Limit Theorems for Markovian Bandwidth Sharing Networks with Rate Constraints

Transcription

1 Lmt Theorems for Markovan Bandwdth Sharng Networks wth Rate Constrants Josh Reed Stern School of Busness New York Unversty Bert Zwart CWI September 9, 214 Abstract Bandwdth sharng networks as consdered by [Massoulé and Roberts (1999] and [Roberts and Massoulé (1998] provde a natural modelng framework for descrbng the dynamc flowlevel nteracton among elastc data transfers n computer and communcaton systems, and can be used to develop traffc prcng/chargng mechansms. At the same tme, such models are exctng from an Operatons Research perspectve as ther analyss requres technques from both stochastc modelng and optmzaton. In ths paper, we develop a framework to approxmate bandwdth sharng networks under the assumpton that the number of users as well as the capactes of the system are large, and the assumpton that the traffc that each user s allowed to submt s bounded above by some rate, as s standard n practce. We also assume that customers on each route n the network abandon accordng to..d. patence tmes. Under Markovan assumptons, we develop flud and dffuson approxmatons whch are qute tractable: for most parameter combnatons, the nvarant dstrbuton s multvarate normal, wth mean and dffuson coeffcents that can be computed n polynomal tme as a functon of the sze of the network. 1 Introducton Bandwdth sharng networks as consdered by [Massoulé and Roberts (1999] and [Roberts and Massoulé (1998] provde a natural extenson for modelng the dynamc nteracton among competng elastc flows that traverse several lnks along ther source-destnaton paths n a network. They offer nsght nto the complex behavor of communcaton networks and have also recently been suggested as a tool n analyzng problems n road traffc (see for nstance [Kelly and Wllams (21]. From an Operatons Research perspectve, bandwdth sharng networks are exctng snce ther statc behavor s governed by nonlnear optmzaton problems, whle understandng ther dynamcs requres a separate set of OR tools, namely stochastc models. Contemporary research has devoted a sgnfcant amount of effort to analyzng bandwdth sharng networks and a consderable amount of ths effort has been devoted to dervng stablty condtons for bandwdth sharng networks. Ths queston s stll not settled n general and t s not the subject matter of the present paper although a varety of results may be found n [De Vecana et al. (1999, De Vecana et al. (21], [Bonald and Massoulè (21], [Mo and Walrand (2], [Massoulé (27], [Gromoll and Wllams (27], and [Chang et al. (26]. Another sgnfcant ssue, whch s more central to the present paper, s concerned wth second order phenomena,.e. methods 1

2 to evaluate the performance of bandwdth sharng models. For the rght combnaton of network topology and bandwdth sharng polcy, t s possble to show that the steady-state dstrbuton of the network not only exsts, but s of product form and s nsenstve wth respect to the flow sze dstrbuton. In some cases, t s even possble to derve necessary and suffcent condtons for steady-state dstrbutons of ths type to exst. Ths work s well summarzed n [Bonald et al. (26]. In general, such nce structure on the network topology and bandwdth sharng polcy as mentoned above cannot be expected to hold and one has to resort to approxmatons. Fundamental papers on flud lmt approxmatons for bandwdth sharng networks are [Kelly and Wllams (24] and [Gromoll and Wllams (29]. Propertes of overloaded bandwdth sharng networks have subsequently been derved by [Borst et al. (29, Ergova et al. (27]. A dffuson approxmaton for bandwdth sharng networks was derved n [Kang et al. (29]. [Ye and Yao (28,Ye and Yao (21] consdered dffuson approxmatons of some bandwdth sharng networks where the servce dscplne per class s FIFO rather than PS, whch concdes wth regular bandwdth sharng networks n the case of exponental flow szes. As we see t, the man message of these works s that the performance of bandwdth sharng networks n heavy-traffc can sometmes be descrbed by a lnear transformaton of a vector of ndependent, exponental random varables. Although ths lne of research s exctng and stll ongong, ths computatonally tractable nsght seems lmted to specfc network topologes and bandwdth sharng mechansms. The current paper proposes a dfferent perspectve leadng to another class of tractable approxmatons, namely multvarate normal approxmatons. As we shall show, such approxmatons arse naturally from the observaton that overall system capacty and ndvdual user download speeds may be of dfferent orders of magntude. It s common n applcatons (see for nstance [Bonald and Proutére (24] that network capacty s measured n GgaBts or TeraBts per second whereas ndvdual user maxmal download speeds are measured n Megabts. In the present paper, we assume that ndvdual user download speeds are bounded above by some maxmum whereas overall system capacty may be arbtrarly large. Ths stands apart from the above mentoned works n whch system capactes and user download speeds are assumed to be comparable. A consequence of our lmt on the ndvdual download speed s that a sgnfcant number of users are requred n order to saturate a lnk. As a result, we consder a system wth large arrval rates and system capactes and vew the system on a fxed tme scale, whereas many of the above mentoned works focus on the large-tme propertes of a network wth fxed arrval rates and capactes. Our framework can be seen as an extenson of the many-server scalng found n the lterature on call center approxmatons. We refer to [Gans et al. (23] for a survey on ths lterature and note that the results n ths paper for the most smple case of a sngle node/class network reduce to the classcal dffuson approxmaton of [Halfn and Whtt (1981] for many-server queues. In fact, the model we consder s a hghly non-trval example of a Markovan Servce Network, as consdered by [Mandelbaum et al. (1998]. Unfortunately, we could not drectly ft our assumptons nto thers, so we verfy the necessary detals from scratch. In the call center queueng lterature, one often makes a dstncton between several qualtatvely dfferent regmes: the Qualty Drven (QD, Effcency Drven (ED, and Qualty and Effcency Drven Regme(QED. As we shall see n the present paper, n a mult-class mult-node bandwdth sharng network t s not a pror clear n whch regme a class wll operate from the outset. Ths s actually determned endogenously (rather than exogenously, as s the case n smple call center models through the dynamcs of the bandwdth sharng allocaton algorthm. We provde a key 2

3 optmzaton problem for a model wth user mpatence that determnes whether n steady-state (on a flud scale the maxmal servce rate of a class of users wll be met or not. To the best of our knowledge, the frst paper to consder dffuson approxmatons of bandwdth sharng networks wth rate constrants s [Ayesta and Mandjes (29]. Ths work begns wth exstng explct schedulng polces wthout ndvdual capacty constrants, and then truncates the capacty constrants at the ndvdual maxma. Our allocaton polces take a more ntegrated approach, allowng users that operate below maxmal capacty to take up bandwdth that s not used by other (rate-constraned users, so that bandwdth allocatons are Pareto optmal. Moreover, rather than formulatng the flud and dffuson approxmatons drectly, we rgorously establsh that these approxmatons arse n a large capacty scalng. We also do not need to make addtonal assumptons that yeld an explct representaton of the bandwdth sharng allocaton functon. In fact, all that s necessary s a drectonal dfferentablty property that s establshed n complete generalty n Secton 3 of ths paper, usng results from the senstvty analyss of nonlnear programs as developed n [Bonnans and Shapro (2]. Ths drectonal dfferentablty result (we actually gve a necessary and suffcent condton for dfferentablty s one of the man techncal results of the paper. In partcular, we hope that the general methodology we use (whch does not seem to be well known n the appled probablty communty wll avod the use of laborous bare hand calculatons n the future. We beleve that ths connecton between stochastc networks and contnuous optmzaton wll be nterestng for other works as well. The lmt theorems obtaned n ths paper are flud and dffuson lmts under Markovan assumptons on the nterarrval tmes, nformaton szes and patence tmes of users. The resultng steady-state dffuson approxmatons often yeld a multvarate normal law, where the means and covarances can be computed by, respectvely, a concave programmng problem wth a polyhedral capacty set, and a set of lnear equatons. Ths results n a computatonal procedure that has complexty whch s polynomal n the sze of the network, and s n prncple vald for any network topology and a large class of utlty based bandwdth allocaton mechansms. Durng the preparaton of the fnal verson of ths paper, our man flud lmt result (Theorem 4.1 was extended to general dstrbutons n [Frolkova et al. (213]. The proof of the flud lmt result of [Frolkova et al. (213] requres the machnery of measure-valued processes, whle the proofs of the flud and dffuson lmt results n the present paper work n a fnte-dmensonal settng. In order to avod unnecessary overlap wth [Frolkova et al. (213], the dscusson at several ponts of the man body of ths paper has been shortened and we smply reference [Frolkova et al. (213]. The remander of ths paper s organzed as follows. A model descrpton s provded n Secton 2. Secton 3 contans a detaled senstvty analyss of the bandwdth allocaton functon. Flud and dffuson approxmatons are presented n Secton 4. Secton 5 focuses on nvarant ponts for the flud model. In partcular, we focus on a model wth user mpatence, for whch we establsh unqueness of an nvarant pont, and we provde suffcent condtons for dfferentablty of the bandwdth allocaton functon at ths nvarant pont, leadng to a multvarate normal law for the dffuson approxmaton. In Secton 6, we llustrate our results wth some examples. Proofs can be found throughout the paper; some of the more techncal proofs of Secton 3 are deferred to Secton 7, whle the proofs of Secton 4 may be found n an accompanyng E-companon. 3

4 1.1 Notaton All random varables and processes n ths paper are assumed to be defned on a common probablty space (Ω, F, P. All vectors x R d are assumed to be column vectors and we denote by x the Eucldean norm of x. We denote by x the transpose of x. We let D([,, R d denote the space of rght-contnuous wth left lmts functons defned on [, and takng values n R d for d 1. Unless otherwse stated, all stochastc processes n ths paper are assumed to be measurable maps from (Ω, F to (D([,, R d, d, where the choce of dmenson d 1 wll be clear from context and d o s the metrc defned n (16.4 of [Bllngsley (1999]. We recall from Theorem 16.3 of [Bllngsley (1999] that the space (D([,, R d, d s separable and complete. Moreover, we recall that f x D([,, R d s a contnuous functon, then x n x n (D([,, R d, d f and only f sup t T x n (t x(t as n for each T. We denote by D 2 ([,, R d the product space D([,, R d D([,, R d, whch we endow wth the maxmum metrc. We denote by e = (t, t the dentty functon. 2 The Model In ths secton, we provde the detals of the bandwdth sharng model that we consder for the remander of the paper. We begn by descrbng what s referred to as the network topology. A bandwdth sharng network conssts of J 1 resources and I 1 routes. Each resource s gven an ndex j {1, 2,..., J} and a route = 1,..., I, s consdered a non-empty subset of {1, 2,..., J}. We then defne the J I ncdence matrx A to be such that A j = 1 f resource j s an element of route and zero otherwse. Intutvely, one may thnk of a resource j as a server on a network and a route as a seres of resources through whch nformaton s passed. Note, however, that a route s an unordered set and so we do not dstngush the order n whch nformaton s beng passed through the resources. A flow represents a specfc transfer of nformaton along a route. Each flow n the network s assgned a processng rate and the sum of all of the processng rates assgned to the flows on a partcular route s referred to as the bandwdth devoted to route, whch we denote by Λ. At the same tme, each resource j s assgned a lmted amount of total bandwdth, < C j <, whch t must dstrbute to each of the routes passng through t. We therefore obtan the matrx nequalty AΛ C, where Λ = (Λ 1, Λ 2,..., Λ I and C = (C 1, C 2,..., C J. In addton, we also assgn to each route a maxmum rate < m at whch flows on that route may be processed. In partcular, f z represents the total number of flows on route, we requre that Λ m z. Thus, lettng I m be the I I dagonal matrx wth the m s on the dagonal, we have that n matrx notaton Λ I m z, where z = (z 1, z 2,..., z I. Note also that n general we requre C j to be fnte but allow for the possblty of m =. We next proceed to descrbe the stochastc assumptons whch we place on our bandwdth sharng network. At tme t =, we assume that there are Z ( flows already present on route, each of whose unprocessed nformaton szes are..d. exponental random varables wth mean 1/µ. Next, for each = 1,..., I, let (E (t, t be a rate one Posson process and let η > be the average arrval rate of flows to route. Also assume that (E (t, t, = 1,..., I, are ndependent of one another. The number of flows that have arrved externally to route by tme t s then gven by E (η t. Hence, flows arrve to route accordng to a Posson process wth rate η. Next, for each k 1, we assume that the nformaton sze of the kth flow to arrve to route s an exponental random varable wth mean 1/µ. Moreover, we assume that the nformaton 4

5 Fgure 1: A bandwdth sharng network wth J = 3 resources and I = 4 routes. Resources are represented by crcles and arcs correspond to routes. szes of flows are ndependent of one another both wthn a specfc route and between routes as well. We also assume that each flow arrvng to route s mpatent and s only wllng to wat an exponentally dstrbuted amount of tme wth rate γ n order to have all of ts nformaton processed before abandonng from the network. Note that ths mples that some flows may depart prematurely from the network before havng all of ther nformaton processed. Ths does not happen f γ =, n whch case flows on route are not mpatent. Now let Z (t be the number of flows on route at tme t and set Z(t = (Z 1 (t, Z 2 (t,..., Z I (t. We refer to Z(t as the user populaton vector at tme t. Next, we suppose that there exsts some bandwdth allocaton polcy Λ : R I + R I + such that f the user populaton vector at tme t s gven by Z(t, then the bandwdth devoted to route by each resource on route s gven by Λ (Z(t. Moreover, we assume that each of the Z (t flows along route must equtably share the total bandwdth allocated to the route. Hence, snce the total bandwdth assgned to route s Λ (Z(t, ths mples that the bandwdth allocated to each of the Z (t flows on route s gven by Λ (Z(t/Z (t, whch we defne to be f Z (t =. In Secton 3, we provde a detaled prescrpton of how the bandwdth allocaton polcy Λ s determned. We now conclude ths secton by provdng a sample-path equaton descrbng the dynamcs of the user populaton process Z = (Z(t, t. Ths equaton wll turn out to be useful n our proofs that follow. Frst note that snce the nformaton szes of each flow on route are assumed to be..d. exponental random varables wth rate µ, t follows that the departure process of servce completons of flows of type from the network s a doubly stochastc Posson process wth nstantaneous rate at tme t gven by Z (t(λ (Z(t/Z (tµ = Λ (Z(tµ. Next, snce the patence tmes of flows on route are assumed to be..d. exponental random varables wth rate γ, t follows that the departure process of abandonments of flows of type from the network s a doubly stochastc Posson process wth nstantaneous rate at tme t gven by γ Z (t. Thus, lettng N s = (N s(t, t and N a = (N a (t, t, = 1,..., I, be ndependent, rate one Posson processes, t follows that the user populaton process Z s gven by the soluton to the system of equatons, Z (t = Z ( + E (η t N s (µ Λ (Z(sds N a (γ Z (sds, t, (1 for = 1,..., I. Note that (1 represents a system of I equatons and t s straghtforward to show 5

6 that there exsts a unque soluton Z. 3 The bandwdth allocaton mechansm and ts propertes In ths secton, we provde a more detaled descrpton of how the bandwdth allocaton functon Λ of Secton 2 s determned. In partcular, we cast Λ as the soluton to an optmzaton problem and we show how one may compute the drectonal dervatve of Λ n the nteror of ts doman. We then proceed to provde a result on the Lpschtz contnuty of Λ as well provdng condtons under whch Λ s fully dfferentable. The results of ths secton are nterestng n ther own rght whle also servng as crucal elements of our flud and dffuson approxmatons n the sectons that follow. Our man reference for ths secton s [Bonnans and Shapro (2] and we note that all of the proofs for the results n ths secton may be found n Secton 7. We begn by assgnng to each flow on a route = 1,..., I, a utlty functon U : (, R. The utlty of a flow s assumed to be a functon of the amount of bandwdth allocated to that flow. In partcular, f Λ s the total amount of bandwdth that has been allocated to route and f there are z > flows on route at a gven pont n tme, then each flow wll receve Λ /z unts of bandwdth and consequently each flow wll have a utlty of U (Λ /z. The total utlty across all flows on route wll therefore be gven by z U (Λ /z. We assume that U s a strctly ncreasng, strctly concave, twce dfferentable functon on (, such that lm x U (x =. We also make the conventon that z U (x/z =, x >, f z =. Fnally, we note that one mportant famly of utlty functons are the weghted α-far utlty functons, whch are gven by the functonal form, U (x = κ x α, wth κ, α >. We now proceed to construct our bandwdth allocaton functon Λ as the soluton to a nonlnear programmng problem. In partcular, gven a fxed number of flows z = (z 1,..., z I on each route and a set of utlty functons U, = 1,..., I, the system wll attempt to allocate bandwdth n such a manner so as to maxmze the total utlty of all flows on the network, subject to the capacty and ndvdual rate constrants descrbed n Secton 2. Mathematcally speakng, the bandwdth allocaton Λ = (Λ 1,..., Λ I s gven by the soluton to the followng global utlty maxmzaton problem, (P z max Λ R I =1 subject to AΛ C, ( Λ z U z Λ I m z, Λ. Note that snce the crterum functon n the above nonlnear program s strctly concave, t follows that for each z = (z 1,..., z I (, I n the nteror of the postve orthant, there exsts a unque optmal bandwdth allocaton, whch we denote by Λ(z = (Λ 1 (z,..., Λ I (z. Moreover, for such z, t s straghtforward from our assumptons on the utlty functons that Λ(z lves n the nteror of the postve orthant, makng the constrant Λ superfluous. Therefore, we can and wll gnore the constrant Λ whenever z (, I. Now, for each pont z (, I n the nteror of the postve orthant and for each drecton 6

7 d R I, defne H d (z = lm t (Λ(z + td Λ(z/t, (2 to be the drectonal dervatve (n the drecton d of Λ at the pont z, assumng that ths lmt exsts. We now proceed n Theorem 3.1 below to show that our choce of the bandwdth allocaton functon Λ( as defned by (P z above leads to a drectonally dfferentable functon at each pont z (, I, n each drecton d R I. Moreover, we also show that H d (z may be determned for each z (, I by solvng a specfc quadratc programmng problem. In order to prove ths result, we wll specalze some of the more general results found n Secton of [Bonnans and Shapro (2]. In partcular, our Theorem 3.1 may be vewed as a relatvely straghtforward applcaton of the perturbaton analyss of nonlnear optmzaton problems n Banach spaces, a theory whch s nowadays well developed. Stll, technques from ths theory do not seem to have found wdespread use n the Appled Probablty communty so far. Before provdng the statement of Theorem 3.1, we must frst adopt some addtonal notaton. For each z (, I, let p(z be a J-dmensonal vector of Lagrange multplers correspondng to the capacty constrants AΛ C, and let q(z be an I-dmensonal vector of Lagrange multplers correspondng to the ndvdual rate constrants Λ I m z, and recall that p(z, q(z and Λ(z jontly form a soluton to the Karush-Kuhn Tucker (KKT condtons for (P z. That s, we have that (AΛ(z Cp(z =, (Λ(z I m zq(z = and U (Λ (z/z = q (z + J p j (za j, for = 1,..., I. One may also consult page 147 of [Bonnans and Shapro (2] for ths result. We now denote by γ(z the set of all possble Lagrange multplers of (P z. Next, suppose that we pck a drecton d = (d 1,..., d I R I. We wll then denote by (p d (z, q d (z, the specfc set of Lagrange multplers of (P z that also solve the optmzaton problem j=1 max d m q ; (3 (p,q γ(z motvaton for ths wll be gven n Secton 7. Fnally, we let I(z denote the set of actve ndvdual rate constrants of (P z, and we denote by J (z the set of actve capacty constrants of (P z. Note that these sets do not depend on the choce of drecton d. The followng s now our frst man result concernng the drectonal dfferentablty of Λ. Theorem 3.1. Let z (, I and for each = 1,..., I, defne =1 v (z = 1 z U (Λ (z/z and u (z = Λ (z z v (z. (4 Then, Λ( s drectonally dfferentable n any drecton d R I, and ts drectonal dervatve H d (z s the unque soluton to the followng quadratc programmng problem, (D z,d max h R I 2 u (zd h + v (zh 2 =1 =1 7

8 subject to (Ah j =, f p d j (z >, (Ah j, f (AΛ(z j = C j, h = d m, f q d (z >, h d m, f Λ (z = m z. We remark that the above result would contnue to hold f m = for some = 1,..., I, so that flows on some routes are not subject to ndvdual rate constrants. Moreover, Theorem 3.1 would also contnue to hold f, rather than assumng that A s an ncdence matrx as descrbed n Secton 2, we smply assumed that the elements of A were nonnegatve, or, f A were assumed to be such that the nteror of the polyhedral capacty set {Λ : AΛ C, Λ } s non-empty. Ths would mply the valdty of several constrant qualfcatons, n partcular the Fromovtz-Mangasaran constrant qualfcaton, see pages of [Bonnans and Shapro (2]. Such extensons are relevant to cover more general network models nvolvng multpath routng, see Secton 5.5 of [Kang et al. (29]. We next proceed to show that Λ as defned by the soluton to (P z above s n fact a locally Lpschtz functon on (, I. Results of a smlar nature may also be found n Lemma A.3 of [Kelly and Wllams (24] and Proposton 4.1 of [Borst et al. (29]. These authors, respectvely, establshed contnuty and Lpschtz contnuty of Λ( as defned above for specal cases, wth no rate constrants. That s, m =, for = 1,..., I. The followng result wll also be helpful n Secton 4 when provng our man flud lmt result, Theorem 4.1. Theorem 3.2. Λ( s locally Lpschtz on (, I. That s, for every compact subset C of (, I, there exsts a constant K C such that Λ(x Λ(y K C x y for all x, y C. It turns out that some of our results n Secton 4 wll also requre the stronger condton of Λ( beng dfferentable at a pont z, whch, n vew of Theorem 3.1, s equvalent to H d (z beng a lnear functon of d. A suffcent condton for dfferentablty s gven n the followng theorem. Recall that the strct complmentarty condton holds for (P z f all of the actve constrants of (P z have strctly postve Lagrange multplers,.e. p j (z > for all j J (z and q (z > for all I(z. Also, recall that a set of lnear constrants are lnearly ndependent f the coeffcent vectors on the lefthand sde of these constrants cannot be wrtten as a lnear combnaton of one another. Theorem 3.3. Λ( s dfferentable at a pont z (, I f the constrants n I(z J (z are lnearly ndependent, and f the strct complmentarty condton holds. In ths case, H d (z s the soluton of (D z,d max h R I 2 u (zd h + v (zh 2 =1 =1 subject to (Ah j =, j J (z, h = d m, I(z. Moreover, H d (z, p d (z and q d (z form the unque soluton of the system of I + I(z + J (z lnear equatons, 2(H d (z v (z = 2u (z + p d j (za j + q d (zi( I(z, = 1,..., I, j J (z 8

9 (AH d (z j =, j J (z, (H d (z = d m, I(z. The proof of Theorem 3.3 follows mmedately from Theorem 3.1, explotng strct complmentarty and lnear ndependence of the constrants. Note also that snce actve constrants are requred to be ndependent of one another, we have that I(z + J (z I and so f the dervatve of Λ( exsts, then t may be found by solvng a system of at most 2I equatons. Thus, from a computatonal pont of vew, fndng the dervatve of Λ s not much more dffcult than fndng Λ tself. 4 Flud and dffuson lmts In ths secton, we provde our man flud and dffuson lmts for bandwdth sharng networks operatng under the bandwdth allocaton functon Λ descrbed n Secton 3. The context n whch our lmts wll be obtaned s an asymptotc regme n whch the network topology, ndvdual user rate constrants, and nformaton and patence sze dstrbutons reman fxed, whle the capactes of the resources n the network and the arrval rates of flows to the network grow arbtrarly large. We refer to ths regme as the large capacty scalng regme and, mathematcally, t s defned as follows. We consder a sequence of bandwdth sharng networks ndexed by some parameter n 1. As mentoned n the precedng paragraph, the topology of each network remans fxed wth respect to n. In partcular, we assume that each network has I 1 routes and J 1 resources wth an ncdence matrx A as descrbed n Secton 2. We also assume that the ndvdual user rate constrants for flows on the network reman fxed wth respect to n. That s, flows on route = 1,..., I, have an ndvdual rate constrant gven by m. Fnally, we assume that the capacty vector of the nth network grows lnearly wth n and s gven by nc, where C = (C 1,..., C I s the orgnal capacty vector as descrbed n Secton 2. We next assume that n the nth network at tme t =, there are Z n ( flows already present on route = 1,..., I, and we set Z n ( = (Z1 n(,..., Zn I (. Flows of type arrve externally to the nth network after tme t = accordng to a Posson process wth rate η n and we assume that η n grows roughly at a lnear rate as n grows large. In partcular, we assume that η n/n η > as n. Fnally, we assume that the nformaton sze dstrbuton and that the patence tme dstrbuton of flows arrvng to the nth network does not change wth n. In partcular, flows of type have nformaton szes whch are..d. exponental random varables wth rate µ and patence tmes whch are..d. exponental random varables wth rate γ. For the remander of the paper, unless otherwse noted, all relevant quanttes assocated wth the nth network wll be denoted by a superscrpt n. One nterestng and useful consequence of the defnton of the large capacty scalng regme s that the bandwdth allocaton functon of the nth network, Λ n, scales n a natural way wth n. In partcular, note that Λ n s gven by the soluton to the orgnal global utlty maxmzaton problem (P z, but where the capacty vector C has been replaced by nc. It s then straghtforward to verfy that ths mples that Λ n (z = nλ(z/n, z R I +. (5 It turns out that the scalng provded by (5 wll play a pvotal role n provng our man lmt theorems of the present secton. 9

10 Now, for each n 1 and t, let Z n (t = (Z1 n(t,..., Zn I (t be the user populaton vector n the nth network at tme t. We then defne Z n (t = Z n (t/n to be the flud scaled user populaton vector at tme t and we set Z n = ( Z n (t, t to be the flud scaled user populaton process. Our frst man result of ths secton, Theorem 4.1 below, provdes a weak lmt, under approprate ntal condtons, for the sequence ( Z n, n 1 n the large capacty scalng regme. We refer to ths weak lmt as the flud lmt of ( Z n, n 1. As we now show, the flud lmt of ( Z n, n 1 may be characterzed as the soluton to an I-dmensonal ordnary dfferental equaton. In partcular, we have the followng. Theorem 4.1. If Z n ( Z( (, I as n, then Z n Z as n, where Z s the unque, strong soluton to the system of equatons gven by, for = 1,..., I. Z (t = Z ( + η t µ Λ ( Z(sds γ Z (sds, t, (6 Our proof of Theorem 4.1 may be found n Secton 8.1 of the E-companon. We remark however for the moment, that by Theorem 3.2 the bandwdth allocaton functon Λ s locally Lpschtz on the nteror of the postve orthant, (, I. Moreover, as s demonstrated n the proof of Theorem 4.1, f Z( (, I, then the soluton to (6 must le n (, I for all t. Hence, a standard successve approxmatons argument may be used to show that there does ndeed exst a unque, strong soluton to the system of equatons gven by (6. Also, a more general, measure-valued verson of Theorem 4.1 may be found n [Frolkova et al. (213] whch relaxes the Markovan assumptons of the present paper. We next note that the flud lmt Z of Theorem 4.1 provdes a frst order approxmaton to the dynamcs of the user populaton process. Indeed, condtonal on Z(, the process Z s entrely determnstc. However, n many stuatons t s desrable to obtan a second order stochastc approxmaton to the user populaton process. Ths s the motvaton behnd our next result, Theorem 4.2, whch provdes a dffuson lmt approxmaton to the user populaton process. In order to state our result, we frst need to setup the followng notaton. As a matter of convenence, for the remander of ths secton we assume that Z n ( Z( as n, (7 where Z( s a constant. By (6, ths then mples that the flud lmt Z n Theorem 4.1 s a determnstc functon. Thus, for each n 1 and t, we may defne Z n (t = n 1/2 ( Z n (t Z(t to be the dffuson scaled user populaton vector at tme t and set Z n = ( Z n (t, t to be the dffuson scaled user populaton process. We now proceed to characterze, under approprate ntal condtons, the weak lmt of ( Z n, n 1 as the soluton to an I-dmensonal stochastc dfferental equaton. We refer to ths weak lmt as the dffuson lmt of ( Z n, n 1. We have the followng. 1

11 Theorem 4.2. If Zn ( Z( as n, and n(n 1 η n η β R as n, for = 1,..., I, then Z n Z as n, where Z s the unque, strong soluton to Z (t = Z ( + ξ (t + β e µ H Z(s t ( Z(sds γ Z (sds, t, (8 for = 1,..., I, where ξ = ( ξ (t, t, = 1,..., I, s a sequence of ndependent, Brownan motons wth nfntesmal varances gven, respectvely, by σ 2 (t = η + µ Λ ( Z(t + γ Z (t, t. (9 The proof of Theorem 4.2 may be found n Secton 8.2 of the E-companon. We remark, however, that by Proposton 7.2 of Secton 7, the drectonal dervatve H d (z of Λ s Lpschtz contnuous n d for each fxed z (, I, wth a Lpschtz constant that s unformly bounded (n terms of z over compact subsets of (, I. Hence, snce by Theorem 4.1 t follows that Z s a contnuous functon whch, over compact ntervals of tme [, T ], les n compact subsets of (, I, a standard successve approxmatons technque may be used to show that there exsts a unque, strong soluton to the system of equatons gven by (8. We now conclude ths secton by conductng both a transent as well as a steady-state analyss of the dffuson lmt Z provded by Theorem 4.2. The results of these analyses may be then be used n a straghtforward manner to obtan stochastc approxmatons to the transent and steady-state behavor of the user populaton process tself. Let us begn by assumng that the bandwdth allocaton polcy Λ( s dfferentable at each pont Z(t along the path of the flud lmt Z. Ths then mples that for each t, the drectonal dervatve H d ( Z(t s a lnear functon of d and so, for each t, we may wrte H d ( Z(t = H( Z(td for a partcular matrx H( Z(t R I I. It then follows that we may apply standard results from the theory of stochastc calculus n order to obtan an explct soluton to (8 and hence the transent dynamcs of Z. One may consult, for nstance, Secton 5.6 of [Karatzas and Shreve (1991] for the proof of ths result. In order to state ths result, however, we frst need to setup the followng notaton. For each t, let ξ(t = ( ξ 1 (t,..., ξ I (t and σ(t = (σ 1 (t,..., σ I (t. Next, set β = (β 1,..., β I and let I µ be the I I dagonal matrx such that (I µ = µ for each = 1,..., I, and, smlarly, let I γ be the I I dagonal matrx such that (I γ = γ for each = 1,..., I. We then have the followng. Proposton 4.3. Suppose that the condtons of Theorem 3.3 hold at Z(t for each t and let Φ(t be the soluton to the matrx-valued ODE dφ dt = (I µ H( Z(t + I γ Φ(t, t, wth ntal condton Φ( = I. Then, the soluton to (8 s gven by ( Z(t = Φ(t Z( + (Φ(s 1 d( ξ(s + βs, (1 for t. Moreover, f E[ Z( 2 ] <, then [ E[ Z(t] = Φ(t E[ Z(] + 11 ] (Φ(s 1 βds,

12 and, for s t, E[(Z(s E[ Z(s](Z(t E[ Z(t] ] [ = Φ(s E[(Z( E[ Z(](Z( E[ Z(] ] + s t ] ((Φ(u 1 σ(u((φ(u 1 σ(u T du Φ (t. Moreover, f Z( s Gaussan dstrbuted, then Z(t s Gaussan dstrbuted as well for each t. For our last result of ths secton, we provde a characterzaton of the steady-state behavor of the dffuson lmt Z. Suppose frst that Z( s such that Z(t = Z( for all t. We refer to such ponts Z( as nvarant ponts for the flud lmt equaton (6 and conduct a thorough analyss of such ponts n the secton that follows. However, for the tme beng, we smply note that f Z( s an nvarant pont for (6 and f Λ( s dfferentable at Z(, then t follows mmedately that we may wrte H( Z(t = H( Z( for all t. Moreover, for each = 1,..., I, we have that σ 2 (t = σ = η + µ Λ ( Z( + γ Z (, t. We therefore conclude from (8 that f Z( s an nvarant pont for (6 and f Λ( s dfferentable at Z(, then the dffuson lmt Z of Theorem 4.2 reduces to a tme-homogeneous, I-dmensonal Ornsten-Uhlenbeck process. In ths case, we may provde an explct characterzaton of the steadystate dstrbuton of Z. In partcular, we have the followng result, whose proof may be found, for nstance, n the proof Theorem of [Karatzas and Shreve (1991]. Proposton 4.4. Assume that Z( s an nvarant pont for (6 and that the condtons of Theorem 3.3 hold at Z(. Then, let A = I µ H( Z( + I γ. If each egenvalue of A has a postve real part, then Z(t Z( as t, where Z( s a normal random vector wth mean and varance-covarance matrx gven by 5 Invarant ponts E[ Z( ] = β E[( Z( E[ Z( ]( Z( E[ Z( ] ] = e ta dt, (11 e ta σσ e ta dt. (12 Recall from Secton 4 that a pont z (, I s sad to be an nvarant pont for the flud lmt equaton (6 f condtonal on Z( = z, one has that Z(t = z for all t. Note, however, that ths condton mples that Z( s an nvarant pont for the flud lmt equaton (6 f and only f one has that d Z (t/dt = for all t, for = 1,..., I. Moreover, upon closer nspecton of the flud lmt equaton (6, t s evdent that d Z (t dt = η µ Λ ( Z(t γ Z (t. (13 Hence, an alternatve characterzaton of the set of nvarant ponts for the flud lmt equaton (6 s those z (, I such that η = µ Λ (z + γ z, for = 1,..., I. (14 12

13 In ths secton, we study the fxed pont equaton (14 and provde a characterzaton of the set of nvarant ponts for the flud lmt equaton (6 both n the case when all flows are patent, that s γ = for = 1,..., I, as well as the case when all flows are mpatent, that s γ > for = 1,..., I. Our man objectve throughout the secton wll be to establsh exstence and unqueness results for such nvarant ponts. We begn wth the case that all flows are patent. 5.1 Patence Suppose that γ = for each = 1,..., I, so that each flow arrvng to the network s wllng to wat an unlmted amount of tme before completng servce. In ths case, the fxed pont equaton (14 for the set of nvarant ponts reduces to the smpler form Λ(z = ρ, (15 where ρ s an I-dmensonal column vector such that ρ = η /µ for each = 1,..., I. The quantty ρ represents the ncomng rate at whch work arrves to route. In addton, recall that by the capacty constrants of the network, we must have that AΛ C. Hence, t s evdent from (15 that n the case where all flows arrvng to the network are patent, there cannot exst an nvarant pont for the flud lmt equaton (6 unless Aρ C. We refer to ths condton as the system not beng overloaded. In other words, the total rate of ncomng work at each resource s no greater than the capacty of the resource. The followng s now our man result of ths secton regardng the case where are flows arrvng to the network are patent. Proposton 5.1. Suppose that γ = for each = 1,..., I, and that Aρ C. Then, z (, I s an nvarant pont for the flud lmt equaton (6 f and only f there exsts non-negatve constants p j, j = 1,..., J, and q = 1,..., I, such that z = ρ (U 1 ( q + J j=1 p ja j, = 1,..., I, (16 and where p j = f (Aρ j < C j for j = 1,..., J, and q = (U (m J j=1 p ja j + for = 1,..., I. Proof. Proof of Proposton 5.1 We begn wth the case of necessty. Suppose that z (, I s an nvarant pont for the flud lmt equaton (6. It then follows by (15 and the KKT condtons for (P z that there must exst a set of non-negatve constants p j, j = 1,..., J, and q, = 1,..., I, such that and U (ρ /z = q + J A j p j and q (ρ m z =, = 1,..., I, (17 j=1 p j (Aρ C j =, j = 1,..., J, (18 where n (17 and (18 above we have dropped the dependency of p and q on z. Now note that the frst equalty n (17 mples (16 and that (18 mples p j = when (Aρ j < C j, j = 1,..., J. In order to verfy that q = (U (m J j=1 p ja j + for = 1,..., I, we reason as follows. Frst 13

14 note that by the frst equalty n (17, we have that q = U (ρ /z J j=1 p ja j for = 1,..., I. Hence, f ρ /z = m, then t s mmedate that the desred relatonshp holds. On the other hand, f ρ /z < m then by the second equalty n (17, we have that q =. Moreover, by the strct concavty of U, we have that U (m < U (ρ /z and so agan the desred relatonshp holds. We next prove suffcency. Suppose that z (, I satsfes the condtons of the proposton. We now show that Λ(z = ρ, whch, by (15, mples that z s an nvarant pont for the flud lmt equaton (6. However, t may be smply verfed that Λ(z = ρ by notng that the condtons of the proposton mply that the KKT condtons for Λ(z hold when choosng (Λ(z, p(z, q(z = (ρ, p, q. We remark that n the underloaded case of (Aρ j < C j for each j = 1,..., J, one has that p j = for each j = 1,..., J, and so by Proposton 5.1 t follows that q = U (m for each = 1,..., I. By the strct concavty of U, ths then mples that z = ρ /m for each = 1,..., I, and so the nvarant pont z = (ρ 1 /m 1,..., ρ I /m I s unque. On the other hand, n the crtcal case for whch (Aρ j = C j for each j = 1,..., J, then any choce of p j s feasble and so there may exst multple nvarant ponts. 5.2 Impatence We next cover the case n whch γ > for each = 1,..., I, so that each flow on the network s mpatent and may abandon from the network f not served wthn a reasonable amount of tme. As t turns out, the proof of the characterzaton of nvarant ponts n the presence of mpatence s more challengng than n the prevous subsecton where all of the flows on the network were assumed to be patent. Our man result s the followng. Proposton 5.2. Suppose that γ > for each = 1,..., I. Then, there exsts a unque nvarant pont for the flud lmt equaton (6. A more general verson of Proposton 5.2 (relaxng the Markovan assumptons s gven by Theorem 2 of [Frolkova et al. (213]. Specalzng that result to our partcular case, we remark that the unque nvarant pont for the flud lmt equaton (6 can be characterzed n terms of the soluton to the followng optmzaton problem. Frst, for each = 1,..., I, defne the functon G by settng G (x = U (xγ /(η µ x for < x < ρ. It then follows that G s a strctly decreasng functon, and hence G s strctly concave. Next, defne the optmzaton problem (Q as follows, (Q max Λ G (Λ =1 subject to AΛ C [ ] m η Λ,, = 1,..., I. γ + m µ The unque nvarant pont for the flud lmt equaton (6 can now be found by frst solvng (Q and obtanng ts soluton Λ and then substtutng Λ nto the fxed pont equaton (14 n order to determne z. We now conclude ths secton by makng a connecton between the results above and those of 14

15 Theorem 3.3 and Proposton 4.4. Frst note that Λ( s dfferentable at the nvarant pont z f the actve constrants of (Q are lnearly ndependent and f the strct complementarty condton s satsfed for (Q. Ths follows from Theorem 3.3 snce the Lagrange multplers for (P z can be expressed n terms of those for (Q, and snce the set of actve constrants are dentcal n both problems, see the proof of Theorem 2 of [Frolkova et al. (213] for more detals. Thus, thanks to the above characterzaton of the unque nvarant pont z n terms of (Q, the condton n Proposton 4.4 can be checked algorthmcally. 6 Examples We now provde two examples llustratng how the methodology of the present paper may be appled to analyzng specfc bandwdth sharng networks. 6.1 Connecton wth queues n the Halfn-Whtt regme We begn wth a basc example showng how our work s related to prevous results from the call center queueng lterature. Let us assume that we have a bandwdth sharng network consstng of a sngle route and a sngle resource. That s, I = J = 1. In addton, set η = C = m = µ = 1 and γ =. Suppose further that n the nth network of the large capacty scalng regme, we have that the arrval rate of flows to the sngle route s gven by η n = n β n + o( n and that the capacty of the sngle resource s equal to n. Moreover, assume that arrvng flows have mean nformaton szes equal to 1 and that all flows are nfntely patent, that s γ =. It s then not dffcult to see that, for any strctly ncreasng utlty functon, U, the bandwdth allocaton functon Λ n ( n the nth system s gven by Λ n (z = mn{z, n}. Consequently, the user populaton process n the nth system, Z n, wll behave as a brth-death process wth constant brth rates η n and death rates equal to mn{n, Z n (t}. In other words, the user populaton process wll behave n an dentcal manner as the number of customers n an M/M/n queue wth an arrval rate of η n and a servce rate of 1. Also note that applyng the results of Theorem 4.1 wth η = C = m = µ = 1 and γ =, t straghtforward to see that ρ = η/µ = 1 and that the resultng set of nvarant ponts for the correspondng flud lmt equaton (6 s [1,. Now note that t s evdent that n the setup descrbed above, the soluton to (P z s gven by Λ(z = mn{z, 1}. Hence, t s straghtforward to deduce that the drectonal dervatve of Λ at the pont z = 1 s gven by H d (1 = d1(d <. Nevertheless, t s nstructve to obtan ths result usng the general theory provded Theorem 3.1. We proceed as follows. For the purposes of llustraton, we assume that the utlty functon of the flows on the sngle route n our network s gven by U(x = log x. It s then evdent that the KKT condtons at z = 1 specalze to 1/Λ(1 = p(1 + q(1, p(1(λ(1 1 = and q(1(λ(1 1 =. (19 We then see by nspecton that the soluton to (19 s gven by Λ(1 = 1 along wth all non-negatve pars (p, q such that p + q = 1. Next, Problem (3, whch s to mnmze dq over all pars (p, q solvng (19, results n an optmal soluton (p(1, q(1 = (, 1 f d < and (p(1, q(1 = (1, f d >. Consequently, H d (1 s the maxmzng value of the quadratc functon 2dh h 2 subject to the set constrants h =, h d f d > and the set of constrants h, h = d f d <. It s then straghtforward to see that ths results n the drectonal dervatve H d (1 = d1(d < gven above. 15

16 Now suppose that we set Z( equal to the nvarant pont z = 1. That s, the lmtng flud scaled ntal number of users on the network s equal to one. Then, applyng Theorem 4.2, we conclude that the dffuson lmt for the user populaton process s gven by the soluton to the SDE d Z(t = (β + mn{ Z(t, }dt + 2d W (t, t, (2 where W = ( W (t, t s a standard Brownan moton. Assumng that β >, the steady-state dstrbuton of ths dffuson s not Gaussan but s stll computable. We refer to [Halfn and Whtt (1981] for further detals. 6.2 A sngle lnk wth multple customer classes We next consder a network wth a sngle resource but wth multple routes passng through t. That s, we asuume that J = 1 but that I 1, see Fgure 2 below. We assume that the sngle resource operates at unt capacty mplyng that C = 1 and that customers on each route n the network are mpatent so that γ > for each = 1,..., I. Moreover, we assume that the utlty functons for the flows on route = 1,..., I, are gven by the weghted proportonal farness utlty functons so that U (x = κ log x for some κ >. Fgure 2: A sngle resource (J = 1 wth I = 3 routes runnng through t. Our man focus wll be on characterzng the unque nvarant pont to the flud lmt equaton (6. Frst, let us set ρ = m η γ +m µ for each = 1,..., I. It s then not too dffcult to see that f ρ η 1, then the unque nvarant pont for the flud lmt equaton (6 s gven by z = γ +m µ wth Λ = ρ for each = 1,..., I. Ths mples that flows on each route are served accordng to ther maxmum possble rate m. On the other hand, f ρ > 1, then the optmzaton problem (Q characterzng the unque nvarant pont may be explctly solved accordng to the followng procedure. 1. Order the ndces = 1,..., I, such that κ 1 m 1 κ 2 m 2 2. Solve for p such that 3. Set = max{ : κ m < p }. 4. Set Λ = η γ max{ κ κ =1 η... κ I m I. γ κ max{ κ m, p } + µ = 1. m,p }+µ and note that Λ = ρ f >. 16

17 5. Set z = (η Λ µ /γ. For the valdty of the above procedure as well as a rgorous proof for the case of ρ 1, we refer the reader to the proof of Theorem 4 of [Frolkova et al. (213], and n partcular equatons (25 (31 of that paper. We now hghlght the mplcatons of the nvarant pont descrbed by the above procedure. Note that flows on routes = 1,..., should be expected to almost always be served at strctly less than ther maxmum possble rate m, whle flows on routes = + 1,..., I should be expected to almost always be served at ther maxmum possble rate m. The only possble excepton to ths rule occurs when p = κ +1 m, n whch case flows on route + 1, wll, dependng upon the number +1 of flows on each route n the system, swtch back and forth between beng served at ther maxmum possble rate m and beng served at some lesser rate. Hence, the rato κ /m plays a crtcal role n the steady-state behavor of the system. Ths observaton may then be used to construct certan prcng schemes. For example, users may be asked to pay a prce for a certan maxmum amount of bandwdth m and a servce provder can then optmze over κ n order to maxmze ts proft. It s also nterestng to note the smlarty between the characterzatons mentoned above and the ED, QD and QED regmes, respectvely, whch were mentoned n the Introducton. Fnally, note that all flows n the network operate n the overloaded regme when κ s set n constant proporton to m. 7 Proofs of propertes of the bandwdth sharng functon In ths secton, we provde the proofs of Theorems 3.1 and 3.2 from Secton 3. We also prove a proposton regardng the drectonal dervatve of the bandwdth allocaton functon Λ. We begn, however, wth the followng prelmnary result regardng the contnuty of Λ(. Lemma 7.1. Λ( s contnuous on (, I. Proof. Proof of Lemma 7.1 Our argument s a smplfed verson of the proof of Lemma 1 n [Frolkova et al. (213]. In partcular, t suffces to show that for each vector z (, I and sequence {z k, k 1} such that z k z as k, we have that Λ(z k Λ(z as k. We proceed by contradcton. Let z (, I and suppose that z k z but that Λ(z k Λ(z. Moreover, note that snce the set of feasble solutons to (P z s a subset of the compact set {Λ, AΛ C}, we may assume wthout loss of generalty that Λ(z k Λ Λ(z. In addton, note that snce Λ s clearly a feasble soluton to (P z, and snce Λ(z s the unque optmal soluton to (P z, we have that l := z U (Λ (z/z > =1 z U (Λ /z =: r. (21 =1 For each k 1, now defne Λ k = (Λ k 1,..., Λk I by settng Λk = mn{(λ(z, m z k } for each = 1,..., I, and note that Λ k s a feasble soluton to (P z k. Moreover, snce z k z, Λ k Λ(z and Λ(z k Λ, t follows that z k U (Λ k /z k l and z k U (Λ (z k /z k r. =1 =1 17

18 Thus, by (21, for k suffcently large we have that z k U (Λ k /z k > =1 z k U (Λ (z k /z k, =1 whch contradcts Λ(z k beng optmal for (P z k. Ths completes the proof. We now proceed wth the proof of Theorem 3.1. Proof. Proof of Theorem 3.1 We follow closely Secton of [Bonnans and Shapro (2]. In partcular, drectonal dfferentablty of Λ( at each pont z (, I wll follow once we have establshed that each condton of Theorem 5.53 n [Bonnans and Shapro (2] as well as each condton of Remark 5.55 n [Bonnans and Shapro (2] are satsfed for each z (, I. We begn by verfyng that each of the fve condtons of Theorem 5.53 of [Bonnans and Shapro (2] are satsfed for (P z for each z (, I. ( Clearly, (P z has a unque optmal soluton Λ(z snce we are optmzng a strctly concave functon over a convex closed set, hence assumpton ( s satsfed. ( Snce the set of feasble solutons {Λ, AΛ C, Λ z m } s non-empty, the Fromovtz- Mangasaran constrant qualfcaton holds (see pages of [Bonnans and Shapro (2], and so, by Proposton 5.5 (v, Gollan s condton holds. Ths verfes that assumpton ( s satsfed. ( Note that snce the crterum functon of our problem s strctly concave, and the set of feasble solutons s convex, the set of Lagrange multplers γ(z s non-empty, and hence assumpton ( s satsfed. (v Let Q = Q(Λ, z, p, q be an I I matrx such that Q j (Λ, z, p.q = Λ j Λ j L(Λ, z, p, q, where L denotes the Lagrangan assocated wth (P z, see (22 below. A smple computaton then shows that Q j (Λ, z, p.q = I( = ju (Λ /z /z, whch s ndependent of p and q. Thus, we have that for any I dmensonal vector h, (Q(Λ, z, p, qhh = =1 h 2 U (Λ /z /z, whch s strctly postve as long as h. Ths mples that the strong second order condtons (5.12 n [Bonnans and Shapro (2] are satsfed, and so assumpton (v s verfed. (v For all z n a neghborhood of z, the set of feasble solutons {Λ, AΛ C, Λ z m } s non-empty and unformly bounded. Note that ths s true even f m = for some, snce Λ max j C j. Hence, assumpton (v s satsfed. 18

19 Havng now verfed that each of the fve condtons of Theorem 5.53 of [Bonnans and Shapro (2] are satsfed at each z (, I, we next proceed to verfy that each of the condtons n Remark 5.55 of [Bonnans and Shapro (2] hold as well. Ths wll then mply the drectonal dfferentablty of Λ( at each pont z (, I. However, note that by propertes ( through (v above, as well as Lemma 7.1, both suppostons of the remark hold for each z (, I and drecton d R I. Hence, Λ s drectonally dfferentable at each z (, I. We next proceed to derve an expresson for the drectonal dervatve of Λ( at each pont z (, I. We begn by ntroducng some addtonal notaton. For each z (, I and d R I, let (P L d max h R I =1 U ( Λ (z z ( ( Λ h + U z subject to (Ah j, j J (z, h m d, I(z, Λ (z U z ( Λ (z z d be the lnearzaton (n the drecton d of (P z at (Λ(z, z (see, for nstance, problem (P L d on page 446 of [Bonnans and Shapro (2]. Next, recall that the Lagrangan assocated wth (P z may be wrtten as L(Λ, z, p, q = =1 ( Λ z U + z J p j ((AΛ j C j + j=1 q (Λ m z, (22 where, n the above, we note that Λ s not a functon of z. Also, let γ(z denote the set of Lagrange multplers of (P z. The dual of (P L d may then be wrtten as =1 (DL d max D zl(λ(z, z, p, qd (p,q γ(z Here, D z L(Λ, z, p, q s the gradent of L(Λ, z, p, q where the dervatve of the Lagrangan s taken wth respect to each coordnate of z when Λ s regarded as not dependng on z, and the multplcaton wth d s taken to be the dot product. The above dual problem may be wrtten explctly as max (p,q γ(z =1 ( ( Λ d U Λ U z z ( Λ z d m q, (23 whch may then be further smplfed to (3. Now let (p d (z, q d (z be the set of Lagrange multplers that solve (23. It then follows that the set of optmal solutons to (P L d may be wrtten as S(P L d = h : =1 (Ah j =, f p d j (z >, (Ah j, f (AΛ(z j = C j, h = d m, f q d (z >, h d m, f Λ (z = m z.. (24 Havng setup the above notaton, we now recall that the drectonal dervatve of Λ at z s a soluton to the optmzaton problem (5.125 of [Bonnans and Shapro (2], whch, n our case, s unque snce (5.126 of [Bonnans and Shapro (2] s always satsfed. We therefore now proceed 19