an ampier and a transmitter. P sat is the interna saturation power of the optica ampier. G max is the maximum sma-signa gain of the optica ampier. The

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1 Optimizing Ampier Pacements in a Muti-Waveength Optica LAN/MAN: The Unequay-Powered-Waveengths Case Byrav Ramamurthy, Jason Iness, and Biswanath Mukherjee Abstract Optica networks based on passive-star coupers and empoying waveength division mutipexing (WDM) have been proposed for depoyment in oca and metropoitan areas. These networks suer from spitting, couping, and attenuation osses. Since there is an upper bound on transmitter power and a ower bound on receiver sensitivity, optica ampiers are usuay required to compensate for the power osses mentioned above. Due to the high cost of ampiers, it is desirabe to minimize their tota number in the network. However, an optica ampi- er has constraints on the maximum gain and the maximum output power it can suppy; thus, optica ampier pacement becomes a chaenging probem. In fact, the genera probem of minimizing the tota ampier count is a mixed-integer noninear probem. Previous studies have attacked the ampier-pacement probem by adding the \articia" constraint that a waveengths, which are present at a particuar point in a ber, be at the same power eve. This constraint simpies the probem into a sovabe mixedinteger inear program. Unfortunatey, this articia constraint can miss feasibe soutions that have a ower ampi- er count but do not have the equay-powered-waveengths constraint. In this paper, we present a method to sove the minimum-ampier-pacement probem whie avoiding the equay-powered-waveength constraint. We demonstrate that, by aowing signas to operate at dierent power eves, our method can reduce the number of ampiers required. Keywords Optica Network, WDM, LAN/MAN, Passive Star, Ampier Pacement, Optimization, Linear/Noninear Programming. Introduction. Network Environment The focus of this study is on a cass of the next-generation optica oca or metropoitan area networks (LAN/MAN) which span distances from fewer than a kiometer to a few tens of kiometers and which provide oop-free communication paths between a source-destination pairs. A argedistance version of such a network is depicted in Figure, and it consists of N = 63 stations and M = passive optica star coupers (\stars"), such that each star is connected to other stars and/or stations via two unidirectiona ber B. Ramamurthy is with the Dept. of Computer Science & Engg., University of Nebraska-Lincon, Lincon, NE (e-mai: byrav@cse.un.edu). J. Iness is with Inte Corp., Hisboro OR (e-mai: iness@cs.ucdavis.edu). B. Mukherjee is with the Dept. of Computer Science, University of Caifornia, Davis, CA 966 (e-mai: mukherjee@cs.ucdavis.edu). This work has been supported in parts by the Nationa Science Foundation (NSF) under Grant Nos. NCR-92-07, NCR , and ECS-9-229, and in part by Advanced Research Projects Agency (ARPA) under Contract No. DABT63-92-C-003. A summarized eary version of this paper appeared in the Proceedings of the IEEE Infocom '97 conference. Such networks have been referred to in the iterature as access networks, passive optica networks (PONs) [22], etc. inks. The passive-star couper provides a broadcast faciity, but it must aso be of the \non-reective" type (to be eaborated beow) in order to prevent oops in the network. Our study wi consider the case where each station in the network has a xed-waveength transmitter and is set to operate on its own unique waveength channe. Each station either has a tunabe receiver or a receiver array in order to receive signas from a of the other stations. The objective is to ensure that a station's transmission can be received by every other station after being subject to osses and gains as the signa traverses through dierent parts of the network. The network consists of optica stars that are non-reective. A non-reective star consists of pairs of inputs and outputs, and each output carries a of the waveengths that were incident on a of the inputs except for the waveengths that were carried on its own paired input (see Figure 2 for an exampe). Such stars have been empoyed in the Leve-0 A-Optica Network (AON) [2]. Non-reective stars are needed in order to avoid interference due to oops (\echoes") in the network. A star in the network with k input bers and k output bers operates such that the power on each waveength on an input ber is divided eveny among the other k? output bers. This is referred to as the spitting oss at a star. Note that the spitting oss can be dierent for dierentsized stars in the network. As the sampe network in Figure shows, these networks can be depoyed as part of a metropoitan area network (MAN). We require that each signa (waveength) be received at a of the other receivers at a power eve greater than a station's receiver sensitivity eve, denoted by p sen. However, apart from the spitting oss due to the stars mentioned above, there is signa attenuation on the bers given by the parameter db/km. Even though attenuation osses for ber are reativey ow (approximatey 0.2 db/km oss 2 ) compared to other transmission media, arger networks (MANs) and networks with numerous spitting/couping osses wi require ampication to aow a transmitted signa to reach the receivers at a detectabe eve. The constraints on the system are shown in Tabe, aong with typica vaues for each parameter. P NONLIN;max denes the power eve, in a ber, above which a signa encounters signicant noninear eects. However, the tota power at any point in the network is usuay bounded by a ower vaue P max, which is the maximum output power of 2 The 0.2 db oss per kiometer of ber is cose to the absoute minimum due to the fundamenta imits of Rayeigh scattering oss and infrared materia oss.

2 an ampier and a transmitter. P sat is the interna saturation power of the optica ampier. G max is the maximum sma-signa gain of the optica ampier. These parameter vaues 3 (ast coumn of Tabe ) wi be used in our iustrative numerica exampes in Section 3. We remark here that the vaue of the parameter p sen can be chosen (by the user) such that it remains much higher than the noise eves at the intermediate ampiers and at the receiver. The vaue of p sen can aso be changed iterativey, after the pacement of the ampiers, in order to obtain the desired signa-to-noise ratio (SNR) [3] or the bit-error rate (BER) at each of the receivers, foowing the approach in [, ]. Thus, in this study, we do not consider system factors such as ampier ASE noise and crosstak at the receivers, expicity; these factors are assumed to be impicity incorporated in the parameter p sen..2 Probem Denition In the network setting described above, it is important to quantify the minimum number of ampiers required to operate the network and to determine their exact pacements in the network. In such a network, when signas on dierent waveengths originating from dierent ocations in the network arrive at an ampier, their power eves coud be very dierent. This phenomenon is known as the near-far eect and it resuts in inecient utiization of the individua ampier. The dierence in power eves of the input waveengths can signicanty imit the amount of ampication avaiabe since the higher-powered waveengths coud saturate the ampier and imit the gain seen by the owerpowered waveengths. Figure 3 shows, at some ocation on a ber ink, a case where three waveengths have dierent power eves and a case where the three waveengths have the same power eve. In Figure 3(a), the tota power is : W, and in Figure 3(b), it is 3 W. Since the perwaveength ampier sensitivity is W (=?30 dbm), in both cases an ampier wi be required before the signas suer any more attenuation. However, since an ampier has a imited tota output power, the amount of achievabe gain is greater when the tota input power is ess. This woud aow the signas in Figure 3(b) to receive a higher gain than the signas in Figure 3(a). Aso, aowing signas in the same ber to be at dierent power eves changes the minima-ampier-pacement probem from a mixed-integer inear program (MILP) [20] into a mixed-integer noninear program, as we sha show ater in this paper. Previous optica ampier-pacement schemes [, 20] bypassed these probems by restricting a of the waveengths at any given point in a ber to be at the same power eve. Unfortunatey, requiring waveengths to be at the same power eve often forces the designer to add more ampi- ers than the minimum necessary in order for the receivers to receive signas at or above the receiver sensitivity eve. Since each optica ampier costs around $2,000, every attempt shoud be made to minimize their number in the 3 The insertion oss at an ampier can be incuded indirecty by requiring a higher sma-signa gain and hence is omitted. network. It is aso desirabe to reduce the number of ampiers used in the network based on noise, maintenance, and faut-toerance considerations. Our study was motivated by the network in Figure. For reasonabe network parameters, this network can operate without using any ampiers. However, if the power eves for a waveengths must be equa on any given ink, as required by the MILP approach in [20], then an ampier (on one of the inks between stars A and B) wi have to be added to the network. This is because, if we x the output power of star A to be some vaue x, then the signas from stations 3 and must reach star B with an output power higher than x. Without an ampier, signas from stations and 2 reach star B at a power ess than x, which means that waveengths on the ink from star B to station 3 (and simiary on the ink from star B to station ) wi have unequa powers. Therefore, requiring equa power on a waveengths adds an unnecessary ampier to this network. As we sha soon observe, aowing waveengths to be at unequa powers eiminates the need for any ampiers in this network. In this paper, we propose a scheme that minimizes the number of ampiers for the network setting described in [] without the restriction that waveengths in the same ber be at the same power eve. The method works as foows: () determine whether or not it is possibe to design the network taking into consideration the imitations of the devices (e.g., the power budget of the ampiers), (2) generate a set of constraints to accuratey describe the probem setting, which turns out to be a noninear program, (3) pass the set of constraints to a noninear sover, such as CFSQP (C code for Feasibe Sequentia Quadratic Programming) [9], in order to sove for the minimum number of ampiers needed for the entire network, and () determine the exact pacements of the optica ampiers. Numerica exampes wi show that this network-wide optimization method without the equa-power constraint often resuts in soutions that require fewer ampiers than the soutions in [, 20]..3 Ampier Gain Mode Currenty, we empoy a simpied mode for the gain of a generic optica ampier. The simpifying assumptions are that the ampier has a at gain over the waveengths being ampied and that the ampier gain is homogeneousy broadened. A at gain can be achieved through various techniques such as () notch ters [23], (2) dierent pump aser powers [7], (3) Mach-Zehnder ters [2], and () demutipexers and attenuators []. However, assuming that optica ampiers are homogeneous is an approximation. For each specic ampier (Erbium-doped ber ampier (EDFA), semiconductor optica ampier (SOA), etc.), we need to deveop a gain mode depending on its degree of homogeneity in order to accuratey sove the ampier- By homogeneous broadening, we mean that a singe high-powered waveength, which saturates the ampier, can bring down the gain avaiabe for a of the waveengths uniformy. 2

3 Parameter Description Range Vaue used p sen Minimum signa power at receiver?30 dbm at Gbps?30 dbm and the ampier sensitivity eve G max Maximum sma-signa gain 2 db { Muti-Quantum 20 db We (MQW) Ampier [8] 30 db { Erbium-Doped Fiber Ampier (EDFA) [] P NONLIN;max Maximum tota power in ber 0-0mW 0 mw P max Maximum tota output power of amp mw and transmitter P sat Interna saturation power of the amp.298 mw Fiber attenuation 0.2 db/km Tabe : Important parameters and their vaues used in the ampier-pacement agorithms. pacement probem. Based on the above assumptions, the gain mode for our ampiers is given by (from [2]) P in = P sat G? n G0 G where P in is the tota input power (across a waveengths) to the ampier in mw, P sat is the interna saturation power in mw, G is the actua gain achieved (in absoute scae, not db), and G 0 is the sma-signa gain (which is the gain achievabe for sma vaues of input power when the ampier does not saturate, again in absoute scae). Since the formua for G is not an expicit formuation, we use an iterative method to sove for the vaue of G. Our ampier mode has been designed into our soution as a generic gain modue that can be easiy repaced when a more accurate mode for a specic ampier is used. Previous studies [] used the gain mode in Figure. In this mode, it is assumed that the fu sma-signa gain of the ampier is reaizabe unti the point at which the ampier output becomes power-imited. At this point, the ampier is assumed to enter saturation and the gain starts to drop. This \point" of saturation occurs in the exampe of Figure at a tota input power of?20 dbm. At ower input powers, the ampier is assumed to be abe to suppy the fu sma-signa gain of G max = 20 db. The moreaccurate mode (Equation ()), which is used in this paper (and aso in [20]), is potted in Figure 6 and shows how saturation does not happen at a specic point but is reay a continuous eect. In fact, we note that, even for sma input powers, the ampier is not abe to suppy the fu sma-signa gain of G max = 20 db. The numerica dierences between the modes are not huge, but are signicant enough so that a network designer may have thought a design was feasibe (based on the mode in Figure ), when in fact it may not satisfy the design specications (based on the more-accurate mode in Figure 6). Notice, aso, that there is a imit on the tota avaiabe output power (P max ) from the ampier. This imit is shown as the dashed ine in The \true" point of saturation occurs when the ampier gain is reduced by 3 db from its maximum []. () Figure 6. Hence, the gain curve used in this paper foows the curved ine (from Equation ()) for ow input powers and the straight dashed ine in Figure 6 for higher powers. 2 Soution Approach Given a network as in Figure, we woud ike to minimize the number of ampiers used in the network without vioating the device capabiities and constraints. Throughout this paper, we assume that the stars are connected together in the form of a tree and that a neighbors have two unidirectiona inks connecting each other. A mathematica formuation of the probem is provided in Section 2.. Unfortunatey, the resuting mixed-integer noninear optimization probem is extremey dicut to sove. Hence, we carefuy avoid the integra constraints by modifying the formuation, specicay the objective function, and sove the resuting noninear optimization probem. The description of the soution strategy is provided in Section 2.2. The output from the sover is fed to an Ampier-Pacement Modue which outputs the exact positions and gains of the ampiers. The functionaity of the Ampier-Pacement Modue is described in Section Formuation In this subsection, the ampier-pacement probem is formuated as a mixed-integer noninear (constrained) optimization probem. First, the notation used in the formuation is introduced, and then the constraints and objective functions are described. 2.. Device Parameters p sen = Minimum power required on a waveength for detection in dbm. This represents both the receiver sensitivity eve and the ampier sensitivity eve, which have been assumed to be equa. P max = Maximum power avaiabe from an ampier in mw = Maximum power of a transmitter in mw. 3

4 It is not necessary that the maximum ampier output and transmitter powers be identica. For simpicity, we have assumed them to be equa. G max = Maximum (sma-signa) ampier gain in db. = Signa attenuation in db/km Probem Variabes This section introduces the variabes used in the probem formuation. Note that, among the variabes representing the power eves, those beginning with owercase (p min;beg, p x;i, p xmit i ) are measured in dbm and those with uppercase, P min ) in mw. Aso, the variabes in owercase represent the per-waveength power eves, whereas the ones in uppercase represent the aggregate power over a the waveengths on the respective ink. (P beg N = number of access stations in the network = number of waveengths in the network. M = number of stars in the network. L = number of inks in the network = 2(N +M?). Note that stars are identied by the indices ; 2; : : : ; M and stations by the indices M + ; M + 2; : : : ; M + N. As we sha soon observe, this provides notationa convenience when we refer to the source/destination of a ink, irrespective of whether it is a station or a star. Aso, the waveengths in the network are identied by the indices M + ; M + 2; : : : ; M + N of the source stations. We associate the foowing parameters with each ink, L. s = Source of ink, s (M + N). d = Destination of ink, d (M + N). = Set of powered waveengths carried by ink. n = Number of ampiers on ink. L = Length of ink in km. SG = Actua tota Suppied Gain on ink in db. p min;beg = Power eve of the east-powered waveength arriving at ink, in dbm. P beg = Tota power at the beginning of ink, in mw. P min = Tota power on ink when a signas are p sen and at east one waveength is equa to p sen, in mw. gmax = Maximum gain avaiabe from an ampier on ink, in db. Consider the star i, i M. D i = in-degree of star i = out-degree of star i. p x;i = power of waveength x at the output of star i, in dbm. Consider the station i, (M + ) i (M + N). p xmit i = Transmitted power of waveength i at station i, in dbm Usefu Functions The foowing functions aow conversion between the miiwatt (reguar) and dbm (og) scaes. T odb() = 0 og 0 () T omw () = 0 =0 They are used to express the constraints convenienty in the appropriate scae. 2.. Basic and Non-Basic Variabes Given a network, the vaues of the topoogy-specic variabes N, M, L, s, d,, L, and D i are xed, irrespective of the ampier-pacement agorithm chosen. The ony basic variabes used in the formuation are p xmit i n. Note that the variabes P beg, p min;beg, SG, and, P min, p x;i, and gmax are non-basic variabes and can be expressed in terms of the basic variabes as foows. For ink, whose source is a star, i.e., s M, we have and we aso have p min;beg = Min x 2 p x;s (2) P beg = X x2 T omw (p x;s ) (3) For ink, whose source is a station, i.e., (M + ) s (M + N), we have and we aso have p min;beg = p xmit s () P beg = T omw (p xmit s ) () For any ink, the tota power drops to its minimum eve when at east one of the waveengths is equa to the sensitivity eve (p sen ). Hence, on ink, starting with an aggregate power eve P beg, when the weakest signa is at a power eve p min;beg, after appropriate scae changes, we have P min = T omw (T odb(p beg )? (p min;beg? p sen )) (6) The equation above is best expained with an exampe. Consider a ink containing three waveengths, 2, and 3. Suppose the power eves on these waveengths at the beginning of the ink were 2 W (?26:99 dbm), 3 W (?2:23 dbm), and W (?23:0 dbm), respectivey. Now, the weakest signa is on waveength, and from Equation (2), we have p min;beg =?26:99 dbm. Aso, from Equation (3), we have P beg = 2 W + 3 W + W = 0 W. Now, with a ink attenuation () of 0.2 db/km, and a sensitivity eve (p sen ) of?30 dbm, this group of waveengths can trave (p min;beg? p sen )= = (?26: )=0:2 = :0 km before the power of waveength drops beow p sen. At this point, the powers on the three waveengths are W (?30 dbm), : W (?28:2 dbm), and 2: W (?26:02 dbm), respectivey. Hence, the aggregate \minimum" power (P min ) is +:+2: W = W. This vaue can be derived from the above equation since T omw (T odb(0 W)? (?26: )) = T omw (?20? 3:0) = W

5 For inks from stations to stars, i.e., (M + ) s (M + N) and d M, we have p s ;d = p xmit s + SG? L? T odb(d d? ) (7) For inks between stars, i.e., s ; d M, we have 8 x 2 p x;d = p x;s + SG? L? For any ink, T odb(d d? ) (8) gmax = G(P min ; G max ; P sat ) (9) We note that various ampier gain modes can be used to obtain this function G. 2.. Constraints Inequaities. Consider the ink, L. The powers on each of the waveengths at the beginning of the ink shoud be at east the sensitivity eve, p sen. This can be ensured by requiring that the weakest signa has a power eve of at east p sen as foows. p min;beg p sen (0) The powers on each of the waveengths at the end of each ink shoud be at east p sen. This is to enabe the receivers to detect the signas correcty. Thus, p min;beg + SG? L p sen () The above inequaities (Equations (0) and ()) ensure that the signa powers remain at or above p sen everywhere aong the ber inks and throughout the network. There are upper imits on the maximum power carried by a the signas in a ink. This vaue P max is the same for transmitters and ampiers, and hence at the beginning of ink, we have P beg P max (2) Simiary, at the end of the ink, we have T odb(p beg ) + SG? L T odb(p max ) (3) Since we need to divide the tota suppied gain SG among the n ampiers on ink, we have SG gmax n () However, the gain SG shoud require no fewer than n ampiers; thus, SG > gmax (n? ) () Integraity Constraints. Consider the ink, L. The number of ampiers, n, on any ink, is an integra vaue. Hence, we require that n is an integer: (6) 2..6 Objective function Minimize 2..7 Compexity LX = n (7) The ony basic variabes used in the formuation are p xmit i, SG, and n. The others can be computed either beforehand from the topoogy or at run-time as a function of the basic variabes. Hence, we have number of variabes = 2 L + N, number of integer constraints = L, and number of noninear inequaities = 6 L Reasons for Noninearities The approach presented in this paper diers from the one in [20] in that it aows the dierent waveengths on a ink to be at dierent power eves. Whereas the method in [20] needed to pace ampiers whenever a the waveengths on the ink were at their owest power eve, now the pacement of the ampier is constrained by the weakest signa on the ink. Hence, on each ink, we need to identify the waveength coming in with the owest power eve (p min;beg ). This introduces a noninear term in the formuation (Equation (2)). Moreover, the maximum gain (gmax ) avaiabe at an ampier on a ink is dependent on the precise mix of the power eves on its incoming waveengths. This computation cannot be performed o-ine and resuts in noninear constraints (see Equations () and ()). 2.2 Sover Strategies The mixed-integer noninear optimization probem resuting from Section 2. is an extremey dicut one to sove and is highy computation-intensive. Surveys on techniques empoyed in soving such probems can be found in [9] and [0]. For such highy-noninear, genera, integer programming probems, branch-and-bound based methods which are empoyed in integer inear programming probems do not work we [6]. In order to reduce the computation compexity, we choose to eiminate the integra constraints atogether. In our case, this can be done by removing the variabes n from the formuation, and hence the constraints in Equations () and () disappear. A simiar approach is described in [6], where the integraity of variabes is expressed as an additiona constraint to the origina probem. So, we dene a new objective function: Minimize LX SG =gmax (8) = which is cose to the origina one, since n = dsg =gmax e. The starting point of the probem space is especiay important for this noninear search. We initiaize the basic

6 variabes of the probem, namey, SG and p xmit i such that SG = 0 p xmit i = T odb(p max ) i.e., the network is initiaized to a state when a the transmitters are operating at their highest powers and a of the inks have zero gain. However, we coud aso use the soution from [20] as a feasibe starting point. Since the new objective function is not identica to the origina one for the integra case, the sover might end up minimizing the function SG =gmax and not the number of ampiers in the network. To hande this situation, we adopt a nonintrusive measurement approach, where, at every feasibe point aong the search path to the optimum soution taken by the noninear program sover, we evauate the origina objective function and remember the point in the search space which resuted in the minimum vaue for the origina objective function thus far. The ensuing heuristic search has the foowing interesting properties.. It contains signicanty fewer variabes and constraints. In fact, it has ony L + N variabes, L inequaities, and zero integer constraints. 2. A the constraints and the objective function are easiy dierentiabe. Hence, the gradients can be fed to the noninear program sover to aid it in its search for the optimum soution. The noninear program sover, CFSQP (C code for Feasibe Sequentia Quadratic Programming) [9], which we have used for this study, consists of a set of C functions for the minimization of the maximum of a set of smooth objective functions, subject to genera smooth constraints. If the initia guess provided by the user is infeasibe for some inequaity constraint or some inear equaity constraint, CF- SQP rst generates a feasibe point for these constraints; subsequenty, the successive iterates generated by CFSQP a satisfy these constraints. Noninear equaity constraints are turned into inequaity constraints and the objective function is repaced by an exact penaty function which penaizes noninear equaity constraint vioations ony. Given a feasibe iterate x, the basic SQP direction d 0 is rst computed by soving a standard quadratic program using a positive denite estimate H of the Hessian of the Lagrangian. d 0 is a direction of descent for the objective function; it is amost feasibe in the sense that it is at worst tangent to the feasibe set if there are noninear constraints and it is feasibe otherwise. The user has the option of either requiring that the objective function (penaty function if noninear equaity constraints are present) decrease at each iteration after feasibiity for noninear inequaity and inear constraints has been reached (monotone ine search), or requiring a decrease within at most a few, say three, iterations (nonmonotone ine search). The user must provide functions that dene the objective function and constraint functions, and may either provide functions to compute the respective gradients or require that CFSQP estimate them by forward nite dierences. Additiona detais on the CFSQP sover can be found in [3]. CFSQP provides the user with some exibiity in the choice of agorithms and vaues for various parameters. We describe beow some of our choices and characteristics unique to the ampier pacement probem at hand. In the formuation presented above, we do not have any noninear equaity constraints and there is ony one objective function (Equation (8)). This enabes CFSQP to empoy the objective function directy (and not any penaty function) in its search. We require that CFSQP use a nonmonotone ine search [8], forcing a decrease of the objective function within at most three iterations. The gradients of the objective and some constraint functions are estimated by CFSQP using forward nite dierences. When there are no noninear equaity constraints (as in our case), CFSQP terminates when the norm for the Newton search direction d 0 fas beow which is taken to be 0?8. Whie eiminating integer variabes greaty simpies the probem, there are, however, imitations to this approach and they are discussed beow.. Loca minima: The noninear program sover might terminate at a point corresponding to a oca minimum for the objective function. This happens, for exampe, when the starting point corresponds to the Linear Program soution (see Tabe 2 and the exampes in Figs. and ). 2. Feasibe point generation: When the starting point is infeasibe, subject to the constraints, the sover may not be abe to ocate a feasibe point in the probem space. With CFSQP, this probem can be xed by using a dierent quadratic programming sover to generate the feasibe point. However, nding a feasibe point becomes increasingy dicut as the number of network eements grows (i.e., more network eements means more variabes). 3. Integer variabes: The noninear program sover (CF- SQP), which we used in this study, is not we-suited to hande integer variabes. Hence, its resuts for this probem coud be improved upon by using speciaized mixed-integer noninear program sovers. The output of the noninear program sover is fed to the Ampier-Pacement Modue which is described next. 2.3 Ampier-Pacement Modue The modue uses the vaues of SG and p xmit i output by the noninear program sover to determine the exact ocation and gain of the ampiers in the network. It operates on a ink-by-ink basis as foows. It computes the maximum vaue of the gain avaiabe from each ampier on a ink (gmax ) using Equation (9) and, hence, the number 6

7 of ampiers (n ) required on that ink 6. It aso computes the power eves of the dierent waveengths at the output of the stars (p x;i ). Severa methods of spitting the gain (SG ) among the n ampiers on a ink are possibe. We describe two methods beow { the As Soon As Possibe (ASAP) method and the As Late As Possibe (ALAP) method. The ASAP method for ampier pacement operates as foows. For a but the ast ampier on a ink, this method paces an ampier on a ink as soon as the input power is ow enough to aow the maximum gain, and for the ast ampier on a ink, it paces the ampier as soon as the input power is ow enough to aow the remaining gain. The ALAP method operates in a simiar fashion, except that it attempts to pace ampiers as cose as possibe to the destination of the ink. Both these methods spit the tota gain on the ink among the ampiers by operating a but one of them at their maximum possibe gain. The dierences between the ASAP method and the ALAP method can be seen in Figs. 7 and 8, where the tota gain of. db is divided among 3 ampiers. The ink shown runs from star to star in the scaed-up MAN network (Figure ) and the gains shown are taken from the LP soution [20] for this network. Severa other methods of spitting the gain, incuding equa distribution among the ampiers on a ink, are possibe. The ALAP method was chosen in our study (see Tabe 3). Further discussions on various approaches to gain spitting can be found in [7]. 3 Numerica Exampes The ink-by-ink method in [] was designed to equaize the powers of the waveengths in the network, as opposed to trying to minimize the number of ampiers in the network. By forcing the powers of a waveengths to be equa to p sen at the beginning of most inks (a inks except those from stations to stars), the agorithm paced ampiers simpy by knowing how many waveengths were on a ink. If the number of waveengths on a ink is precomputed, this aowed the agorithm to operate on each ink individuay (ocay) without knowing what was happening on other inks. This ed to a very simpe ampier-pacement agorithm. Unfortunatey, as was shown in [20] and can aso be seen in Tabe 2, this approach does not minimize the number of ampiers needed in the network. The transmitter powers can be adjusted to avoid pacing ampiers on the inks which originate at a station. However, since signas on a other inks start o with the minimum power (p sen on each waveength), we know that the agorithm wi pace an ampier on every singe ink not originating at a station in the network. We note that there are L? N such inks in the network which originate at a star (reca that L = number of inks, N = number of stations, and M = number of stars); thus, we obtain the ower bound 6 The noninear program sover coud possiby come up with a soution with negigibe gains (SG ) at certain inks. We use n = dsg =gmax? e, where is a sma number to hande this situation ( = 0:0 in our numerica exampes). of L? N = 2 (N + M? )? N = N + 2 (M? ) on the number of ampiers used by the method in []. This agorithm performs the poorest, in comparison to other pacement schemes, on networks that have short inks because the other agorithms can usuay avoid pacing an ampier on a short ink simpy by exiting the originating star with enough power to traverse the short ink. We show the resuts of this agorithm for various networks in coumn 2 of Tabe 2. The goba method in [20], aowed waveengths at the beginning of the inks to be above the absoute minimum aowed, p sen. However, the powers on a of the waveengths at any given point in the network was required to be equa; this equay-powered-waveengths constraint enabed the computation of the maximum gain (gmax ) avaiabe on a ink, by knowing just the number of waveengths on the ink. The ampier-pacement probem can be formuated as a mixed-integer inear program and soved exacty. Consider a pair of adjacent stars in the network. Taking into account the attenuation oss aong the inks connecting the stars and the spitting osses at the stars, we require that there be at east one ampier on either of these inks. The ower bound on the number of ampiers required using the Linear-Program (LP) method in [20] is thus M?, where M is the number of stars in the network. We show the resuts of this agorithm for various networks in coumn 3 of Tabe 2 (see [20] for detais). The method described in this paper (see Section 2) is a goba one too; however, unike the LP method in [20], it aows the waveengths at any point in the network to operate at unequa powers. 7 The soution obtained to the ampier-pacement probem is not guaranteed to be the optimum because of the presence of oca minima. We show the resuts of this agorithm for various networks in coumn of Tabe 2. The absoute ower bound was deveoped in [] by rst utiizing the number of waveengths on each ink and the physica constraints on the ampiers to derive the maximum gain avaiabe from each ampier on a given ink. These vaues were then incuded in a LPsovabe soution to derive the ower bound on the number of ampiers required in the network. We show the resuts of the ower bound computation for various networks in coumn of Tabe 2 (see [] for more detais). Next, we compare the resuts of these three approaches to ampier pacement on certain sampe networks (see Tabe 2). As mentioned earier, the network in Figure motivated this study. Whie both the earier approaches (the ink-byink method and the LP method) required a few ampiers to operate the network, the NLP method described in this paper does not require any. The network in Figure 9 is the motivating network, described above, taken to the extreme. This network has many stars and yet it needs no ampiers to function. Tabe 2 reveas that the new method was indeed abe to 7 Reca that, in these experiments, the NLP sover's starting point is chosen such that a transmitters are operating at the maximum power without any ampiers in the network. From this possiby infeasibe starting point, the sover reaches a feasibe point for a the exampe networks, except for the Previous MAN (Figure ). 7

8 Network Link-by-ink LP NLP Absoute CPU time method method method ower for NLP [] [20] (this work) bound [] (this work) Simpe 2 star (Figure ) s Tree (Figure 9) 0 0 9s MAN (Figure 0) h 6m 2s Scaed-up MAN (Figure ) h 7m 2s Scaed-down MAN (Figure 2) m 9s Previous MAN (Figure ) m 0s Denser MAN (Figure 3) 0 7 2h m 9s Tabe 2: Reative performance of the various ampier-pacement schemes. A \*" in coumn indicates that the NLP sover coud not perform better than the LP soution, even when it was given mutipe feasibe starting points, incuding the soutions found in [] and [20]. Coumn 6 shows the tota CPU time taken by the noninear sover running on an otherwise-unoaded DEC 000/20 to sove each probem. come up with the soution of not needing any ampiers. This is the type of network where the unequay-poweredwaveengths soution is ceary superior to the previous two ampier-pacement methods. Athough it is arguabe whether this network is reaistic or not, we have presented it here in order to give the reader some insight as to the conditions in which the new method performs best. The network in Figure 0 is meant to be a reaistic design of a MAN. This network was designed in a semi-random fashion with some heuristics to guide the design. Tabe 2 shows that the new method was abe to nd a soution which required fewer ampiers than the methods in [] and [20]. Figure 0 aso provides an insight into how the actua pacements of ampiers dier between the LP method and the NLP method. The trianges that are ed back are the ocations at which the equay-powered-waveengths method paced the six ampiers it deemed necessary to operate the network. The empty, or ed-white, trianges are the ocations where the unequay-powered-waveengths method paced the two ampiers it deemed necessary. The numerica information on exact gains and exact pacements of the ampiers can be seen in Tabe 3. The power eves of the signas at the transmitters and receivers can be found in Tabe. Note that the equay-powered-waveengths constraint resuts in more ampiers and a higher overa gain in the network. Note aso that the transmitters are unabe to operate at their maximum power for the same reason. However, when waveengths are aowed to operate at different power eves, we nd that the NLP soution requires just the minimum overa gain to operate the network. This network serves as the reference point for a study into the eects of scaing network distances up and scaing network distances down, which wi be discussed beow. As previousy noted in Section, an ampier becomes ess ecient when mutipe waveengths passing through it are operated at dierent power eves. If a ink were ong enough, we woud expect that this ineciency woud start to require the addition of more ampiers. On the other hand, we woud expect that, if inks were short, then waveengths at dierent power eves might not require the addition of more ampiers and might aow us to potentiay save even more ampiers at critica points in the network. The network in Figure is meant to study the eects on the soution when we have inks that span onger distances and the network in Figure 2 is meant to study the eects on the soution when a network has shorter inks. Both of these networks are the same as the network in Figure 0 except that the distances have been scaed up and down, respectivey, by a factor of 0. As we see in Tabe 2, the resuts seem to verify our earier predictions. The new method is not abe to nd a better soution than the equay-powered-waveengths soution for the arger network in Figure, even when it was given mutipe feasibe starting points (incuding the soutions found in [] and [20]). Our method's soution is not guaranteed to be the best because it coud have become stuck at a oca minimum. If our new method is stuck at a oca minimum, we potentiay can miss the goba minimum soution. This diers from the LP soution which does nd the goba minimum soution (subject to the equaypowered-waveengths constraint). On the other hand, the new NLP method is abe to come up with a better soution for the smaer network (Figure 2). In fact, as we predicted, our new method was abe to take advantage of the smaer network environment. The unequay-poweredwaveengths soution was abe to use 0 ampiers compared to for the equay-powered-waveengths soution, which was a savings of ampiers. In the reference network (Figure 0), the unequay-powered-waveengths soution was abe to use 2 ampiers compared to 6 for the equaypowered-waveengths soution, which was aso a savings of ampiers. The network in Figure is aso examined here because both of the previous studies [, 20] examined this particuar network 8. This network has many nodes and we predicted that our new method might not perform better than the equay-powered-waveengths soution. We predicted this because the more nodes a network has, the more variabes the sover is manipuating and the more oca minima 8 The number of nodes for group 3 was reduced from 3 to 28 nodes because the origina network, as proposed in [], was infeasibe because signas exited the star of degree 3 with power beow p sen =?30 dbm. 8

9 Link (Star!Star) LP (gain and distance NLP (gain and distance from start of ink) from start of ink)! Gain.2 db at km Gain 0.9 db at km! Gain 0.80 db at km Gain 0.9 db at km 2! Gain 6. db at.00 km!2 3! Gain.76 db at.96 km!3! Gain 9.76 db at 0.00 km! Gain 9.0 db at 0.00 km Tabe 3: Exact ampier pacements for the network depicted in Figure 0. Stations LP method NLP method Transmitter power Receiver power Transmitter power Receiver power {9 (Group )?3:02 dbm?22:96 dbm 0.00 dbm From G:?9:9 dbm From G2:?27:9 dbm From G3:?26:80 dbm From G:?30:00 dbm 0{ (Group 2)?2:82 dbm?30:00 dbm?0:0 dbm From G:?27:8 dbm From G2:?8:9 dbm From G3:?23:9 dbm From G:?27: dbm 6{20 (Group 3)?2:82 dbm?29:2 dbm?0:0 dbm From G:?26:79 dbm From G2:?23:9 dbm From G3:?7:0 dbm From G:?26:36 dbm 2{30 (Group )?2:6 dbm?23: dbm 0.00 dbm From G:?30:00 dbm From G2:?27:6 dbm From G3:?26:37 dbm From G:?0:80 dbm Tabe : Transmitter and receiver powers for the network depicted in Figure 0. the sover can get stuck at. As Tabe 2 shows, the sover was unabe to come up with a better soution than the LP soution, even when given mutipe feasibe starting points incuding the soutions found in [] and [20]. The \denser-man" network in Figure 3 diers from the MAN network in Figure 0 in that there are 2 additiona stations in it, 3 in each of the four groups of stations. This exampe shows the eect of scaing up the network by adding more stations on the number of ampiers needed. A three ampier-pacement schemes require additiona ampiers to operate. Note, however, that the NLP method performs better than the other two schemes and remains cosest to the absoute ower bound on the number of ampiers. For each of the previous exampe networks, coumn 6 in Tabe 2 shows the tota CPU time taken by the noninear sover running on an otherwise-unoaded DEC 000/20. In genera, the running time is found to increase with () increasing number of network components (which eads to more constraints) and (2) increasing ink spans (which eads to a greater choice in feasibe soutions). The running time of the sover can be potentiay reduced by modifying the stopping criteria (see section 2.2); however, this can aso aect the quaity of the soution. Note that, due to the characteristics of our NLP soution process, no poynomiaform time compexity can be specied. Future Work. Switched Networks The agorithms described in this paper were designed to operate on \oopess" networks where there is ony one path from a source to a destination. In a switched network, there can potentiay be mutipe paths between a source and a destination. Since the above agorithms operate knowing how many waveengths are on a given ink, they assume that a waveengths that can possiby reach a ink coud a be present on that ink simutaneousy. This approach has the potentia to pace more ampiers in the network than is absoutey necessary. The exampe switched network given in Figure incudes mutipe paths between any source-destination pair. When examining the permutation of connections that use the WR-PS3 ink, notice 9

10 that a 8 of the stations coud each have a connection set up that use this \haf" of L. Actuay, there is aways at east one permutation of connections that woud cause any of the \haves" of ink L, L2, L3, and L to carry 8 connections. Now, if ampiers were paced in this network to aow any possibe conguration of connections, a \haves" of inks L-L woud have to be designed with enough ampiers to carry 8 connections in the worst case. Now, it is fairy easy to see that, if the connections are setup in a \smart" fashion, a ink never has to carry 8 connections. In fact, a ink shoud never have to carry more than 2 connections in this network. Designing inks to carry 2 connections instead of 8, since the network wi then potentiay need ony one-fourth the power on these inks, can resut in a signicant savings in the number of ampiers. We beieve that it wi be possibe to modify our current agorithms to aow them to expoit this phenomenon that occurs in switched networks. This is a topic of our future work..2 Modeing Device Characteristics In the near future, we pan to try and further improve on the optica ampier gain mode. We expect to be abe to create a reasonaby-accurate gain mode of the popuar Erbium-doped ber ampier (EDFA). Anaytica methods for modeing the ampier gain, gain saturation, and noise described in [6] wi be incorporated in the mode. We aso pan to expand our ampier gain mode to hande perwaveength gain. This woud aow us to mode an ampier that has a non-at gain spectrum. It woud aso aow us to mode the sma gain for waveengths that are normay considered to ie outside of the \ampier bandwidth". The formuation of the probem woud have to be changed to hande per-waveength gain too. Further, we pan to consider the eect of crosstak and the received bit-error rate (BER) of the signas on the ampier pacement in such networks. A study describing the computation of BER in the presence of crosstak and ampier-generated ASE noise in switched networks can be found in []. Concusion We considered the probem of minimizing the number of optica ampiers in an optica LAN/MAN. This study departed from previous studies by aowing the signa powers of dierent waveengths on the same ber to be at dierent eves. Athough this increases the compexity of the ampier-pacement agorithm, numerica resuts show that certain networks do benet signicanty from this method by requiring fewer ampiers. Acknowedgment We thank Professor Jonathan P. Heritage of the Eectrica and Computer Engineering Department, UC Davis, for his assistance in our understanding of the optica ampier gain mode. We aso thank the reviewers for their comments which heped improve this paper. References [] G. P. Agrawa, Fiber-Optic Communication Systems. New York: Wiey, 992. [2] S. B. Aexander et a., \A precompetitive consortium on wide-band a-optica networks," IEEE/OSA Journa of Lightwave Technoogy, vo., no. /6, pp. 7{ 73, May/June 993. [3] A. R. Chrapyvy, J. A. Nage, and R. W. Tkach, \Equaization in ampied WDM ightwave transmission systems," IEEE Photonics Technoogy Letters, vo., no. 8, pp. 920{922, Aug [] D. Datta, B. Ramamurthy, H. Feng, J. P. Heritage, and B. Mukherjee, \BER-based ca admission in waveength-routed optica networks," in Optica Fiber Communication (OFC '98) Technica Digest, vo. 2, (San Jose, CA), pp. 92{93, Feb [] A. F. Erefaie, E. L. Godstein, S. Zaidi, and N. Jackman, \Fiber-ampier cascades with gain equaization in mutiwaveength unidirectiona inter-oce ring network," IEEE Photonics Technoogy Letters, vo., no. 9, pp. 026{03, Sept [6] C. R. Gies and E. Desurvire, \Modeing erbiumdoped ber ampiers," IEEE/OSA Journa of Lightwave Technoogy, vo. 9, no. 2, pp. 27{283, Feb. 99. [7] C. R. Gies and D. J. Giovanni, \Dynamic gain equaization in a two-stage ber ampier," IEEE Photonic Technoogy Letters, vo. 2, no. 2, pp. 866{868, Dec [8] L. Grippo, F. Lamparieo, and S. Lucidi, \A nonmonotone ine search technique for newton's method," SIAM Journa on Numerica Anaysis, vo. 23, pp. 707{76, 986. [9] O. K. Gupta and A. Ravindran, \Noninear integer programming agorithms: A survey," OPSEARCH, vo. 20, pp. 89{206, 983. [0] P. Hansen, \Methods of noninear 0- programming," in Discrete Optimization II (P. L. Hammer et a., eds.), New York: North Hoand, 979. [] J. Iness, Ecient Use of Optica Components in WDM-based Optica Networks. PhD dissertation, Dept. of Computer Science, University of Caifornia, Davis, CA, Nov [2] K. Inoue, T. Kominato, and H. Toba, \Tunabe gain equaization using a Mach-Zehnder optica ter in mutistage ber ampiers," IEEE Photonic Technoogy Letters, vo. 3, no. 8, pp. 78{720, 99. [3] C. Lawrence, J. L. Zhou, and A. L. Tits, User's Guide for CFSQP Version 2.: A C Code for Soving (Large Scae) Constrained Noninear (Minimax) Optimization Probems, Generating Iterates Satisfying A Inequaity Constraints. Institute for Systems Research, Feb Technica Report No. TR-9-6r. 0

11 [] C.-S. Li, F. F.-K. Tong, C. J. Georgiou, and M. Chen, \Gain equaization in metropoitan and wide area optica networks using optica ampiers," in Proceedings, IEEE INFOCOM '9, (Toronto, Canada), pp. 30{37, June 99. [] C.-S. Li, F. F.-K. Tong, C. J. Georgiou, and M. Chen, \A near-far compensation scheme for aoptica WDMA/WDM networks with arbitrary topoogy," tech. rep., IBM, 99. [6] H.-L. Li, \An approximate method for oca optima for noninear mixed integer programming probems," Computers & Operations Research, vo. 9, no., pp. 3{, Juy 992. [7] H.-D. Lin, \Gain spitting and pacement of distributed ampiers," Tech. Rep. RC 626 (#7200), IBM, Oct [8] K. Magari, M. Okamoto, and Y. Noguchi, \. m poarization insensitive high gain tensie strained barrier MQW optica ampier," IEEE Photonic Technoogy Letters, vo. 3, no., pp. 998{000, Nov. 99. [9] E. R. Panier and A. L. Tits, \On combining feasibiity, descent and superinear convergence in inequaity constrained optimization," Mathematica Programming, vo. 9, pp. 26{276, 993. [20] B. Ramamurthy, J. Iness, and B. Mukherjee, \Optimizing ampier pacements in a muti-waveength optica LAN/MAN: The equay powered-waveengths case," IEEE/OSA Journa of Lightwave Technoogy, vo. 6, no. 9, pp. 60{69, Sept [2] A. E. Siegman, Lasers. Mi Vaey, CA: University Science Books, pp. 298{30, 986. [22] V. Tandon, M. Wiby, and F. Burton, \A nove upgrade path for transparent optica networks based on waveength reuse," in Proceedings, IEEE INFOCOM '9, (Boston, MA), pp. 308{3, Apr. 99. [23] A. E. Winer and S. M. Hwang, \Passive equaization of nonuniform EDFA gain by optica tering for megameter transmission of 20 WDMA channes through a cascade of EDFA's," IEEE Photonics Technoogy Letters, vo., no. 9, pp. 023{026, Sept Byrav Ramamurthy (S'97) received his B.Tech. degree in Computer Science and Engineering from Indian Institute of Technoogy, Madras (India) in 993. He received his M.S. and Ph.D. degrees in Computer Science from University of Caifornia (UC), Davis in 99 and 998, respectivey. Since August 998, he has been Assistant Professor in the Department of Computer Science and Engineering at the University of Nebraska-Lincon. His research interests incude high-speed computer networks, distributed systems, and teecommunications. He was a recipient of the Indian Nationa Taent Search schoarship and was a feow of the Professors for the Future program at UC Davis. His e-mai address is: byrav@cse.un.edu. Jason Iness received the B.S. degree in computer science and engineering (highest honors) from the University of Caifornia, Davis, in 992, He received his M.S. degree and Ph.D. degree in computer science from the University of Caifornia, Davis, in 99 and 997 respectivey. At UC Davis he conducted research in WDM-based ightwave networks, focusing primariy on ecient design of such networks (incuding sparse ampier pacement, and sparse waveength conversion). Other research interests incuded ATM networks and wireess communications. Whie at UC Davis he received the UC Davis feowship in the and academic years. In 997 he joined the Inte Corporation and currenty works in Hisboro, OR. Biswanath Mukherjee (S '82 - M '87) received the B.Tech. (Hons) degree from Indian Institute of Technoogy, Kharagpur (India) in 980 and the Ph.D. degree from University of Washington, Seatte, in June 987. At Washington, he hed a GTE Teaching Feowship and a Genera Eectric Foundation Feowship. In Juy 987, he joined the University of Caifornia, Davis, where he has been Professor of Computer Science since Juy 99, and Chairman of Computer Science since September 997. He is co-winner of paper awards presented at the 99 and the 99 Nationa Computer Security Conferences. He serves on the editoria boards of the IEEE/ACM Transactions on Networking, IEEE Network, ACM/Batzer Wireess Information Networks (WINET), Journa of High-Speed Networks, and Photonic Network Communications. He served as the Technica Program Chair of the IEEE INFOCOM '96 conference. He is author of the textbook \Optica Communication Networks" pubished by McGraw-Hi in 997, a book which received the Association of American Pubishers, Inc.'s 997 Honorabe Mention in Computer Science. His research interests incude ightwave networks, network security, and wireess networks. His e-mai address is: mukherjee@cs.ucdavis.edu.