Design of WDM Mesh Networks with Sparse Grooming CapabilityKeyao Zhu

�

, Hui Zang�

, and Biswanath Mukherjee�

�

Dept. of Computer Science, Univ. of California, Davis, CA 95616, USA

zhuk@cs.ucdavis.edu, mukherje@cs.ucdavis.edu�

Sprint Advanced Research Laboratories, Burlingame, CA 94010, USA, hzang@sprintlabs.com

Correspondence Author: Keyao Zhu – Tel: +1-530-752-5129; Fax: +1-530-752-4767

Abstract – In a WDM optical network, the bandwidth require-ment of a customer’s connection can vary over a wide range, and manyof these connections could have a capacity that is much lower than thecapacity of a wavelength channel. Efficiently grooming low-speedconnections onto high-capacity wavelength channels can significantlyimprove the bandwidth utilization and minimize the network cost. Ourresearch shows that it is not neccessary to have traffic-grooming capa-bility at every network node. We call a network which has only a fewgrooming nodes to be a sparse-grooming network. Through propernetwork design and traffic engineering, it is possible for a sparse-grooming network to achieve similar network performance as a net-work which has grooming capability at every node. We investigatethe problem of designing such a sparse-grooming WDM mesh net-work. The problem is mathematically formulated and several designschemes are proposed. Illustrative numerical results from the mathe-matical formulation as well as heuristics show that, by properly choos-ing the grooming nodes, a network with sparse-grooming capabilitycan achieve good network performance and the network cost can besignificantly reduced.

I. INTRODUCTION

Wavelength-division multiplexing (WDM) technology pro-vides huge amount of link bandwidth to today’s telecom net-works. As WDM technology keeps maturing, there is a band-width gap between a wavelength channel’s transmission speed(over a gigabit per second (Gbps)) and the capacity require-ment of customers’ connections, which may vary in range fromSTS-1 (51.84 Mbps) (or lower), up to full wavelength-channelcapacity. In order to ensure the most efficient utilization ofnetwork resources, and to maximize revenue from existingcapacity, the low-speed traffic streams need to be efficiently“groomed” onto high-capacity optical channels (commonly re-ferred to as lightpaths).

Most previous research on traffic grooming was conductedon SONET/WDM ring networks (e.g., [1] - [4]). It is wellaccepted that WDM mesh networks are more scalable, flexi-ble, and resource efficient than WDM/SONET ring networks.Hence, WDM mesh networks are very suitable for long-haulbackbone networking applications, and they are the focus ofour current research. With the emergence of grooming opticalcrossconnects (G-OXCs), traffic grooming in WDM mesh net-works becomes an extremely important problem and it has been

This work has been supported by the US National Science Foundation underGrant Nos. NCR-9508239 and ANI-9805285, and by Sprint Advanced Tech-nology Laboratories.

gaining more attention in industry and academe [5] - [8]. In allprevious research on traffic grooming in WDM mesh networks,it is assumed that every network node has traffic-grooming ca-pability, which may not be practical or cost-effective in a na-tionwide WDM backbone network.

Figure 1 shows a sample G-OXC architecture. There aretwo switching fabrics in this OXC, a wavelength-switchingfabric (W-Fabric) and a grooming fabric (G-Fabric). Becausea grooming OXC may be more costly than an OXC withoutgrooming capability (i.e., the OXCs which only have the W-Fabric), and the G-OXCs from different vendors may have dif-ferent grooming capability, in a multi-vendor inter-operationalWDM mesh network, only a few network nodes may havetraffic-grooming capability. We call this type of network a“sparse-grooming network”, and we call a node which hastraffic-grooming capability to be a grooming node (G-Node).

Hence, the problem of designing a sparse-grooming WDMmesh network is a very important and practical problem. Tothe best of our knowledge, no previous work have addressedthe sparse-grooming problem in the literature. In this study,we investigate the problem of efficiently designing a sparse-grooming WDM mesh network for a given set of traffic re-quests via theoretical formulation as well as simulation exper-iments. The results from our research indicate that, throughcareful network design, a sparse-grooming WDM network canachieve similar network performance as a full-grooming net-work, while significantly reducing the network cost.

Fig. 1. A sample architecture of a G-OXC. Note that the G-Fabric is presentonly in a grooming node, while the W-Fabric is present in all nodes.

Figure 2 shows an example of designing a sparse-groomingWDM mesh network. Figure 2(a) shows a six-node network,where each edge represents a pair of unidirectional fiber links.For simplicity of illustration, let us assume that each fiber sup-ports one wavelength channel, which can carry two low-speed

1

3

5

2

0

4

(a) Network 1: A 6-node network

��������������������������������������������������������������������������������������������������

1

5

2

0

3 4

(b) Design I

��������������������������������������������������������������������������������������������������

1

5

2

0

3 4

(c) Design II

Fig. 2. A sample network and two sparse-grooming network designs.

connection requests. And only one G-OXC is allowed to beused in the network. Assume that there are four low-speed re-quests, among which three are between node pair

����� �and

one between node pair������ �

. Two network designs are shownin Fig. 2. The shaded nodes represent the G-Nodes, and thedashed lines represent established lightpaths.

In Design I (Fig. 2(b)), node

is the G-Node. Two light-paths –

������ �and

����� �– are established. Three requests can

be satisfied by these lightpaths: two between����� �

, and onebetween

������ �. In Design II (Fig. 2(c)), three lightpaths are set

up:����� �

,������ �

, and��������

. Note that in Design II all requestscan be satisfied. Thus, this simple example illustrates that im-proved performance can be achieved by carefully chosing theG-Node and engineering the traffic in the network.

In this paper, we study the problem of how to efficiently de-sign such a sparse-grooming WDM mesh network. SectionII gives the formal problem statement. Then, mathematicalformulations for two objective functions are presented. Fastheuristic algorithms, which can be used to handle large net-work topologies, are proposed in Section III. Then, illustrativenumerical results from mathematical formulations as well asheuristics are shown and analyzed in Section IV. Finally, Sec-tion V concludes the study.

II. PROBLEM STATEMENT AND MATHEMATICAL

FORMULATION

We formally define the problem as follows: given a networktopology � ��������� , where

�represents the node set and

�rep-

resents the link set of the network; given traffic matrix � , whereeach element represents the number of low-speed requests be-tween a node pair; assume that each wavelength channel cansupport � low-speed requests (where � is known as groomingratio); design the network such that either one of following twoobjectives can be optimized:

1) For a given certain amount of network resources (numberof G-OXCs ��� , and number of wavelengths � on eachfiber), maximize the network throughput.

2) Carry all traffic requests, whilesimultaneously minimiz-ing the overall network cost, which is determined by thenumber of wavelength channels and G-OXCs used in thenetwork.

Figure 3 shows a WDM network in which two low-speedrequests ( � � and � � ) are being carried. Both requests are fromnode

�to node � . In Fig. 3, nodes

�and � are G-Nodes, and

only their G-Fabrics are shown. Nodes�, , and � are equipped

with OXCs without grooming capability. Figure 3 shows that,

in a sparse-grooming mesh network, a low-speed request canbe carried either by a single lightpath ( � � ) or by traversingmultiple lightpaths and G-Nodes ( � � ).

Fig. 3. A sample sparse-grooming WDM network which carries two requestsusing four lightpaths.

We formulate the problem mathematically and it turns out tobe an integer linear program (ILP). Due to space limitations,only the most important equations of the formulation are pre-sented here. The following notations and variables are used:! �#"$�&%'�

,�#()�+*,�

, and��-���.,�

denote the end nodes of a fiber, alightpath, and a low-speed request, respectively.! There are four types of lightpaths, represented by

�#()�+*,�,�#(0/1�+*,�

,�#()�+* /2�

and�#(�/��+* /2�

.�#()�+*,�

denotes a lightpath whichdoes not connect with any G-Fabric at its end nodes, e.g.,3 ����� � � in Fig. 3.

�#( / �+*,�,�#()�+* / �

, and�#( / �+* / �

denote thelightpaths which are connected to G-Fabrics at the sourcenode, the destination node, and both nodes, respectively,e.g.,

3 � ,3 �

, and3 �

in Fig. 3(b). We use�54 6

,��4 7 6

,��4 6�7

and�84 7 6�7

to denote the number of each type of lightpaths.!:9 4 6;=< , 94 7 6;=< , 9

4 6 7;=< , and 9 4 7 6 7;=< denote the number of wave-

lengths which have been used to support each type oflightpaths, on fiber

�#"$�&%'�. 9 ;=< denotes the number of

fiber links between node pair�#"$�&%'�

.!?>A@�B4 6 , >A@�B4 7 6 , >A@�B4 6 7 , and >C@�B4 7 6 7 denote the total amount of trafficbetween node pair

��-���.,�, which are being carried by the

lightpaths�#()�+*,�

,�#(&/1�+*,�

,�#()�+* /2�

, and�#(�/1�+* /2�

, respectively.> @�B denotes the successfully carried traffic between��-���.,�

.� @�B denotes the total offered traffic between

��-���.,�. Note

that > @�BED � @�B .!GF 4'HI�if node

(is a G-Node; otherwise, F 4'HJ

.

A. Maximize Network Throughput

Given � wavelengths per fiber link and ��� as the numberof G-OXCs which can be used in the network, the problem canbe formulated as follows:

! Objective Function:

F���� (+" (������@�� B

> @�B (1)

! Constraints:

;

9 4 6;� H� <9 4 6 < (2)

4 � 6� 9 4 6;=<�� 9 4 7 6;=<�� 9 4 6 7;=<�� 9 4 7 6 7;=< � D ��� 9 ;=< (3)

> @�B@ 7 � 4 > @�B4 7 7 H > @�B 7 B � 6 > @�B 7 6�7 (4)

@�� B

> @�B4 7 6 7 D ��� ��4 7 6�7 (5)

> @�B@�B � > @�B@ 7 B � > @�B@ 6�7 � > @�B@ 7 6�7 H > @�B (6)

> @�B@�B � > @�B@�B 7 � > @�B4 7 B � > @�B4 7 B 7 H > @�B (7) 6 ����4 7 6 � ��4 7 6�7 � �,6�4 7 � �,6�724 7�� D F 4 ��� (8)

4 F 4 D � � (9)

Equation (2) is the flow-conservation equation at node � forestablishing lightpaths

�#()�+*,�, which may use node � as an inter-

mediate node. There are similar equations for lightpaths�#( /1�+*,�

,�#(0/1�+*,�, and

�#(�/1�+* /2�. Equation (3) is the resource-constraint equa-

tion for a fiber link�#"$�&%'�

, i.e., the total number of lightpaths(four types) carried by fiber

�#"$�&%'�cannot exceed the total

number of wavelength channels in fiber�#"$�&%'�

. Equation (4)is the flow-conservation equation at any intermediate node �for routing the requests between

��-���.,�, which may use node

� as an intermediate G-Node. Equation (5) is the resource-constraint equation for lightpaths

�#( /1�+* /2�, i.e., the total number

of low-speed connections carried by the lightpath�#( /1�+* /2�

can-not exceed the overall capacity of lightpath

�#( /1�+* /2�. There are

similar equations for lightpaths�#()�+*,�

,�#( /1�+*,�

, and�#()�+*,/2�

. Equa-tion (6) guarantees that, for a given node pair

��-���.,�, the amount

of traffic successfully flowing out from the source node shouldbe equal to the amount of traffic that can be successfully carriedbetween

��-���.,�. Equation (7) captures the similar constraint at

the destination side. Equation (8) ensures that node(

is a G-Node if there is a lightpath connected to its G-Fabric (i.e., node(

initiates or terminates a lightpath at its G-Fabric). � is a largeconstant, which can be the upper bound on the maximum num-ber of lightpaths that can originate from any node. Equation (9)ensures that there are at most � � G-Nodes in the network.

B. Minimize Network Cost

Let ��� denote the cost of supporting one wavelength chan-nel in the network, and � � denote the extra cost of employinga G-OXC instead of an OXC without grooming capability. Thesecond network design objective can be achieved using the fol-lowing formulation. Most equations are the same as those inSection II-A, and we only present the different ones below:

! Objective Function:

F (+% (+" (������ ��� ��� � � ��� � � (10)

! Constraints: > @�B H � @�B (11)

III. HEURISTIC APPROACHES

The computational complexity makes the ILP formulationsonly suitable for designing small or moderate-sized WDM net-works. When the network size is large, i.e., there are aboutseveral tens of network nodes (e.g., Fig. 4), efficient heuristicsare needed to solve the problem.

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6

7 9

11

123

810

13

1418

17

16

15

19

20

21

22

23

24

Fig. 4. Sample network topology used in this study.

For a given network topology and given values of � � and � ,we design a heuristic � � � � � � �

which chooses G-Nodes androutes the traffic requests to maximize the network throughput(for the first objective function). The second objective functioncan be achieved by starting with small values of � � and � , andgradually increasing � � and � until all the traffic requests aresatisfied. In this section, we concentrate on the development ofsuch a heuristic � � � � � � �

for the first objective function.Given a WDM network topology � ��������� and a list of traffic

demands � , the heuristic � � � � � � �is composed of the follow-

ing two steps.1) Choose � � nodes as G-Nodes from

�, based on certain

cost function ��� �! � for a given node

(to be elaboratedbelow).

2) Route the traffic requests on the network subject to thenetwork resource constraints. Let �#" � � � denote the al-gorithm to be used to route the traffic requests. Whenthe traffic is static (known a priori), the algorithm maycontain some backtracking procedure; when the traffic isdynamic or incremental, no backtracking is allowed, andthe requests will be carried in a first-come first-servedmanner. In this study, we consider static traffic.

A. Grooming-Node-Selection Schemes

We propose and study the characteristics of three differentcost functions for selecting G-Nodes, namely nodal-degree se-lection, bypass-traffic selection, and random selection. Notethat some ideas on these node-selection schemes are borrowedfrom sparse-wavelength-converter-placement studies [9], [10].

! Nodal-Degree Selection: In this scheme, the first � �nodes which have the maximum nodal degree are pickedto be G-Nodes. If several nodes have same nodal degreeand only some of them can be chosen, random selection isused to break any ties.! Bypass-Traffic Selection: For a given node

, �� �! � is

computed as the total amount of traffic which may bypassthe node, assuming each traffic request is routed on thephysical network topology using a standard shortest-pathrouting algorithm. The � � nodes which have maximumamount of bypass traffic can be selected as the G-Nodes.Instead of routing the traffic requests between a node pair��-���.,�

using a single shortest-path route, it may be alsopossible to compute � � ��� ���

alternate paths between��-���.,�and bifurcate the traffic among these � alternate

paths.! Random Selection: In this scheme, � � nodes are ran-domly picked to be G-Nodes.

One can also design other schemes to select the G-Nodes.Similar heuristics exist in the study of sparse wavelength con-version in WDM networks [9], [10].

B. Traffic Routing Scheme

A simple algorithm is used in our study to perform trafficrouting after the network resources have been determined andallocated. Given a set of traffic requests, a request list is gen-erated based on a random permutation of all the requests. Therequests are then served sequentially following their order inthe list.

3such random lists can be generated and tried. The

best one among these3

results will be taken as the final result.In our study, we choose

3as � . If

3is equal to

�, the traf-

fic pattern is equivalent to incremental traffic where connectionrequests arrive one at a time.

IV. ILLUSTRATIVE NUMERICAL EXAMPLES

Figure 5 shows an illustrative numerical example based ona sample network shown in Fig. 2(a). For simplicity of expo-sition, let each fiber in this example support two wavelengths( � H �

), each wavelength can carry two low-speed connec-tion requests ( � H �

), and there is one G-OXC which needsto be placed in the network ( � � H �

). Figure 5(a) shows arandomly-generated traffic matrix, in which each element rep-resents the number of low-speed connection requests between anode pair. The total number of low-speed requests is � � in thisexample. We use “CPLEX”, a commercial optimization soft-ware package, to solve the ILP formulation. For this example,our study shows that node

�is the best candidate node to be the

G-Node. Then, we force our formulation to artificially selecteach of the other nodes separately to be the G-Node and com-pare the network throughput with that of the best design (i.e.,selecting node

�). The performance comparison is shown in

Fig. 5(b). The horizontal axis in Fig. 5(b) represents the nodechosen as the G-Node. The vertical axis in Fig. 5(b) shows theoptimal network throughput (obtained via ILP formulation) bychoosing the corresponding node as the G-Node. It is observedthat 100% network throughput can be achieved if the G-Node

NODE 1

NODE 0

NODE 2

NODE 3

NODE 4

NODE 1

NODE 5

NODE 5NODE 0

0

0

0

0

0

0

0 0

0

0

0

1 1 1

11 1

1

1

0 1

1

1

1

0

02 2

2

2

2

2

2

2

2

2

NODE 2 NODE 3 NODE 4

(a) Traffic Matrix

(b) Network Throughput vs Grooming Node Id

Fig. 5. Illustrative Result from ILP Formulation for Topology 1 assumingonly one node has grooming capability.

is node�; at the other performance extreme, if node is cho-

sen as the G-Node, the network throughput is below 85%. Theresults indicate that a network operator can increase the net-work throughput as well as reduce the network cost (using lessgrooming equipment in the network) by carefully designing asparse-grooming WDM mesh network.

When a network has several tens of nodes, the ILP ap-proach becomes computationally intractable; hence, heuristicwill need to be used. Figures 6 and 7 show the results obtainedon the network topology of Fig. 4, using the heuristic algo-rithms proposed in Section III. The network contains 24 nodesand 43 bidirectional fiber links. Traffic demands are randomlygenerated between each node pair with uniform distributionbetween

�1 � � � . Each fiber link can support eight wavelengthchannels ( � H�� �

. Different values of the grooming ratio �will be investigated below.

Figure 6 shows the performance comparison between differ-ent G-Node selection schemes. It can be observed that ran-dom selection does not perform as well as the others. Selectingthe G-Node by bypassing traffic achieves better performancethan selecting the G-Node by nodal degree in most cases. Thisis because the Bypass-Traffic-Selection Scheme considers thenetwork topology as well as the traffic intensity at each node.We can also observe that, if the grooming ratio is equal to eight( � H�� ), when the number of G-Nodes exceed a certain bound( � � H ���

in this case), no additional performance gain can beachieved by having more grooming nodes in the network. Wecan also observe that, when the grooming ratio is large (e.g.,� H��

), having more grooming nodes will achieve better net-work performance than the case when the grooming ratio issmall (e.g., � H ).

Figure 7 shows different network costs based on different

Fig. 6. Performance comparison between different G-Node selection schemesapplied to the network in Fig. 4.

values of the cost ratio � , which is defined as follows:

� H ����� � � (12)

where �#� denotes the cost (equipment cost, operational cost,etc.) to support one wavelength channel in the network, and � �denotes the cost to employ grooming capability in a networknode. The network cost � < can be represented as:

� < H � ��� � � � � � ��� HI� � � � � � � � � � � (13)

The horizontal axis� � � � � � in Fig. 7 represents, for a given

number of wavelength channels on each fiber link (W), howmany G-Nodes

� � � � are needed to satisfy all the requests. Bynormalizing the cost � � to be one unit, the vertical axis repre-sents the overall network cost computed using Eqn. (13).

We can observe in Fig. 7 that, for a given cost ratio � ,an optimal design with minimal overall network cost can beachieved. This optimal configuration reflects the cost trade-offbetween a grooming node and a wavelength channel. When thecost of a wavelength channel is cheaper compared to the costof a grooming node (for � H �� �

, and � H �� � ), more wave-length channels and less grooming nodes should be used, andvice versa (for � H �

). Note that, although the cost of a wave-length channel will decrease as WDM technology keeps ma-turing, supporting more wavelength channels in a nationwideWDM backbone network may still be expensive because of thelarge geographic distance between the network nodes, the num-ber of fiber links a backbone network may have, and the costfor network monitoring, maintainence, and management.

V. CONCLUSION

This study is devoted to the problem of designing a WDMmesh backbone network with sparse traffic-grooming capabil-ity. The mathematical formulations (ILPs) for two design ob-jective functions were presented. Due to the large computa-tional complexity of ILPs, three heuristic algorithms were alsoproposed to solve large instances of the problem. Our resultsfrom both the mathematical formulations and heuristics show

Fig. 7. Network cost vs. network resources based on different cost ratio � .

that, by employing a limited number of grooming nodes, thenetwork capacity can be used more efficiently and the networkperformance can be improved significantly. We also showedthat it is possible to find a balance between the number of wave-length channels and the number of grooming nodes used in thenetwork. This balance will eventually reduce the network cost.Further study is needed on the effect of sparse grooming on dy-namic traffic, and more intelligent heuristic algorithms couldalso be developed.

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