348 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 3, MARCH 1996

ractical Constraints in Growt twave Networks

Kenneth A. Falcone, Member, IEEE, and Ozan K. Tonguz, Member, IEEE

Abstract-The amount of fiber required, propagation delay, and length of the longest link are significant design constraints in spa- tially large networks. This paper examines these characteristics from the viewpoint of growth and compares basic networks with hierarchical ones in terms of these characteristics. Results show that, when considering growth from three nodes, a star network randomly placed with a uniform distribution uses Iess fiber than a dual ring until there are 57 nodes. As the networks become large, the star has the smallest propagation delay and the clual ring uses the least amount of fiber. A two-level network having a star on the upper level and dual rings on the lower network level performs well in both categories by using 1.38 times as much fiber as the dual ring and having 1.65 times the propagation delay of a star as the number of nodes becomes large.

I. INTRODUCTION ASIC characteristics of interconnection networks where the number of nodes is fixed have been studied by several

authors [ 11-[ lo]. When examining the properties of intercon- nection networks from the viewpoint of growth, however, the basic characteristics change. A ring usually uses less fiber than a star, but as links are removed to add new nodes obsolete fiber is created. While the cost of installing this fiber has already been paid, it is no longer used. The offered load, amount of fiber, and propagation delay for a network will increase as it grows larger. After a certain point, it may become beneficial to break up this network. Previous work has considered the layout and connection of networks from the perspective of fault- tolerance [l], [2], average number of hops [8], [9], or physical implementation [lo]. The main objective of this work is to find the general tendencies of growing networks and consider the advantages and disadvantages of using hierarchical networks to reduce the shortcomings of single-level networks.

This paper considers two fault tolerant physical-network topologies, the star and dual ring, and analyzes them with respect to the amount of fiber required and the length of the longest link as the network grows. The amount of fiber used in a ring grown from three to 100 nodes is compared to that which may be obtained by optimization, and it is shown that the

Paper approved by M. S. Goodman, the Editor for Optical Switching of the IEEE Communications Society. Manuscript received April 5, 1994; revised March 24, 1995. This work was supported by the National Science Foundation Research Initiation Award under Grant ECS-9309701. This paper was presented in part at the 7th Annual Meeting of the IEEE Lasers and Elctro-Optics Society, Boston, MA, November 1994.

K. A. Falcone was with the Photonics Research Laboratory, Department of Electrical and Computer Engineering, State University of New York at Buffalo, Buffalo, NY 14260 USA. He is now with the Radar/Communications Research Division, GE Inc., Schenectady, NY 12301 USA.

0. K. Tonguz is with the Photonics Research Laboratory, Department of Electrical and Computer Engineering, State University of New York at Buffalo, Buffalo, NY 14260 USA.

Publisher Item Identifier S 0090-6778(96)01770-9.

difference is expected to be less than 10%. When comparing to a dual ring grown from three nodes, the total amount of fiber required for a star is less until there are 57 nodes for a unifom distribution and 89 nodes for a Gaussian distribution.

The basic networks are then compared to hierarchical net- works in terms of the amount ofjiber required, and the average propagation delay by assuming that the nodes are uniformly spaced throughout a square area and neglecting growth. It is shown that, as the networks become large, the dual ring uses the least amount of fiber. Results indicate that a single-level star network requires at least twice as much fiber as a dual ring for 28 or more nodes, a two-level star requires twice as much at around 450, and a three-level star at about 18 000. The star has the smallest propagation delay for large networks. A single-level dual ring network has at least twice the average propagation delay of a star for 36 or more nodes, the two-level ring has twice as much at about 280, and the three-level ring at around 1900. In contrast, it is shown that a properly optimized two-level network having a star on the upper level and rings on the lower level will never require twice as much fiber as the ring or have twice the propagation delay of a star.

The remainder of this paper is organized as follows. Section 11 analyzes the basic networks from the viewpoint of growth. Section I11 compares the basic networks to hierarchical networks. Results are discussed in Section IV. Finally, Section V contains the conclusions of the study.

11. BASIC NETWORKS The amount of fiber required can be the dominant monetary

cost in a large network. The first installation cost of 1 km of fiber in the downtown Buffalo area, for example, is about $16000 + the cost of right of way for an existing trench and $360000 + the cost of right of way when a trench must also be installed [ll], [12]. While the actual installation costs will depend on what trenches exist and how much can be shared by the fiber links, the results here serve to predict what the needs of a network will be as it grows. Average propagation delay is a significant design parameter, especially when the bit. rate is high 161, 1131. The size of the longest link in the network is another important characteristic. It determines what the maximum values of bit rate [3] and propagation delay are. All variables used in the analysis are described in Table I.

To perform the analysis, assumptions must be made about the distribution of the nodes. The first distribution assumed is deterministic, with the nodes uniformly spaced throughout a square area. This is similar to what would be seen looking at houses in a residential area. For random distributions, a uniform distribution is used to represent a network which serves a given area. A Gaussian distribution may represent an

0090-6778/96$05.00 0 1996 IEEE

FALCONE AND TONGUZ: PRACTICAL CONSTRAINTS IN GROWTH OF LIGHTWAVE NETWORKS 349

Variable

v

1

N

F

P

L

R

k

i

n,

f,

p ,

Definition

velocity of light in the fiber

uniform spacing distance between nodes

total number of nodes in the network

amount of fiber required in the network

average propagation delay in the network

length of the longest link in the network

average traffic load offered by one node

total number of levels

level of a network between 0 and k

number of nodes in a subnetwork at the ith level

amount of fiber required in a network at the ith level

average propagation delay of a subnetwork at the ith level

urban area. The idea is that any characteristic that behaves the same for deterministic (uniform spacing), constrained (uniform distribution), and infinite space (Gaussian) distributions should be valid for any network.

Four different levels of topologies are required for this work. The placement of fiber links between the nodes defines the physical-network topology. These fiber links must be contained within the trench level, which will not be discussed in detail in this paper. Fiber links that share trenches could be increased by algorithms such as that in [14] to reduce costs. Traffic flow between nodes is called the physical-path topology [15]. Amount of fiber and longest link are deter- mined for the physical-network topology. Since the passive star and dual ring have the lowest propagation delay [3], the average propagation delay calculations assume that these are the physical-path topologies. The logical topology is the allowable communication between nodes, and must usually be a complete graph [15]. The logical topology represents all of the OS1 layers above the physical layer, which will not be considered in this work.

A. Star There are many characteristics of a star network which may

make it the best choice of topology for a specific network application. It can tolerate faults in the fiber, for example, whereas other topologies require additional hardware to do so. If a star coupler is used at the center, then the average propagation delay is lower than in other networks. Also, the bit rate which can be supported for a small number of nodes is usually higher than for other network types. When the network grows, adding one node is not difficult since this only involves running additional fiber from the new node to the star coupler. If the network becomes too large, however, breaking it up into two networks serving different regions is not practical since all existing fiber links meet in a central location.

-A -P

(a) (b)

Fig. 1. (a) An example of a star network where the nodes are uniformly distributed throughout a square area. (b) An example of a ring network where the nodes are uniformly distributed throughout a square area. The addition of the sixteenth node is depicted.

Consider a node layout where the nodes are deterministi- cally distributed with equal spacing throughout a square area. The number of nodes on a side is fl and so the spacing between nodes is

2A 1 = - - fl

where 2A is the length of a side of the square as shown in Fig. 1. By reversing the variables

The amount of fiber pair required is given by [3]

The average propagation delay between any two nodes is proportional to this and is given by

2F N u

P = - = 1.5304A/~. (4)

The distance of the corner node on one axis is -1 = (1 - 1 / f i ) A , and so the total distance of the longest link is given by

In a large network, it is not realistic to expect the nodes to be arranged in a deterministic pattem such as that previously analyzed. The calculations, however, are simple and in closed- form. This may make them useful as an approximation to what the actual expected values are. A more realistic way to examine the characteristics of a network may be to assume that the network must serve a given spatial area. The nodes in the area are then uniformly distributed, as shown in Fig. l(a). By assuming that the central coupler is at the center of the distribution, the characteristics (amount of fiber required in the network, average propagation delay in the network, and the length of the longest link in the network) can be found using order statistics as [16]

E { F } = 0.7652NA (6)

E { P } = 1.5304NA/u (7)

350 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 3, MARCH 1996

. ~ z sin-l(z-l) - sin-’ ( d a ) ) dr] A.

(8)

Note that expressions (6) and (7) are identical to expressions (3) and (4).

If the network is not constrained inside an area, a spatial two-dimensional Gaussian distribution is a reasonable assump- tion. The characteristics can be found as [16]

(

E { F } = 1.2533Na E { P } = 2.5066Na/v

E { L } = L1Na2/ -2In( l - z)zN-’dr. (11)

B. Ring A ring network also has desirable characteristics which may

make it the best choice of topology. A ring uses less fiber than a star network. If two counter-rotating rings are implemented, it can tolerate faults in the fiber [17]. By taking advantage of the counter-rotating nature of the network, the propagation delay can be reduced considerably [3]. A high bit rate can be supported by using regeneration at the nodes. Adding one node to the network requires that the ring be cut so that the new node, with new fiber links, can be inserted. This makes some fiber obsolete, and its installation costs are not accounted for if growth is neglected. Breaking up the ring into two networks serving different regions is fairly easy to do, albeit this also creates some obsolete fiber.

The first layout considered is again where the nodes are

represents the penalty incurred by not knowing where future nodes will be located. The average propagation delay for a dual ring can also be found from the amount of fiber, and is given by 131

if two counterrotating rings are used and the direction is taken advantage of in routing. The size of all links are the same, and so the longest link in the network is simply

(15)

When random placement is considered, the optimum ring interconnection network must be found via “traveling salesman algorithms” [l]. In such a case it is not possible to find the distributions of the characteristics being sought. The expected values are found by simulating uniform and Gaussian node distributions, with an example shown in Fig. l(b).

The first method implemented was to start with three nodes, which can only be connected in the optimal way. The network grows as nodes are added one at a time. The location of the node in the network is found by inserting it between each pair of two consecutive nodes until the amount of fiber in use is minimized. The node is added at this point in the ring, and the amount of fiber which connected the pair before the new node was inserted is then considered to be obsolete. One thousand growth simulations were run from three to 100 nodes.

The layout of the network for the growth simulation is not the optimum. To discover how much difference this makes as the network becomes large, results were compared with locally optimum connections, found using the three-opt method of solving the traveling salesman problem [18]. One hundred simulations were run for each multiple of 10 nodes.

2A L = l = - - ma

C. Results deterministically distributed with equal spacing. The spacing between nodes is given by expression (l), and the amount of fiber in w e is

Figures 2 and 3 show the results for the normalized amount of fiber required. When considering the amount of fiber in use for a ring, the largest deviation of the growth simulation -

F = Nl = 2JNA. (12) from the three-opt values is less than 10%. This indicates that

Uniform spacing requires that the number of nodes be a perfect square, but by examining Fig. 1(b) one can estimate the effect of adding one node at a time. Consider that adding each node to the network adds the length of two links to the amount of fiber required. The total amount of fiber required considering growth from i to j nodes can then be found by

= 8 4 A - 61hA (13)

which indicates that there is a penalty of a factor of four for large growth. This ratio is supported by simulations, and

growing a network does not have a substantial effect on the amount of fiber in use. When considering the total amount of fiber, which includes the obsolete fiber, the ring can use more fiber than a star. Starting from three nodes, the star uses less fiber until the size of the network reaches 57 nodes for a uniform distribution and 89 nodes for Gaussian (see Fig. 3). If the initial size of the network is at least seven nodes for a uniform distribution or 17 nodes for Gaussian, then the dual ring will always use less fiber. The effect is less significant in large networks because, as the ring grows large within a fixed area, the amount of fiber in use becomes large while the links become shorter. The amount of obsolete fiber created by adding a node is then small compared to the total amount of fiber used in the ring. The uniform spacing values are exact for the star and within 21.1% for the ring growth simulation results for up to 1000 nodes.

FALCONE AND TONGUZ: PRACTICAL CONSTRAINTS IN GROWTH OF LIGHTWAVE NETWORKS 351

Umform Spacing Rmg U N f O ~ Izlw - Total

Umfonn Spacmg - Total

1 ‘ I 1 10 100 1000

Number of Nodes

Fig. 2. area where A = 1.

Normalized amount of fiber in single-level networks within a square

2.5

2.0

B .j

P

8 1.5

1

0.5

0.0 10 100 lo00

Number of Nodes

Fig. 4. Normalized longest link in single-level networks within a square area where A = 1.

a uniform distribution for this parameter. Part of the difference for a star with a small number of nodes is due to the fact that the central coupler is located at the center of the distribution rather than the mean location of the existing nodes. For a passive star the important parameter would be the distance from the farthest node to the second farthest node. The joint distribution of these two distances for the star can also be found 1191.

111. HIERARCHICAL NETWORKS

I 1 10 100 lo00

Number of Nodes

Normalized amount of fiber in single-level networks with a Gaussian Fig. 3. distnbution where (T = 1.

While it is difficult to compare the three distributions directly, the ratio of the mean amount of fiber required by a star to that required by a ring is fairly consistent for the three distributions over the range of number of nodes considered. The mean * one standard deviation values were calculated to indicate the confidence that a value for a single random network will be near the mean, but results for the ring are not plotted to improve readability. The standard deviation for the total amount of fiber required for a Gaussian distribution having three nodes is 37% of the mean, at 20 nodes it reduces to 16% of the mean, and at 1000 nodes it is only 2.8% of the mean. These are the largest values for the ring, and the other curves have smaller standard deviations at these points. Values for the star are 28% and 66% for uniform and Gaussian distributions, respectively.

Figures 4 and 5 show results for the normalized longest link in the network. Integrals from expressions (8) and (11) were solved by using a commercially available (Macsyma) Newton-Cotes eighth order polynomial adaptive quadrature routine. The longest link in a ring network is larger than that in a star until there are at least 11 nodes for a Uniform distribution and nine nodes for Gaussian. It is clear that the uniform spacing does not act as a reasonable approximation to

There are a number of limitations on the size that a network can grow to before problems develop. These concerns may be related to the needs of the access method such as bit rate, propagation delay, or the number of wavelengths as in 131-[7], and 1201. An increase in bit rate or number of wavelengths may be required as more nodes-and their corresponding traffic loads-are added to the network. Beyond a certain point, further increases will not be practical. The propagation delay will also increase with the number of nodes, perhaps beyond a value which is acceptable to the protocol.

Limitations may also be related to physical implementation of the network. As more nodes are added, the excess capacity or tolerance to faults may decrease to a level which is no longer capable of providing a sufficient quality of service [5 ] . Another limitation on network size is the capability of individual components, such as a star coupler. Since the existing coupler has a fixed number of ports, switching to a larger coupler may reduce the power budget enough to degrade performance below the desired bit error rate or may not even be possible.

To offset these problems, a large network may be broken up into smaller subnetworks. These small subnetworks can then be connected by a higher level subnetwork, as shown in Fig. 6. The assignment of nodes to these lower level subnetworks can be optimized to reduce propagation delay in a ring network, or the amount of fiber in a star. Note that for a star, the hierarchical layout could represent shared trenches for a single-level network. In this case we can see the penalty in propagation delay caused by decreasing the cost of the trench level, although this is not the best way to share conduit.

352 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO 3, MARCH 1996

Number of Nodes

Fig 5 distnbution where u = 1

Normalized longest link in single-level networks with a Gaussian

(b)

Fig. 6. ring.

A. Amount of Fiber The total amount of fiber in a multilevel network will

be found by using the uniform spacing approximations of Section 11. Multiplying the amount of fiber in a lower level subnetwork by the number of lower level subnetworks and adding the amount in the upper level subnetwork gives the total amount of fiber. This can be used to find the amount of fiber in the composite network as

(16)

Block diagrams of two-level networks: a) star over star; b) ring over

F = n1fo + f l

F = n2n1fo + n 2 f 1 + f2

for a two-level network, and

(17)

for a three-level network. Expression (3) can be modified to give the amount of fiber for a star network having two levels so that [16]

where the minimum amount of fiber is used when n1 = ( $ ) 2 / 3 . For three levels the value is

F = (-no + 6 ~ ~ 1 + n2)0.7652A. (19)

TABLE Il AMOUNT OF FIBER IN MULTILEVEL NETWORKS

Levels (k) 1 Network I Amount of Fiber (2) 1 I Sta r I N2

Ring I ,

I I

I Ring/Star I 0.76525 -t 2 6

The same calculations can be done for the dual ring, and the results are given in Table 11. Note that the length of the longest link will be shorter for a multilevel network using stars, and longer for one using rings.

Hybrid networks may also be of interest. Two types of two-level hybrid networks will be considered. The first is a network having a star on the upper level, while the lower level subnetworks are rings. The amount of fiber can be expressed as

(20) F = ( 2 f i + 0.7652ns)A

where ns is the number of nodes in the upper level star subnetwork and the minimum amount of fiber is used when n, = 0. The other example is a network with a ring on the upper level, with star networks forming the lower level subnetworks. The amount of fiber in this case is

where nr is the number of nodes in the upper level ring network and the minimum amount of fiber is used when n, = (0.7652N)2/3.

B. Average Propagation Delay

The average propagation delay also uses the uniform spac- ing approximation. All traffic in a ring must pass through at least one lower level subnetwork, and a factor of ( N - no) / (N - 1) must pass through the upper level subnetwork and another lower level subnetwork. The average propagation delay due to fiber is then given by

N - no P = p o + - (P1 + P O ) N - 1

for a two-level network and N - no

P=po+--- N - l (PI +PO)

N - no nlno - n1

N - 1 n1n0 - I +- ( P 2 + P1) (23)

for a three-level network. For a multilevel star where the router is located at the central coupler as in Fig. 6(a), each

FALCONE AND TONGUZ PRACTICAL CONSTRAINTS IN GROWTH OF LIGHTWAVE NETWORKS 353

Levels ( I C )

1

2

3

TABLE 111 AVERAGE PROPAGATION DELAY IN MULTILEVEL NETWORKS

I Network I Average Delay (5)

Star I 1.5304

packet travels through an entire lower level subnetwork if its destination is a node within its own subnetwork, but only travels through half of the subnetwork when traveling to higher network levels. As a result, the propagation delay for a three level network is found by

Although it can easily be added, no additional latency due to packets being switched between the subnetworks will be considered here.

The average propagation delay for a multilevel dual ring is found by using integers for the number of nodes in the upper level subnetworks, and a rational number to represent the average number of nodes in the lower level ones. As a result of this, the propagation delay in the upper subnetwork levels is represented by the exact expression (14), while the lower level is represented by the average of the propagation delay for odd and even values. The propagation delay for the composite network is then [16]

1 N - no

1 N - n

for a two-level dual ring network, and

N - no 1

for a three-level one. The other average propagation delays are given in Table 111.

C. Fault Tolerance How does dividing a network into multiple levels affect

the fault tolerance of the network? The star and dual ring topologies are both tolerant to faults in the fiber. They will also operate if a single access coupler fails. The difference between the two is that, in either case, the dual ring still provides a connection to the affected node while the star does not. The star provides the same service to other nodes in the network, while the dual ring suffers a one-half reduction in capacity. This reduction, however, will only occur if full advantage is taken of the topology by sending traffic in only one direction.

354 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 44, NO. 3, MARCH 1996

10 100 1000 loo00 1OOOOO Number of Nodes (N)

Fig. 7. Normalized amount of fiber in multllevel networks.

If these topologies are used in multilevel networks, the performance on the lowest level would be about the same. When examining performance on the highest level, it is clear that a fault in a star would isolate a large number of nodes from the rest of the network. A fault in a dual ring can reduce traffic handling ability in a part of the network that the majority of packets must use. Another limitation is that a dud ring can only compensate for one fault; when there are multiple faults, the network is divided into fragments. This seems to indicate that additional equipment-such as a four-fiber ring [17] or dual homing [21]-is required to provide adequate protection from faults in the higher network levels.

IV. DISCUSSION

Figures 7 and 8 show the results of the expressions from Section 111. The number of nodes in each level for two- and three-level star networks have been optimized to reduce the amount of fiber required. For the two- and three-level ring networks, optimization is with respect to the propagation delay. The two-level starhng and ring/star networks use no = n1 = a. It is assumed that a subnetwork must have at least three nodes.

Figure 7 shows results for the normalized amount of fiber required in the networks. At 100 nodes a single-level star network requires the most fiber at 3.8 times as much as a dual ring. When considering 100000 nodes the star is even worse, and uses 121 times as much fiber as a dud ring. The second worst is the ring/star; the third worst is the two-level star with 4.9 times as much fiber. The stadring, two-level ring, and three-level ring are close to that required by a single-level ring. It is obvious from this that the amount of fiber required for a network that uses stars on the lowest level will eventually become much more than that of a ring as the network grows. This is because the total number of nodes in the lowest level networks is much larger than that in the higher levels.

Figure 8 shows the normalized average propagation delay in the networks. The single-level dual ring has the largest propagation delay, which is 3.3 times as much as that of a single-level star for a 100 node network. At 100000 nodes the propagation delay for a dual ring is 103 times as much

Ring Star - 2 level Ring - 2 level - Star - 3 level Ring - 3 level

--- Star/Ring Rin /Star

1 10 100 1000 10000 100000 Number of Nodes (N)

Normahzed average propagation delay in multdevel networks. Fig. 8.

as that of a star. The star/ring, two-level ring, and three-level

the majority of the propagation delay occurs in the highest network level shows that a star should be used if propagation delay is an important parameter r61.

Now consider at what point the difference in network characteristics become significant. The star uses twice as much fiber as the dual ring at 28 nodes, the two-level star uses twice as much at about 450 nodes, and the three-level star around 18000. By using n, = in expression (20) it can be shown that, as N goes to infinity, the starhing network uses 1.38 times as much fiber as the dual ring. The dual ring has twice the propagation delay of the star at 36 nodes, the two- level ring has twice as much at about 280, and the three-level ring around 1900. The stadring, on the other hand, has 1.65 times the propagation delay of the star as N goes to infinity. By using the starkng two-level network, performance in both propagation delay and amount of fiber required are near the ideal values for any size network. This indicates that if both of these parameters are important, the stadring network should be considered. The only reasons for needing more levels of hierarchy would then be the other limitations on network size, such as bit rate, number of wavelengths, or quality of service.

V. CONCLUSION Analysis of the star and ring networks from the point of

view of growth indicates that the star network can actually use less fiber if the starting size of the network is less than seven nodes for a Uniform distribution or 17 nodes for Gaussian. As the network grows larger, the amount of obsolete fiber created by addition of a node becomes insignificant compared to that of the existing fiber. Although the amount of fiber required is considered a major disadvantage of the star, this shows that a star may be practical for networks which cover a large area if the number of nodes is expected to grow, but remain relatively small.

When considering hierarchical networks, a network com- posed of stars can be designed to have fiber requirements near that of a ring. A network composed of rings can be designed to have propagation delay characteristics near that of a star. As

FALCONE AND TONGUZ: PRACTICAL CONSTRAINTS IN GROWTH OF LIGHTWAVE NETWORKS 355

the number of nodes becomes larger, the number of network levels required to succeed in this becomes larger as well. If both characteristics are important, the solution may be to use a hybrid network composed of a star on the higher level and a number of ring networks on the lowest level.

The analysis in this paper did not consider the access methods which would be used. If a network is expected to grow from a small to an extremely large number of nodes, logical topologies that are both modular and scalable must be developed which can also meet this additional demand for capacity [16], [23].

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[11] D. P. Malley, private communication, Oct. 1992. [12] D. P. Malley and 0. K. Tonguz, “Fiber in the loop: Where and when is it

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[14] H, Pirkul, J . Current, and V. Nagarajan, “The hierarchical network design problem: A new formulation and solution procedures,” Trans- portation Science, vol. 25, no. 3, pp. 175-182, Aug. 1991.

[15] P. E. Green, Fiber Optic Networks. Englewood Cliffs, NJ: Prentice- Hall, 1993.

1161 K. A. Falcone, “Characteristics of lightwave networks,” Ph.D. disserta- tion, State University of New York at Buffalo, Buffalo, Nov. 1993.

[17] T. H. Wu and W. I. Way, “A novel passive protected sonet bidirectional self-healing ring architecture,” IEEE/OSA J. Lightwave Technol., vol. 10, no. 9, pp. 1314-1322, Sept. 1992.

[I81 S. Lin, “Computer solutions of the traveling salesman problem,” Bell Syst. Tech. J. , vol. 44, pp. 2245-2269, Dec. 1965.

[19] R. H. Randles and D. A. Wolfe, Introduction to the Theory of Nonpara- metric Statistics. Malahar, FL: Geiger, 1991.

[20] C. A. Brackett, “A perspective on scalability and modularity in multi- wavelength optical networks,” presented at Workshop on WDM Tech- nologies, Systems and Network Applications, Optical Fiber Communica- tions Conference, San Jose, CA, Feh. 1992.

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[23] See the Special Issue of IEEE/OSA J. Lightwave Technol., vol. 11, no. 5/6, May/June 1993.

Kenneth A. Falcone (S’92-M’95) was born in Buffalo, NY, on November 14, 1965. He received the A.A.S. degree in electrical technology from the State University of New York at Alfred, in 1984, the B.S. degree in electncal engineenng from the University of Illinois at Urbana-Champagn, in 1989, and the M.S. and Ph.D. degrees from the State University of New York at Buffalo, both in electrical engineering, in 1992 and 1994, respectively.

He served in the U.S. Air Force from 1984 to 1990 as an electronic technician. In 1991, he started

graduate school at the State University of New York at Buffalo. While at SUNY/Buffalo he was a teaching assistant from 1991 to 1992, a research assistant from 1993 to 1994, and a lecturer in the Fall of 1994. He is currently working at GE Corporate Research and Development in the Digital and Network Systems Program.

Ozan K. Tonguz (S’86-M’90) was born in Nicosia, Cyprus, in May 1960. He received the B.Sc. de- gree in electronic engineering from the University of Essex, Essex, U.K., in 1980, and the M.Sc. and Ph.D. degrees from Rutgers University, New Brunswick, NJ, both in electrical engineering, in 1986 and 1990, respectively.

In 1981, he returned to Cyprus. After two years of mandatory military service (from 1981 to 1983), he was an Assistant Lecturer at the Eastern Mediter- ranean University of Northem Cyprus, during the

academic year of 1983-1984. In September 1984, he joined-the Depmment of Electrical and Computer Engineering, Rutgers University. Between January 1988 and May 1990, he was a visiting doctoral student at the Advanced Lightwave Systems Division of Bell Communications Research, Red Bank, NJ, where he conducted research on coherent lightwave technology. From May 1990 to August 1990, he was a Member of Technical Staff at Bellcore, and continued to do work on coherent lightwave systems technology and optical amplifiers. He joined the Department of Electrical and Computer Engineering, State University of New York at Buffalo (SUNY/Buffalo), as an Assistant Professor in September 1990, where he was granted early tenure and promoted to the rank of Associate Professor in June 1995. At SUNY/Buffalo, he leads substantial research activity in the broad area of telecommunications. His current research interests are in mobile radio and personal communication systems (PCS), fiber optic communication systems and networks, and high- speed networking. He has published in the areas of optical communication systems and networks, mobile radio and personal communication systems and networks, and is the author or co-author of more than 50 technical papers in leading technical journals and conference proceedings. While his research on optical networks has been sponsored by National Science Foundation via a Research Initiation Award, his research in wireless communications is currently being funded by several companies active in the mobile radio and PCS industry.

Dr. Tonguz frequently acts as a reviewer for various IEEE and IEE TRANSACTIONS and Journals, has served on the Technical Program Committees of IEEE Lasers and Electro-optics Society (LEOS), is a Guest Editor of the Joint Special Issue of the IEEE/OSA JOURNAL OF LIGHTWAVE TECHNOLOGY and IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS on Multiwavelength Optical Networks and Technologies, and was the Chair of the Optical Networks and Systems Session of the IEEE LEOS’95 Annual Meeting. He is a member of the Optical Society of America and Eta Kappa Nu.