Performance Impact Of Partial Reconfiguration On ... - ECE UC Davis

IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 5, NO. 3, JUNE 1997

351

Performance Impact of Partial Reconfiguration on Multihop Lightwave Networks Jean-Fran¸cois P. Labourdette, Senior Member, IEEE Abstract— Rearrangeable multihop lightwave networks can be dynamically reconfigured to adapt the connectivity among network stations to prevailing traffic conditions. With no limitations on the tunability range of the transmitters and/or receivers allocated to each network station, all connection diagrams can be realized among network stations, within the constraints of the number of transmitters and receivers at each station. Since the number of wavelengths in use may be very large, the tunability range of transmitters/receivers will be restricted, in practice, to some limited set of contiguous wavelengths, or waveband. As a result, all connection diagrams might not be achievable and the network may loose some of its ability to adapt to traffic changes through reconfiguration. In this paper, we derive bounds on the performance degradation experienced by the network as a function of the tunability restrictions. We then propose and analyze a waveband decomposition, illustrating how networks with good performance can be designed with tunability constraints on the transceivers. We also describe other architectures (subcarrier division multiplexing, multi-fiber physical topology) to which our model and analysis readily apply. Index Terms— Limited turnability, optical networks, performance analysis, reconfiguration.

I. INTRODUCTION

R

EARRANGEABILITY is emerging as a key feature of today’s telecommunications networks—in particular lightwave networks—providing the ability to dynamically optimize the network for changing traffic patterns, or to cope with equipment failures. This is achieved by embedding a logical connectivity, or virtual topology, in the underlying physical infrastructure, which can be optimized with respect to the traffic conditions, and subject to availability of network equipments. Reviews of logically rearrangeable multihop lightwave networks can be found in [1]–[3]. We consider a network of stations with transmitters and receivers per station. The transmitters and/or the receivers are tunable. The stations are physically attached to, and communicate over, a passive, purely broadcast, optical infrastructure (e.g., star, bus, tree) without wavelength-routing nodes and without wavelength reuse. Let the spectrum of distinct wavelengths in use be decomposed into nonoverlapping wavebands of equal size.1 The receivers and/or the transmitters Manuscript received November 27, 1995 revised June 24, 1996; approved by IEEE/ACM TRANSACTIONS ON NETWORKING Editor R. Ramaswami. An earlier version of this work was presented at IEEE Infocom ’95. The author was with the Center for Telecommunications Research at Columbia University, New York, NY 10027 USA. He is now with AT&T Laboratories, Holmdel, NJ 07733 USA (e-mail: [email protected]). Publisher Item Identifier S 1063-6692(97)04610-4. 1 The assumption of nonoverlapping wavebands of equal size is made to simplify the discussion at this point. The assumption does not necessarily hold throughout the paper, and will be mentioned when it is made.

have a common tunability range (the number of contiguous wavelengths in a waveband they can tune to). In the extreme case (i.e., there is one waveband of and no tunability limitations on the transmitters and/or size receivers. The performance of such networks has been studied (see [2], [3] for a review). In the other extreme case (i.e., each waveband reduces to a single wavelength, yielding a fixed connection diagram. The performance of networks with regular connection diagrams has been studied for both uniform traffic [4]–[6] and nonuniform traffic [7], [8]. Of interest to us are the cases between these two extremes for which all connection diagrams will not be achievable [9] and performance will be impacted. Consider a network operator who has deployed transmitters/receivers restricted to certain wavebands, and assigned wavelengths to the transceivers to achieve a topology for the offered traffic. If the traffic pattern changes significantly, limitations on retuning the transceivers will limit the topologies that can be realized to handle the new offered traffic, and the performance will degrade: How fast, and by how much? And how do the network planners deploy transceivers operating within a limited waveband in an efficient and flexible manner? The results derived in this paper apply to three network architectures: (A1) networks with partially tunable transceivers; (A2) networks combining subcarrier division multiplexing (SDM) with wavelength division multiplexing (WDM); and (A3) multi-fiber networks. The optical medium can support a very large number of wavelengths, and the tunability range of transceivers will be limited in practice to a contiguous subset of all the available wavelengths (A1). There is a trade-off between the cost of a laser and the spectrum range over which the laser can operate. Economic considerations may dictate the use of cheaper lasers with tunability restrictions if the performance of the network remains within some bounds. Tunability restrictions, on the other end, increase the complexity of controlling the network and of determining a new configuration. In particular, the optimization problem that determines the assignment of wavelengths to connections between stations and the routing of traffic over the resulting connection diagram becomes more complicated as it needs to account for the constraints [9]. In [10], the authors propose a SDM/WDM-based architecture (A2) where the wavelengths shared among several stations through SDM are logically equivalent to the wavebands in architecture (A1). The size of a single-fiber based system is limited by the power budget and by the finite number of wavelengths. Therefore, to build large optical networks, multifiber implementations (A3) such as multi-star networks [11], [12] are desirable. Here, the stars are equivalent to the wave-

1063–6692/97$10.00  1997 IEEE

352

Fig. 1. Multi-star physical topology.

bands in architecture (A1), and the transmitters and receivers connected to a star similar to the transmitters and receivers whose tunability is restricted to a waveband (see Fig. 1). Under the assumptions that and that all wavebands are of equal size, our first set of results shows that when there exists an assignment of wavebands to transceivers such that all connection diagrams can still be achieved. We also show that for even though not all connection diagrams can be achieved, all connection diagrams with at most one connection between any two nodes (out of possible connections without constraints) can still be achieved (see also [11], [12]). This is however not the case for The ratio of the number of connection diagrams that can be realized with wavebands to the total number of connection diagrams, decreases polynomially with However, connectivity power may not be a good measure of the network performance such as delay or throughput. Instead, we show how performance, defined as number of hops between two arbitrary stations, increases with a decrease in the tunability range of transmitters/receivers, or equivalently, with an increase in the number of wavebands. To that effect, we compute lower bounds on 1) the worst case minimum number of hops2 and 2) the average case minimum number of hops to communicate between two arbitrary stations, for all waveband assignments. While a network operator would like to assign wavebands to achieve a range of connection diagrams for a set of traffic patterns, traffic information is not necessarily available. The network operator may thus want to initially set a regular connectivity which provides simple routing, many alternate routes, low diameter, and is robust to failure. Furthermore, the traffic patterns may indeed be uniform most of the time, and nonuniform only unfrequently. For those reasons, it appears to be desirable to assign wavebands so that some regular 2 Without tunability constraints, any two arbitrary stations can always be directly connected so that the worst case minimum number of hops is one.


connectivity, simple to operate, and with good performance for uniform traffic, can always be achieved (e.g., de Bruijn graph). We thus propose a waveband assignment and grouping scheme which starts with a de Bruijn connection diagram and creates larger wavebands by merging contiguous wavelengths or wavebands together, at a time. This recursive scheme decomposes the entire spectrum into nonoverlapping wavebands of equal size. SDM/WDM-based architectures and multi-star implementations are adequately modeled by nonoverlapping equal-sized waveband decompositions. In the context of partial tunability, operational considerations may justify an architecture of nonoverlapping wavebands of equal size, even though such an architecture does not fully exploit transceivers that are tunable over arbitrary ranges. The resulting network will be easier to operate, maintain, and upgrade. Indeed, the waveband decomposition proposed in this paper allows to upgrade the transmitters/receivers by increasing their tunability range by a factor of This waveband construction achieves an average case minimum number of hops between two arbitrary stations and a worst case minimum number of hops that increase only logarithmically with decreasing tunability range This paper is organized as follows. In Section II, we propose sets of principles for waveband assignment and decomposition. In Section III, we study the connectivity power of networks with partial reconfiguration capabilities. In Section IV, we derive lower bounds on the worst case minimum number of hops and the average case minimum number of hops for these networks. In Section V, we propose an efficient waveband assignment and grouping scheme and derive network performance for it. Section VI concludes the paper. II. PRINCIPLES FOR WAVEBAND ASSIGNMENT AND DECOMPOSITION Assigning wavebands to transmitters and receivers that have limited tunability is a critical aspect of designing optical networks. Waveband assignments where transmitters and/or receivers at a given station are assigned to some of the same wavebands are not desirable because they can allow, for example, for self-loop connections that are of no use and prevent potentially useful connections to be set. The following principles achieve “good” waveband assignments: (P1) transmitters, respectively, receivers, at the same station are assigned to as many different wavebands as possible,3 and (P2) for a given station pair transmitters at station and receivers at station are assigned to as many different wavebands as possible, except for possibly one waveband common to a transmitter and a receiver at most. Principles (P1)–(P2) do not imply a waveband decomposition into nonoverlapping wavebands, nor wavebands of equal size. Two other principles will be used in Section V to derive a specific waveband assignment. These principles restrict the space of waveband assignments but present several advantages (scalability, ease of upgrade, capability to achieve a regular connection diagram) and allow us to analyze the performance of the resulting waveband assignment: (P3) a splitting divides each waveband into 3 Principle (P1) insures that all stations can transmit to, receive from, all p: wavebands when W

=

LABOURDETTE: PARTIAL RECONFIGURABILITY IN MULTIHOP LIGHTWAVE NETWORKS

353

equal-sized and contiguous wavebands, and (P4) iteratively dividing the spectrum until there are wavebands (each of size one), the resulting fixed connection diagram is regular (e.g., de Bruijn graph). Principles (P3)–(P4) imply (P1)–(P2) and are maintained after each waveband splitting. They also imply that the wavebands are nonoverlapping, of equal size and that and are powers of The waveband assignments in Fig. 2 meet principles (P3)–(P4). III. THREE-STAGE SWITCH REPRESENTATION In most of this section, we assume that the spectrum is decomposed into wavebands and that transmitters/receivers are assigned to those wavebands according to principles (P1)–(P2). We further assume that the wavebands are nonoverlapping, of equal size, and that the total number of wavelength is We introduce a three-stage switch representation as follows. The first and third stages are made of crossbar switches with inputs and outputs, which represent the network stations with transmitters and receivers. The middle stage is made of crossbar switches which represent the wavebands, each of size wavelengths. Switch representations for a network with stations, transmitters/receivers per station are shown in Fig. 2 for different waveband decompositions. Setting all switches in this three-stage switch representation achieves a permutation of inputs to outputs which yields a connection diagram among the network stations. Each connection diagram can be realized by many settings/permutations.

(a)

A. Bounds on the Number of Connection Diagrams Let denote the total number of connection diagrams for a network with nodes and transmitters and receivers per node. is independent of principles (P1)–(P2) and (P3)–(P4) and of any waveband decomposition. The threestage switch representation is now used to derive bounds on The number of permutations realizable by the switch and several permutations yield the same connection is diagram. Specifically, permuting the inputs to any of the switches in the first stage or the outputs of any of the switches in the last stage of the three-stage switch representation yields the same connection diagram, but in general different permutations of the inputs into the outputs. They are not necessarily different permutations because the simultaneous permutings of the inputs to one of the switch in the first stage and of the outputs of one of the switch in the last stage may cancel each other and result in the same permutation (e.g., if a first stage switch has more than one connection to a third stage switch). Thus, is a lower bound on Permuting the inputs to any of the switches in the first stage, or permuting the outputs of any of the switches in the last stage of the three-stage switch representation, but not doing both, yields the same connection diagram but different permutations, although not all permutations that realize the connection diagram. Thus, is an upper bound on and (1)

(b)

N

= 8 and Fig. 2. Three-stage switch representations for two wavebands and (b) four and eight wavebands.

p = 2 with (a)

B. Connectivity Power for Different Waveband Decompositions Each permutation that can be passed by the three-stage switch corresponds to a connection diagram that can be achieved by the network. Recall that several permutations yield the same connection diagram. However, for some waveband decompositions, all connection diagrams may not be achieved by the network as all permutations may not be realizable by the three-stage switch. Let be the number of connection diagrams achievable in a partially reconfigurable multihop network with wavebands assigned according to principles (P1)–(P2). Let us first consider the case By principle (P1), the three-stage switch is a Clos network [Fig. 2(a)], and thus rearrangeable nonblocking. According to the Slepian-Duguid theorem [13], it can therefore pass any permutation. The corresponding multihop network can achieve all connection diagrams, since every one of them can be realized by at least one permutation through the three-stage switch. Thus, by equipping stations with transmitters able to tune over one of wavebands of equal size and with fixed receivers,

354


we can assign wavebands and build a reconfigurable multihop network capable of realizing all connection diagrams, including those with up to connections between stations. Such connection diagrams can be achieved because there are distinct paths between any two stations in the threestage switch, one through each of the middle-stage switches (wavebands). Thus We now consider the case where there are more wavebands than transceivers per node. Because of principle (P2), and since there is at most one middle stage switch (waveband) to connect a first stage switch to a last stage switch in the three-stage switch representation. Therefore, there is at most one way to set up a connection between any two nodes in the multihop network. It follows that for we can compute by looking at the three-stage switch representation shown in Fig. 2(b) in dashed line for and in solid line for Each of the middle stage switches can be set up in ways, yielding for For each assignment of wavebands to transmitters and wavelengths to receivers, we would like to assess the ability of the resulting multihop network to achieve as many connectivity diagrams as possible. We use the following connectivity power measure,4 similar to the combinatorial power of a connecting network [13]: of achievable connection diagrams total of connection diagrams (2) Using (1), the expression for approximation and dropping the term bounds:

and the Stirling for large we obtain the following

(3) and imply that and it follows from inequality (3) that goes to zero polynomially fast as grows large. Therefore, such multihop networks can only achieve a very small fraction of the connection diagrams achievable in the absence of tunability constraints. For all connection diagrams cannot be achieved: the connection diagram with more than one connection from one node to another cannot be realized.5 According to principle (P1), each node reaches different wavebands, and each 4 A notion similar to connectivity power called “reconfigurability” was introduced in [11] and [12]. 5 There can be up to p directed connections between two nodes in general.

destination cannot be reached in or less hops in from originating station

of stations that cannot be reached in

waveband is reached by receivers belonging to different nodes. Therefore each node reaches different nodes, reaching every node once. If it did not, two nodes would have more than one waveband in common, which violates principle (P2). Since there is one unique path through the three-stage switch representation between any two stations when and the middle stage switches (wavebands) are nonblocking, all connections diagrams with at most one connection between any two nodes can be realized. This result was previously reported in [11], [12] in the context of multistar lightwave networks. For there is not always a path between any two stations and more connection diagrams, beyond those with more than one connection between any two nodes, are not achievable. The conclusion we draw from this section is threefold. First, splitting the spectrum in wavebands (i.e., still allows the network to achieve all connection diagrams and thus does not cause any performance degradation. Second, splitting the spectrum in (i.e., wavebands allows the network to achieve all connection diagrams with at most one directed connection between any two nodes, and therefore will not, in most cases, cause any performance degradation, with the exception of traffic patterns which warrant two or more connections between some nodes. Third, as the number of wavebands increases, (equivalently, as decreases, the ratio of connection diagrams that can be achieved to the total number of connection diagrams decreases polynomially fast. This may lead us to think that the performance degrades rapidly as increases. We show in the next section that this is not necessarily the case.

IV. BOUNDS

ON

PERFORMANCE IMPACT

When there are no tunability restrictions, a connection diagram can always be selected such that two arbitrary stations are directly connected. If each transmitter/receiver can only be tuned over a single waveband out of it may now take several hops to reach a destination station from an originating station through tuning of the transmitters/receivers. We derive a lower bound on the worst case minimum number of hops to reach a destination station, and a lower bound on the average case minimum number of hops to reach a destination station, over all waveband assignments. We do not assume that principles (P1)–(P2) or (P3)–(P4) hold, and we do not assume that the spectrum can be decomposed into nonoverlapping wavebands nor that We do however assume that all the transmitters have the same tunability range Let be the random variable representing the minimum number of hops between an originating station and a des-

destination is among stations that cannot be reached in or less hops in from originating station

or less hops in

(4)


tination station for a network of size of degree with transmitter tunability range for a given waveband assignment. Let be the random variable representing the minimum number of hops between an originating station and a destination station for a nonnecessarily feasible network described as follows. An arbitrary station can reach wavebands, and each waveband can provide access to receivers or stations. Therefore, a station can reach stations6 in one hop out of a total of stations (all stations excluding the originating station), assuming no two of the receivers belong to the same station. A station can then reach at most7 new stations in two hops. More generally, the number of stations that can be reached in exactly hops from any station in network is given by We now write the equations shown in (4) at the bottom of the page. Similarly, Let resp., and resp., be the average and worst case minimum number of hops between two arbitrary stations in resp. . By construction of network is maximum, and thus for all Therefore for all From the theory of stochastic ordering [14, Ch. 8, Lemma 8.1.1], it follows that Stochastic ordering is also preserved for increasing functions [14, Ch. 8, Proof of Proposition 8.1.2], and thus Therefore, , respectively, is a lower bound on the worst case, respectively, the average case, minimum number of hops between two stations in any network of size degree and tunability range for all waveband assignments. is the smallest integer that satisfies the following inequality: (5) so that

(6)

denotes the smallest integer greater than or equal where to Note that the expression for is similar to the Moore bound on the diameter of a regular graph with nodes and degree [5]. Reporting , for and in yields, after some algebraic manipulations:

(7) Note that for for i.e.,

i.e., and

f

355

increase with Note also that independent of for Note finally that i.e. i.e., yields As the tunability range decreases (i.e., increases), the performance may not degrade as fast as the connectivity power does. Even though a lot of direct connections may not be achievable any more, the traffic is still very likely to be carried in a small number of hops. V. WAVEBAND ASSIGNMENT

AND

GROUPING

In this section, we propose a waveband decomposition that follows principles (P3)–(P4) defined in Section II. Given a network with nodes, and transmitters and receivers per node that can be tuned over some subset of size of all the wavelengths in use in the spectrum, we propose an assignment of wavebands to the stations and a recursive waveband grouping scheme such that a de Bruijn regular graph can always be realized. Specifically, for the resulting fixed connection diagram is a de Bruijn graph. It is known that the de Bruijn graph achieves the optimal diameter and an average number of hops which comes very close to the optimum, but not necessarily achievable, average number of hops as given by the Moore bound [6]. Since de Bruijn graphs exist only for networks of size where is the diameter of the graph, the waveband decomposition presented below can be carried out in only a limited set of networks. A. Wavelength Assignment for Fixed de Bruijn Graph An example of the wavelength assignment for fixed de Bruijn graph appears in Fig. 2 for We identify each of the nodes that appear in the first stage of the three-stage switch representation by its row coordinate where Addresses are assigned in a natural way: rows are numbered top to bottom from 0 to Row addresses are numbered using base digits and are made of digits,

with for all Similarly, each node that appears in the third stage of the three-stage switch representation is with identified by its row address, for all We identify each of the wavelengths that appears in the middle stage of the threestage switch representation by its row coordinate where Addresses are assigned in a natural way: rows are numbered top to bottom from 0 to Row addresses are numbered using base digits and addresses are made of digits,

and and

0g

6 This number should really be min pT ; N 1 ; but we omit this obvious R fact for simplicity of the notations. 7 “At most” because some stations reached in two hops could already have been reached in one hop. This assumes that wavebands have been assigned to transmitters an receivers in some optimal, not necessarily feasible, way.

with for all A specific addressing of stations and wavelengths in the three-stage switch representation is shown in Fig. 2(b) for and We define the 1-to-

356


TABLE I WAVEBAND DECOMPOSITION

MULTI-HOP REACHABILITY

FOR

k

p

TABLE II WAVEBANDS OF SIZE

mapping

between nodes in the first stage of the three-stage switch representation and wavelengths in the second stage of that switch representation. This mapping is a Shuffle Exchange from nodes to wavelengths (see Fig. 2). We define the -to-1 mapping

between wavelengths in the second stage and nodes in the third stage of the switch representation. This mapping is an Inverse Shuffle Exchange from wavelengths to nodes with the first wavelength assigned to the first node (see Fig. 2). The resulting mapping between nodes in the first stage and nodes in the last stage of the three-stage switch representation is the 1-to- mapping

D+10k 2

p

;

k

D

+1

shows, for all waveband decompositions, a representation of the set of wavelengths that can be reached from the arbitrary station as well as a representation of the set of stations that can be reached from that arbitrary station in one hop. This can be proved by induction on using the results from Section IV. One can repeat the same argument recursively for each of the nodes reached at subsequent hops and construct Table II, which represents the set of new stations reached after each additional hop.8 C. Performance Evaluation From Table II, the worst case minimum number of hops between two arbitrary stations, is the smallest integer that satisfies the following inequality, whose left side is obtained by summing the numbers of new stations reached at each hop (from the rightmost column of Table II):

(8)

This mapping is precisely the definition of a de Bruijn graph [6], which shows that our construction achieves such a graph. B. Waveband Grouping We show how, Let us consider an arbitrary station starting from a de Bruijn graph, one can achieve a waveband decomposition in any number of wavebands of equal size with wavebands made of contiguous wavebands/wavelengths. This is done by iteratively increasing the tunability range of transmitters/receivers by a factor merging contiguous wavebands into larger wavebands. The number of wavebands follows the sequence The This waveband size follows the sequence waveband decomposition is shown in Fig. 2 for a network with and Table I

This reduces to immediately that the number of wavebands is given by we report the expression for and obtain

which gives Now, recalling and that into

(9)

0

8 The “ 1” term in the first row and last column of Table II would appear on a different row for a different originating station and a different value of W (i.e., TR ): But the “ 1” term always appears on the first row if the first or last station is the originator because of the self-loops on these stations in de Bruijn graphs.

0


357

Using the rightmost column of Table II, one can write the following expression that upperbounds9 the average case minimum number of hops, between two arbitrary stations:

(10) After some algebraic manipulations, it can be reduced to:

(11) where is given by (9) and unity given by

is a positive number less than

Fig. 3. Worst case and average case minimum number of hops as a function of the (logarithm of the) tunability range TR (N = 1024; p = 2):

(12) . Also for Note that for (i.e., fixed de Bruijn connection diagram), and which yields the known result [6]. In both cases, equals the lower bound Finally, and increase logarithmically with decreasing tunability range A comparison of the worst case and average case minimum number of hops with the their respective lower bounds and derived in Section IV is shown in Figs. 3 and 4 for networks with and as a function of the logarithm of Although can be three times as large as the lower bound in the case it differs by at most one for It appears from the curves that as increases for fixed values of and the performance of the proposed waveband decomposition approaches the lower bounds. This can be explained by the fact previously noted that for becomes independent of while decreases as the inverse of the logarithm of Note also that for , and and grow at most logarithmically with the size of the network, which indicates good scaling capabilities for reasonable tuning ranges. VI. CONCLUSION The use of partially agile transmitters and receivers in multihop lightwave networks may restrict the set of connection diagrams that can be achieved among network stations. This also occurs with network architectures based on subcarrier and wavelength-division multiplexing as well as with networks based on multi-fiber implementation of the physical topology. We have presented a waveband assignment and grouping scheme which lets us achieve a regular connection diagram, specifically a de Bruijn graph (when such a graph exists). Although the assignment does not achieve lower bounds on the worst case and average case minimum numbers of

0

9 When

new stations are counted from a station with self-loops, the “ 1” term appears in the first row of Table II (last column), and g = g 3 : In other cases, the “ 1” term would appear in some other row of Table II, and g < g 3 :

0

Fig. 4. Worst case and average case minimum number of hops as a function of the (logarithm of the) tunability range TR (N = 1024; p = 4):

hops required to connect two arbitrary stations, worst and average case minimum number of hops increase at most logarithmically with decreasing tunability range of transmitters/receivers (i.e., increasing number of wavebands). It is an open problem whether there exist waveband assignment and grouping schemes whose performances are closer to the lower bound than the scheme proposed here. In this paper, we analyzed the performance degradation in the extreme case of highly nonuniform traffic, where we are only concerned about the traffic and connectivity between two stations. Once we fix one or more connections, the results would not apply to any two other stations. ACKNOWLEDGMENT The author would like to thank the reviewers and the editor whose valuable comments greatly improved the quality of this paper, and Dr. I. Rouvellou for her help with the proof used in Section IV.

358


REFERENCES [1] R. Ramaswami, “Multiwavelength lightwave networks for computer communication,” IEEE Commun. Mag., Feb. 1993. [2] B. Mukherjee, “WDM-based local lightwave networks, Part II: Multihop systems,” IEEE Network Mag., July 1992. [3] J.-F. P. Labourdette, “Traffic optimization and reconfiguration management of broadcast multihop lightwave networks,” submitted for publication. [4] A. S. Acampora, “A multichannel multihop local lightwave network,” in Proc. Globecom’87, Tokyo, Japan, Nov. 1987. [5] M. G. Hluchyj and M. J. Karol, “ShuffleNet: an application of generalized Perfect Shuffle to multihop lightwave networks,” IEEE J. Lightwave Technol., vol. 9, Oct. 1991. [6] K. Sivarajan and R. Ramaswami, “Lightwave networks based on De Bruijn graphs,” IEEE/ACM Trans. Networking, vol. 2, Feb. 1994. [7] M. J. Karol and S. Shaikh, “A simple adaptive routing scheme for ShuffleNet multihop lightwave network,” in Proc. Globecom’88, Hollywood, FL, Nov. 1988. [8] M. Eisenberg and N. Mehravari, “Performance of the multichannel multihop lightwave network under nonuniform traffic,” IEEE J. Select. Areas Commun., vol. 6, Aug. 1988. [9] J.-F. P. Labourdette and A. S. Acampora, “Partially reconfigurable multihop lightwave networks,” in Proc. Globecom’90, San Diego, CA, Dec. 1990. [10] R. Ramaswami and K. Sivarajan, “A packet-switched multihop lightwave network using subcarrier and wavelength division multiplexing,” IEEE Trans. Commun., Feb./Mar./Apr. 1994. [11] A. Ganz, B. Li, and L. Zenou, “Reconfigurability of multi-star based lightwave lans,” in Proc. Globecom’92, Orlando, FL, Dec. 1992. [12] P. P. To, “Reconfigurability of shufflenets in multi-star implementation,” in Proc. Infocom’94, Toronto, Ont., Canada, June 1994.

[13] V. E. Benes, Mathematical Theory of Connecting Networks and Telephone Traffic. New York: Academic, 1965. [14] S. Ross, Stochastic Processes, 2nd ed. New York: Wiley, 1983.

Jean-Francois P. Labourdette (M’91–SM’97) received the Diplome d’Ingenieur from Ecole Nationale Superieure des Telecommunications, Brest, France, in 1986, and the M.Sc. and Ph.D. degrees in electrical engineering from Columbia University, New York, in 1988 and 1991, respectively. From 1987 to 1991, he was a Graduate Research Assistant at the NSF Center for Telecommunications Research, Columbia University, working on reconfigurable lightwave networks. He spent the summer of 1990 at Motorola Codex, Mansfield, MA, investigating the design of fast packet networks for integrated traffic. He joined AT&T Laboratories, Holmdel, NJ, in 1991. From 1991 to 1996, he conducted core network planning activities, working on circuit-switched network routing, dynamically reconfigurable T1/T3 network architectures and ATM VP networking. He was promoted to Principal Technical Staff Member in 1996. He is currently with the Frame Relay department, working on service development, operations planning, and engineering for international Frame Relay services. His research interests include lightwave network architectures, network performances, traffic optimization and reconfiguration management of circuit-switched and packet networks. Dr. Labourdette was a recipient of a Lavoisier Scholarship from the French government in 1986–1987, and a Motorola Scholar in 1989–1990 and 1990–1991. He received the Eliahu I. Jury award for excellence in Systems, Communications or Signal Processing at Columbia University in 1992. He has served as a member of the Technical Program Committee for the IEEE INFOCOM conference since 1994.