging curves of the k-ENA algorithm in relation to the minimisation of the maximum link flow. The vertical axis represents the ratio of the maximum link flow with respect to the final solution and the horizontal axis represents the number of refinements k. For networks of size 7, IO, 20, 50 and 100 nodes the Figure shows that the performance of the k-ENA algorithm is always within 2.5% of the final solution in any value of k. On the other

374

hand, the converging curves in relation to the mean hop distance is similar to Fig. 4 and the performance is always within IS%. Obviously, proper choice of the starting point in each refining process often helps to improve the solution. More particularly. we use the greedy inethod to choose a starting point among n - k remainder nodes in the kth refinement. Form these converging curves we see that the solutions of the k-ENA algorithm can be rapidly converged on a few number of rcfinements.

1.020 I f

iterations

-e- ?nodes ~ -m- ~ IO nodes ~ -+- ~ 20 nodes - -X- - 5U nodes -f-- 100nodcs

Tables I and 2 summarise the maximum link flow and the mean hop distance for an n-nodes lincar multihop network under symmetric matrix with uniformly distrib-

Table 1: Maximum link flow for symmetric random traffic

Nodes Algorithm (n) (Complexity)

7 Man. link flow

Total linkflow

10 Max. link flow

Total linkflow 20 Max. link flow

Total link flow 50 Max. link flow

Total link flow

100 Max. linkflow

3-ENA an? - 4.98 45.61

10.08

137.46 42.24

1156.85 278.83

18828.2

1151.36

mENA a+) 4.97

45.58

10.06

137.14 42.08

1150.82

277.41

18779.6

1147.00

Min-max [51 Iterative 161 a+) a n 4 )

5.19 4.97 46.78 46.02

10.59 10.21

138.44 139.61 44.51 42.98

1177.29 1181.11

289.54 281.93

19231.19 19247.7

1183.91 1164.06

LBound log n)

4.33

14.8%

7.99

25.9% 31.13

35.2%

175.37

58.2%

664.14

Total link flow 155282 154482 157299 157144 72.7%

Arrangements from set of 500 random traffic patterns

Table 2 Mean hop distance for symmetric random traffic

Nodes Algorithm 3-ENA mENA Iterative 161 LBound In) [Complexity) an21 a+) a n 4 ) an2 log n)

7 Mean hops 2.15 2.14 2.19 1.87 Total linkflow 45.35 45.25 46.12 14.4%

10 Mean hops 2.98 2.97 3.10 2.45

20 Mean hops 5.92 5.90 6.19 4.43

50 Mean hops 15.20 15.14 15.60 10.40

Total linkflow 132.85 132.54 138.44 21.2%

Total linkflow 1126.74 1121.82 1176.09 33.2%

Total link flow 18616.70 18535.46 19084.11 45.6%

100 Mean hops 31.07 30.97 31.62 20.51

Total linkflow 153650.18 153136.39 156338.72 51.0%

uted random traffic. The columns under '3-ENA and 'n- E N A show the time complexity and the results obtained by the k-ENA algorithm for running 3 and n refinements_ respectively. Columns min-max [5] and iterative [6] are the results produced from the best algorithms in [5, 61. From these results the k-ENA algorithm generally yields the best results for both maximum link flow and mean hop distance. Moreover, the constant number of refinements (practically 3 - 6) performed quitc well for each network Sire. It is noteworthy that the k-ENA algorithm guarantees to produce k locally optimal solutions and it is able to make a better choice of candidatc nodes than previous algorithms by estimating the future traflic at each selection moment.

The last column, labelled 'Lbound', in Tables 1 and 2 shows the lower bounds on the maximum link flow wLB and the mean hop distance hLB, derived from theorems I and 2, respectively. Now we denote the percentage differ- ence as the result of k-ENA algorithm above the corre- sponding lower bound. Then the dilkrence between the result or the Ir-ENA algorithm and the lower bound on the maximum link flow wLR can be obtained from Table I , e.g. 14.8% for n = 7, 35.2% for n = 20, and 72.7% for n = 100. We can also obtain the diircrence between the results of the k-ENA algorithm and the lower hound on the mean hop distance zLB from Table 2, e.g. 14.4"/, for n = 7, 33.2%) for ii = 20, and 51.0% for n = 100. Although the lower bound is better than the immediate bound in eqn. 6 there still

exists a gap between the k-ENA algorithm and the lower bound. This may be due to the inexact lower bound in which we only consider the two-way traffic generated by a particular sourcedestination pair. As the network sire has grown much of the unidirectional traffic information between nodes was ignored.

The next set of experiments was carried out under the nonuniformly distributed traffc patterns. A network of 50 nodes with four clusters is considered and the correspond- ing results are shown in Tables 3 and 4. In the clustered case the k-ENA algorithm also provides the best results. Clusters are better defined for larger values of aCLFand this helps the k-ENA algorithm to properly identify them.

5.2 Asymmetric traffic patterns The case of asynunetric traffic patterns was claimed as an open problem in [6]. Among the various flow-based algo- rithms studied in [q, only the iterative algorithm is well suited to the asymmetric case. The same parameters and experimental models are employed except that the offered flow from node i to j is not necessary the same as the flow from j to i. The corresponding results are shown in Tables 5-8. Since the problems are becoming diflicult for the asymmetric traffic patterns, the minimum link flow and the mean hop distance are higher than the corresponding symmetric results. Under the asymmetric case our k-ENA algorithm produced the better results as the network size grows.

Table 3: Maximum link flow for clustered random traffic

Cluster loading factor = 10 Cluster loading factor = 50

Algorithms Max. link flow Total link flow Max. linkflow Total link flow VENA 437.84 31 186.84 1364.14 60741.79

3-ENA 384.55 28083.04 1330.83 60425.17

n-ENA 381.23 27826.79 1314.78 60263.80

Min-max 151 442.16 28459.98 1371.01 64477.75 Iterative 161 401.70 29329.69 1334.61 97669.94

Table 4 Mean hop distance for clustered random traffic

Cluster loading factor = 10 Cluster loading factor = 50

Algorithms Mean hops Total link flow Mean hops Total link flow 1-ENA 8.93 27710.80 5.19 59403.54 3-ENA 8.92 27679.77 5.19 59403.54

mENA 8.92 27679.77 5.19 59403.54 Iterative I61 9.07 28145.23 5.64 64554.13

Table 5: Maximum link flow for asymmetric random traffic

Nodes Algorithm 3-ENA n E N A Min-max I51 Iterative I61 LBound in) (Complexity) ad) a+) a+) an ' ) a+ log n)

7 Max. link flow 5.76 5.66 5.93 5.65 4.57 Total linkflow 51.61 50.30 50.13 51.27 23.9%

10 Max. link flow 11.56 11.33 11.97 11.33 8.11 Total link flow 150.46 148.53 149.01 151.72 39.5%

20 Max. linkflow 46.33 45.49 47.13 45.92 32.01 Total linkflow 1296.85 1218.85 1234.82 1249.73 42.1%

50 Max. link flow 293.96 m.55 301.20 294.13 185.77 Total link flow 19828.20 19505.47 19817.79 19896.19 56.4%

100 Max. linkflow 1193.01 1184.07 1213.76 1196.22 679.68 Total link flow 159882 158642 160584 160818 74.2%

375

Table 6: Mean hop distance for asymmetric random traffic

Nodes Algorithm 3-ENA mENA Iterative I61 LBound (nl (Complexity) Mr?) Mr?) an41 Mr? log n)

7 Mean hops 2.35 2.34 2.32 2.12

10 Mean hops 3.23 3.22 3.22 2.81

Total link flow 49.03 48.86 48.47 10.4%

Total link flow 145.48 145.00 144.96 14.6%

20 Mean hops 6.32 6.30 6.34 5.38

Total link flow 1203.28 1197.97 1204.70 17.1%

50 Mean hops 15.86 15.18 15.90 13.18

Total link flow 19427.24 19330.42 19477.31 19.7%

100 Mean hops 31.99 31.91 32.11 26.23

Total link flow 15814865 157718.72 158755.95 21.7%

Table 7: Maximum link flow for asymmetric clusters

Cluster loading factor = 10 Cluster loading factor = 50

Algorithms Max. link flow Total link flow Max. link flow Total linkflow

1-ENA 467.58 32409.23 1484.26 64243.33

3-ENA 410.78 29103.21 1452.81 63652.81

mENA 402.55 28469.80 1419.31 60263.80

Min-max I51 465.18 28983.42 1513.74 67319.29

Iterative 161 450.08 33505.40 1483.74 90219.97

Table 8: Mean hop distance for asymmetric clusters

Cluster loading factor = 10 Cluster loading factor = 50

Algorithms Mean hops Total link flow Mean hops Total link flow

1-ENA 9.15 28465.62 5.41 62747.48

3-ENA 9.14 28410.91 5.41 62747.48

mENA 9.14 28410.91 5.41 62747.48

Iterative I61 9.68 30076.74 6.16 70690.86

6 Conclusion

For better utilisation of network bandwidth an efficient node arrangement algorithm is indispensable. Good arrangement of nodes also reduces the possibility of net- work congestion. A general k-ENA algorithm has k e n proposed. Applying this algorithm, an initial node arrange- ment of linear multihop lightwave networks can be rapidly set up in O(n') time. Our result have shown that perfom- ance of the 3-ENA algorithm is generally better than previ- ous results in [5 , 61. Since feasibility is maintained throughout the refining process, we may wish to stop the algorithm no matter how many refinements have been exe- cuted. In a real-time environment. this is a major advan- tage, since feasibility is more important than optimality.

Furthermore, there is considerable work to improve the lower bounds on thc maximum link flow and the mean hop distance. We only consider the two-way traffic gener- ated by a particular source4estination pair, hence the pro- posed lower bounds do not approach the larger networks closely. For further investigation, it is worthwhile looking into ways of making better use of the unidirectional traffic information between nodes for obtaining a tighter lower bound.

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