Robust Bandwidth Allocation Strategies

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1 Robu Bandwidh Allocaion Sraegie Oliver Heckmann, Jen Schmi, Ralf Seinmez Mulimedia Communicaion Lab (KOM), Darmad Univeriy of Technology Merckr. 25 D Darmad Germany {Heckmann, Schmi, Abrac. Allocaing bandwidh for a cerain period of ime i an ofen encounered problem in nework offering ome kind of qualiy of ervice (QoS) uppor. In paricular, for aggregae demand he required bandwidh a each poin in ime may exhibi coniderable flucuaion, random flucuaion a well a yemaic flucuaion due o differen aciviy a differen ime of day. In any cae, here i a coniderable amoun of uncerainy o be deal wih by raegie for effecively allocaing bandwidh. In hi paper, we ry o devie ocalled robu raegie for bandwidh allocaion under uncerainy. The noion of robune here mean ha we look for raegie which perform well under mo circumance, bu no necearily be for a given iuaion. By imulaion, we compare he differen raegie we propoe wih repec o he robune and performance hey achieve in erm of (virual) co aving. We how ha robune and good performance need no be conradicory goal and furhermore ha very good raegie need no be complex, eiher. Keyword. Bandwidh managemen, demand uncerainy, VPN, robu algorihm. 1. Inroducion Many deciion and opimizaion in he area of nework deign, raffic engineering and oher reource allocaion problem are baed on uncerain daa due o he relaively long imecale on which hee mechanim operae. In hi paper, we argue ha a deciion maker i ypically inereed in robu oluion and we derive everal fairly general raegie for a recurring ub-problem of he above area - bandwidh allocaion. The differen raegie for he bandwidh allocaion problem wih renegoiaion and reervaion in advance beween a cuomer and a nework provider are implemened and heir robune and performance i eed in a erie of numerical imulaion. In order o be able o guaranee a baic level of qualiy o a cuomer he provider ha o know a lea he upper limi of he cuomer raffic, allowing him o proviion he righ amoun of reource and perform admiion conrol, independen of he qualiy of ervice archiecure, e.g., In- Serv [3] or DiffServ [2], ued. In hi paper, we look a a cuomer ha need a coniderable, varying amoun of nework reource (e.g., bandwidh) over long imecale, for example for a provider proviioned virual privae nework (ee IETF working group ppvpn, [4, 9]), poenially in uppor of buine-criical applicaion. The demand flucuae heavily over he coure of a day wih peak in he lae morning and afernoon hour and far lower demand in he nigh hour a well a over he coure of he week wih up on he weekday and down a he weekend. Previou reearch work [11, 22, 35, 15] ha hown ha i i generally favourable for boh cuomer and provider o allow renegoiaion of bandwidh allocaion. The cuomer ave co during phae of low demand and he provider can make beer ue of he capaciy of he nework. Among oher finding, he imulaion in hi paper confirm ha wihou renegoiaion he co increae coniderably (a lea by a facor of 3 in our eing). A lo of reearch in he area of virual privae nework i done o increae he flexibiliy of VPN [6, 21, 17, 18, 24], a rend which make renegoiaion eay. On he downide, for buine criical applicaion renegoiaion can be a dangerou mechanim becaue cuomer are given no guaranee ha hey obain he higher amoun of bandwidh hey need for heir peak demand a he provider could run ou of reource in uch ime leading o a rejecion of he reque. Thi problem can be avoided if renegoiaion i combined wih reervaion in advance. Cuomer can now reque heir increaed bandwidh ahead of ime. They can hu avoid he rik of running ou of bandwidh for buine criical applicaion. We will how in hi paper ha hey will uually ill ave co. So here are rong argumen for cuomer o ue reervaion in advance. On he oher hand wih reervaion in advance he provider ha a beer prognoi of he uilizaion of he nework in advance which may allow him in urn o poenially allocae bandwidh more efficienly a furher provider, ye he laer recurion i no in he cope of hi paper. We aume ha if here i no enough bandwidh for a reervaion in advance ha eiher he provider allocae he miing bandwidh a anoher provider or he cuomer change provider. In hi paper, we ake he viewpoin of a (e.g., VPN) cuomer menioned above ha reerve bandwidh (e.g., for one of he runk of hi VPN) in advance a a provider (e.g., offering a bandwidh-aured VPN ervice). The problem for he cuomer i ha i demand foreca i necearily uncerain. We will ue mehod from ochaic programming. Sochaic programming deal wih opimizaion under uncerainy and wa inroduced in 1955 by Danzig [5]. Good overview on ochaic programming are given in [19, 29, 33, 34]. A cae udy ha ue ochaic programming for capaciy planning in he emiconducor indury can be found in [20]. In [31], ervice proviioning for diribued communicaion nework wih uncerain daa i udied. Several ervice proviioning model are preened ha accoun for everal ype of uncerainy. However, no efficien oluion algorihm are preened and no imulaion are carried ou. Anoher relaed work i [7], here a ervice provider offer compuaional ervice and rie o maximize profi. In our work we conider a nework ervice

2 and ake he perpecive of he cuomer. Some of he mehod preened in hi paper were alo uccefully applied o a differen problem domain, he planning of a producion program [28]. A a remark, he menioned problem can be conidered a an inance of he MPRASE (Muli-Period Reource Allocaion a Syem Edge) framework [12, 14]. Thi framework model he edge beween wo nework. Our recen work [27, 13, 15] ha hown ha many reource allocaion problem a an edge are imilar o a cerain degree which make i eay o reue algorihm or o reduce problem o oher, already olved one. Thi background come in handy when deriving oluion algorihm in hi paper. In erm of he MPRASE axonomy [13], he bandwidh allocaion problem deal wih here i FV * D D a i deal wih an uncerain edge (dicree ochaic demand) beween one cuomer and one provider, ue a one-dimenional reource model and a linear co model wih fixed and variable co. The paper i rucured a follow: In he nex ecion, he deerminiic verion of he bandwidh allocaion model we ue a applicaion example i inroduced and decribed. In he hird ecion, uncerainy in planning problem i dicued. In paricular, we how everal way of modeling uncerainy, define he robune of a plan and how ome general raegie ha deal wih uncerainy in model conrain. Thoe raegie are evaluaed in he fourh ecion baed on heir robune and general performance, before in he la ecion we ummarize our finding and poin oward fuure reearch direcion in hi area. 2. Applicaion Example: Ordering a VPN Service 2.1 Bandwidh Allocaion Model In order o have a realiic background we ue a applicaion example a VPN for which bandwidh i reerved in advance. We aume ha a cuomer reque bandwidh for a provider proviioned VPN for a longer period. The level of bandwidh r i flexible and can be changed (ahead of ime). In order o give incenive no o change he level of bandwidh oo ofen, fixed co c which are incurred by each change in he level of bandwidh are inroduced. Thee co can be real co or ju calculaory ficive co o accoun for he renegoiaion overhead. Variable co are incurred depending on he level of reerved (no necearily ued) bandwidh. Thi bandwidh allocaion model i formulaed a a MIP (mixed ineger programming [16]) problem in M1. M1 i a deerminiic problem, all of he parameer are aumed o be known exacly - an obviouly unrealiic aumpion, which i why we inroduce uncerainy in he nex ecion. The objecive funcion (1) of M1 minimie oal co. (2) enure ha demand i fully aified in each period. Whenever r and r -1 differ, i.e., a new bandwidh allocaion ake place and i forced o become 1. Thi i expreed in (3) and (4). Noe ha i e o 0 in all oher cae auomaically becaue of he non-negaive enry c in he objecive funcion. M1 Deerminiic Bandwidh Allocaion Problem Variable: r Amoun of reerved capaciy in period = 1,..., T. Binary variable, 1 if a (re)allocaion i made a beginning of period = 1,...,T and 0 oherwie. Parameer: b Demanded capaciy in period = 1,...,T. Demand i aumed o poiive (b > 0). c Fixed allocaion co, co per allocaion. We aume poiive co ( c > 0). r c Variable allocaion co, co per reerved capaciy uni per period. r 0 Allocaion level before he beginning of he fir period. M M i a ufficienly high number (e.g., max {b }). Minimize c + c r r (1) ubjec o r b (2) r r 1 M (3) r 1 r M (4) { 01, } (5) 2.2 Soluion Algorihm In [14], everal exac and heuriic algorihm for he problem above are preened and evaluaed. In hi paper, we ue he cheape exac algorihm from ha work which i baed on he dynamic programming paradigm [1] and ha a complexiy of O(T 2 ). The funcion C( 1, 2 ) i defined a he minimal co for a ingle allocaion beween period 1 and 2. I can be calculaed a C ( 1, 2 ) = c r c τ τ = 1 max( b { 1,..., 2 }). (6) The algorihm exploi he rucure of he problem which caue C ( y, x ) C ( y, x + 1 ) ( y, x ) x > y. The algorihm i depiced in Figure 1. Preparaion: Prepare an empy array cmin and an empy array pred, each wih T enrie. Sar: cmin(1) = C( 1, 1 ) pred(1) = 1 Ieraion = 2,..., T: cmin() = min{c(i, ) + cmin(i-1) i = 1,..., } pred() = argmin{c(i, ) + cmin(i-1) i = 1,..., } Reul: cmin(t+1) conain he minimal co while array pred ore he hop oward ha oluion. Figure 1: Dynamic programming algorihm for he deerminiic bandwidh allocaion problem.

3 3. Bandwidh Allocaion under Uncerainy 3.1 Modeling Uncerainy If here i no uncerainy wih regard o a parameer he value of ha parameer i known a he ime he deciion i made. We hen call ha parameer deerminiic. The deerminiic cae of he bandwidh allocaion problem ha been briefly preened in he previou ecion and i reaed in deail in [14], where he baic problem i alo advanced oward he cae of muliple provider. Type of Uncerainy. Parameer like fuure bandwidh demand which form he bai for a deciion or opimizaion proce can be and in pracice ofen are uncerain. Several degree of uncerainy for a parameer can be diinguihed: Toal uncerainy: Nohing i known abou which value he parameer will ake. The be hing one can do in hi cae i o ry o reac flexibly and learn from pa value he parameer ook. [27] deal wih he ingle provider ingle cuomer bandwidh allocaion problem under oal uncerainy. Sochaic uncerainy: The exac value he parameer will ake i no known bu he deciion maker know he probabiliy diribuion of he parameer and can hu make ome predicion abou he parameer. [8] and [26] are ypical work ha deal wih ochaic uncerainy for bandwidh allocaion problem from a provider poin of view by auming ource wih on-off raffic. Dicree ochaic uncerainy: The parameer i drawn from a dicree e of value, each value ha a cerain probabiliy. The e i ypically modeled a a number of cenario. Thi approach i dicued below in more deail a i i he approach aken in hi paper. Modeling Uncerainy wih Scenario. The idea of modeling uncerainy wih cenario ha i roo in cenario analyi [25, 23]. Scenario analyi i a mehod for long-range planning under uncerainy. Conforman and plauible combinaion of he realizaion of all uncerain parameer yield a number of cenario. Thee cenario form he bai for he following deciion proce (e.g., a producion plan i baed on he aumpion ha one of he hree cenario will occur: price and demand go up, price fall lighly and demand remain equal, demand goe back and price fall heavily ). An applicaion example and lieraure overview i given in [20]. However, decribing uncerainy wih a range of cenario i alo enible for hor- and mid-range planning and ofen ued for ochaic programming [19, 5, 29] a i ha ome crucial advanage over uing a paramerized probabiliy diribuion: I i eay and inuiive for he deciion maker o creae he cenario, hey could alo be creaed auomaically [10]. Scenario are eay o analyze, heir plauibiliy can be approved eaier han by creaing a mahemaical probabiily diribuion. Scenario are flexible, every kind and number of poible even can be eaily accouned for in he cenario. Finally, cenario can be ued a a dicreizaion of probabiliy diribuion for numerical algorihm. Due o he advanage of he cenario mehod we apply i in hi paper o model he uncerainy of he demand b for period =1,...,T. We aume ha we have a number S of cenario wih he demand foreca b for period and cenario, each cenario ha a probabiliy p wih S p. (7) 3.2 Robune The noion of robu plan em from deciion heory [29]. Deciion maker are ypically evaluaed ex po by how good heir propoed plan performed in realiy (i.e., in he cenario ha acually occured). A hey can looe heir job and career when heir plan perform badly in he occuring cenario and hi ypically ouweigh he praie if he plan perform well, clever deciion maker are rik-avere o a cerain degree and biaed oward robu plan. A robu plan i a plan ha i judged poiive in mo of he cenario and doe no perform oo badly in any of he cenario. The deciion making inance in he VPN applicaion example i alo inereed in robu plan, no (corporae) cuomer run high rik ha here are inufficien reource in criical ime ju for aving ome communicaion co. We now derive raegie ha can deal wih he uncerain parameer b and evaluae heir robune laer in imulaion. 3.3 Sraegie for Dealing wih Uncerainy In general, uncerain parameer can occur in he objecive funcion and he conrain of an opimizaion problem. If he objecive funcion i affeced he deciion maker run he rik of no achieving opimal reul becaue of he uncerainy. If, however, he conrain are affeced he deciion maker rik creaing plan ha are no valid or realizable in realiy. Dealing wih uncerainy in he conrain i uually harder and more complex, ye more imporan han dealing wih uncerainy in he objecive funcion [29]. In he bandwidh allocaion problem conrain (2) i affeced by he uncerain parameer b.we now preen ome general raegie how o deal wih problem ha have uncerain conrain. Deerminiic Subiuion Sraegie. For he deerminiic ubiuion raegie we ubiue he uncerain (cenario dependen) parameer b wih a deerminiic (cenario independen) parameer bˆ and hen olve he reuling deerminiic problem M1 wih he algorihm preened in Secion 2.2. Several ubiuion can be ued. An obviou one i o ue he expeced value

4 S 1 bˆ = -- p (8) S b a ubiue, we call hi raegy DED (deerminiic wih expeced demand). To avoid undereimaing he demand a urcharge α can be added o he ubiue. We call hi raegy urcharge raegy (DSUα): S 1 bˆ = ( 1 + α) -- (9) S p b For he deerminiic wor-cae raegy DWC we ue he highe value of all cenario a ubiue: bˆ = max{ b } (10) A plan baed on he wor cae value yield a oluion ha aifie all conrain for all cenario, hi i why uch a raegy i alo called fa oluion raegy [19, 29]. Chance Conrained Sraegie. The deerminiic raegie have no real conrol over he chance ha heir plan violae he uncerain conrain wih he excepion of DWC which make ure ha he plan i valid for 100% of he cenario. The chance conrained raegy CC allow finer conrol over he chance ha a plan i valid by inroducing a facor α and forcing he uncerain conrain o be aified in a lea α percen of he cenario. M2 Chance Conrained Bandwidh Allocaion (CC) Variable ee M1 and: ζ Binary Variable, 1 if all demand aified i aified for cenario and 0 oherwie. Parameer ee M1 and: b Demanded capaciy in cenario,...,s for period = 1,...,T. p Probabiliy of cenario,...,s. α The probabiliy ha he plan i valid. Minimize (1) (11) ubjec o (3), (4), (5) and r + M( 1 ζ ) b, (12) S p ζ α ζ { 01, } (13) (14) The chance conrained raegy i much harder o implemen han he deerminiic ubiuion raegie, a can be een from he complexiy of he MIP model M2: The binary variable ζ i ued o indicae if he demand i aiffied for all period of cenario (conrain (12)). (13) force a number of cenario o be aified wih a chance of a lea α. An efficien algorihm o olve he chance conrained raegy CC i o reduce i o a number of deerminiic problem: For all poible permuaion of for =1,...,S ζ we denoe Ω he e of all cenario for which ζ =1. Now look a all Ω ha aify (13) excep hoe Ω ha have a ube Ω' Ω ha aifie (13) 1. A deerminiic problem can be formulaed wih bˆ = max{ b Ω}. (15) The deerminiic problem can be olved wih he algorihm from Secion 2.2. For all he deerminiic problem, elec he one ha yield lea co, i opimal oluion i he opimal oluion of he CC raegy. If all S cenario have he ame probabiliy hen he number of deerminiic problem ha have o be olved i S. (16) αs For 20 cenario and a chance α of 0.8 hi, e.g., lead o 4845 deerminiic problem. Becaue of he high complexiy we alo look a a modificaion of he idea behind he CC raegy which make he calculaion coniderably eaier. Inead of requiring ha a plan i valid wih a chance of α for all period we require a plan o ju accoun for he demand of α percen of he cenario in each period. We call hi raegy he eparaed chance conrained raegy SCC. I i quie eay o implemen. Aume ha b' ζ are he parameer b ored over all cenario by increaing value and le p' ζ be heir probabiliie. For he SCC raegy we pick ζ bˆ min b' ζ p' α = ν (17) ν = 1 a ubiue and can hu reduce he SCC problem o a ingle deerminiic problem. Recoure Sraegie. The CC raegy conrol he rik ha a oluion i invalid o ome exen. Recoure raegie conrol he rik in a differen way. In M3 a recoure raegy wih expeced recoure (RER) i given. M3 Bandwidh Allocaion wih Expeced Recoure (RER) Variable ee M1 and f Recoure for cenario,...,s for period = 1,...,T. Parameer ee M1 and f c Recoure co for cenario,...,s for period = 1,...,T. b Demanded capaciy in cenario,...,s for period = 1,...,T. p Probabiliy of cenario,...,s + + Minimize c c r r f p c f (18) ubjec o (3), (4), (5) and r + f b, (19) f 0, (20) 1. Thee e canno yield beer oluion han heir ube, hi i why hey do no have o be looked a.

5 In conrain (19) he new variable f meaure by which amoun he demand remain unaified in cenario for he reuling planned allocaion in period, r. The CC raegy only ake ino accoun ha demand i unaified or no, he recoure raegy alo ake ino accoun how much demand i unaified in a given cenario. The recoure f ha o be penalized in he objecive funcion. The RER doe hi by weighing f wih c and add- f ing he expeced value over all cenario o he objecive funcion(18). In order o implemen he recoure raegy he algorihm of Secion 2.2 can be reapplied wih ome modificaion.. The modified algorihm i preened in Figure 2. Preparaion: Prepare empy array cmin, niveau and pred, each wih T enrie. Sar: niveau(1) = r op (1, 1) cmin(1) = C op (1, 1) pred(1) = 1 Ieraion = 2,..., T: cmin() = min{c op (i, ) + cmin(i-1) i = 1,..., } pred() = argmin{c(niveau(i) i, ) + cmin(i-1) i = 1,..., } niveau() = r op (pred(), ) Reul: cmin(t+1) conain he minimal co while array pred ore he hop oward ha oluion. and array niveau he opimal reervaion niveau. Figure 2: Dynamic programming algorihm for he recoure raegie. I ue a co funcion Cr ( 12, 1, 2 ) = c 1 2 = 1 + c r r = 1 S p c f f ( r 12, 1, 2 ) (21) (22) he opimal rae r op (ha lead o minimal co C op ( 1, 2 ) r op ( 1, 2 ) = = Cr ( op ( 1, 2 ), 1, 2 ) beween 1 and 2 ) rcr (, 1, 2 ) = (23) min{ C( r, 1, 2 ) r [ 0, max{ b [ 1, 2 ]}]} and he recoure f ( r, 1, 2 ) which i defined a f ( r, 1, 2 ) = max{ 0, b r} (24) A c 1 i fixed, he minimum co Cr ( 12, 1, 2 ) from (21) can alo be wrien a 2 S C ( r, 1, 2 ) = c r r + p c f max{ 0, b r} (25) = 1 2 = 1 which can be rewrien a 2 r C ( r, 1, 2 ) c 2 S = r p (26) c f min { 0, r b } = 1 = 1 C ( r, 1, 2 ) = C 1 (27) Funcion C r 1 = c r (28) = 1 i a linear ricly monoonic increaing funcion of r. Funcion = p c f min{ 0, r b } (29) C 2 i a wide-ene increaing piecewie linear funcion ha ar wih negaive value. I lope i decreaing and become zero for all r > max {b =1,...,S, [ 1, 2 ]}. For a local minimum he lope of he difference of hee wo funcion C ha o be zero 2. A he lope of C i he difference beween he conan poiive lope of C 1 and he decreaing lope of C 2 i i zero only for a ingle poin a or a ingle inerval [ a, b ]. C herefore only ha one local minimum which i hen a he ame ime he global minimum. If here i only a ingle minimum i can be eaily found wih a binary earch over all r = b wih [ 1, 2 ] and =1,...,S. Thi reul in a wor-cae complexiy of O(T 2 log(ts)). 4. Simulaion 2 C 2 2 S = 1 A imulaive comparion i ued o ae he meri of he differen raegie preened above. Fir in hi ecion, he imulaion eup and he generaion of he cenario are decribed. Afer ha he robune of he raegie i examined. Ye, robune i no he only imporan crierion, he average performance of he raegie i alo very imporan, herefore, i i evaluaed in a econd erie of imulaion run. Afer ha ome furher reul from oher imulaion are preened horly. 4.1 Seup In order o generae realiic demand paern for he cenario he following mehod i ued o generae a baic demand paern for one day: A day i divided ino 48 period of 30 minue each. A curve wih peak in he lae morning and afernoon and down during he nigh in accordance wih [30] and [32] i ued o decribe empirically found raffic paern. The average demand i 170 bandwidh uni. Baed on hi curve random flucuaion of up o +/- 20% are generaed for all period. Thi i done for every day in he week, aurday are decreaed by 60% and unday by 80% o reflec decreaed buine aciviy during hoe day. The reul i a baic demand paern which i hen muaed o creae he differen cenario for he problem inance. The following muaion are made independenly for each generaed cenario: Wih a probabiliy of 80% he demand of 1 o 4 whole day i caled up or down by up o 20%, wih a probabiliy of 80% repreening buy or calm day. The ame i done for he whole week wih a chance 2. The lope in a local minimum or maximum i zero. The difference funcion here obviouly ha no maximum.

6 Demand Demand Demand of Scenario 1 Demand of Scenario 2 Demand of Scenario Figure 3: Demand of one cenario for one week (lef) and demand of hree cenario for one day (righ). of 75%. In addiion, 15% o 35% of he demand of 8 o 12 period i hifed 1 period earlier or laer, repreening a ligh hif in working chedule (e.g., a videoconference half an hour laer a uual). For each imulaion run 20 cenario were generaed baed on he baic demand. Each cenario wa aigned he ame probabiliy p. The bandwidh demand of one ample cenario i depiced for a whole week in Figure 3 (on he lef). In he ame figure hree example cenario are depiced for a ingle day (on he righ). The fixed co were drawn from a uniform diribuion beween 700 and 1000 and are equal for all period, he variable co were e o 5 for all period. The raegie ha were eed are lied in Table 1. Abbrev. CERT DED DSUα DWC CCα SCCα RER c Sraegy Soluion of he deerminiic bandwidh allocaion model M1 for he bandwidh allocaion problem wihou uncerainy Deerminiic raegy wih expeced demand Deerminiic wih urcharge α=0.05, 0.1, 0.2, 0.3, 0.4 Wor-Cae raegy Chance-Conrained raegy wih chance α=0.8, 0.85 and 0.9 Separaed Chance-Conrained Sraegy wih chance α=0.8, 0.85 and 0.9 Recoure raegy wih recoure co c=38, 50 and 75 (he recoure co are in he ame order of magniude a he penaly co below) Table 1: Overview of he eed raegie. 4.2 Evaluaing he Robune In order zu evaluae he robune we have o evaluae he performance of a plan for diadvanageou cenario. In order o do o we evaluae he plan reuling from he differen raegie for each cenario. I i poible ha a plan doe no allocae ufficien bandwidh for he demand of ome period for a given cenario. See, e.g., Figure 4 for he allocaion of he RER 50 raegy and he demand of a cerain cenario. To accoun for uch failure of he bandwidh allocaion raegie he unaified demand i penalized wih penaly co ha are 10 ime a high a he variable co. For comparion he deerminiic problem wihou uncerainy, denoed CERT, i olved (baed on he acual demand) - i naurally alway lead o he be reul. A raegy i good if i come cloe o he co of CERT, a a meauremen we ue he relaive deviaion ( co dev X co CERT ) X = for each cenario. In order co CERT o evaluae he robune he maximum relaive deviaion mu no be oo large. Table 2 how he aggregaed plan and penaly co for he differen raegie, averaged over 10 imulaion run (10 differen problem inance). The ranking of he raegie i alo lied, baed on he maximum relaive deviaion. Demand Demand Bandwidh Allocaion of RER Figure 4: Demand for one week of one cenario

7 algorihm av. co av. dev x min. dev x max. dev x rank CERT 321' DED 422' % 17.19% 57.27% 16 DSU ' % 17.23% 45.97% 13 DSU ' % 14.47% 36.32% 9 DSU ' % 13.67% 27.12% 5 DSU ' % 14.99% 32.29% 8 DSU ' % 19.98% 39.16% 10 DWC 432' % 22.82% 51.49% 15 SCC ' % 12.59% 26.45% 4 SCC ' % 12.90% 28.16% 6 SCC ' % 14.61% 31.43% 7 CC ' % 21.71% 43.53% 11 CC ' % 22.66% 44.74% 12 CC ' % 22.28% 46.48% 14 RER ' % 9.92% 24.06% 2 RER ' % 10.31% 22.62% 1 RER ' % 10.56% 25.75% 3 Table 2: Aggregaed Plan and Penaly Co. A one can ee from Table 2, he RER raegie how he be wor-cae behavior, followed by SCC and DSU 0.2. DED and DSU wih lower or higher urplu perform very badly, a doe DWC and CC. Thoe raegie canno be conidered robu. The RER and SCC raegie are more robu concerning he variaion of heir parameer α repecively c. Inead of penalizing unaified demand he cuomer could alo ry o hor-erm allocae he miing bandwidh if hi on-demand renegoiaion feaure i uppored by he provider. Table 3 how he planned co plu he adapaion co if hor-erm allocaion i allowed (and he provider alway ha enough free capaciy). The fixed and variable co for hor-erm allocaion are e o wice he co for reervaion in advance. Noe ha even if hor-erm allocaion in ha fahion i poible, i i ill beer o ue reervaion in advance. Fir of all, i avoid he rik ha here migh be no hor-erm reource lef and econd reervaion in advance i ill cheaper, becaue o compleely rely on hor-erm reervaion canno be beer han wice he co of he comparion raegy wihou uncerainy CERT ( ) and hee co are far higher even han he wor raegy wih reervaion in advance. algorihm av. co av. dev x min. dev x max. dev x rank CERT 321' DED 409' % 23.23% 34.70% 10 DSU ' % 21.16% 33.14% 9 DSU ' % 17.36% 30.11% 8 DSU ' % 16.65% 30.01% 7 DSU ' % 18.09% 35.75% 11 DSU ' % 20.94% 41.66% 12 DWC 432' % 22.82% 51.49% 16 SCC ' % 14.37% 24.94% 1 SCC ' % 14.52% 25.48% 2 SCC ' % 16.38% 29.56% 6 CC ' % 21.71% 43.53% 13 CC ' % 22.74% 44.74% 14 CC ' % 24.21% 46.48% 15 RER ' % 15.93% 27.95% 5 RER ' % 16.32% 27.42% 4 RER ' % 15.07% 26.90% 3 Table 3: Aggregaed Plan and Adapaion Co. Looking a he aggregaed plan plu adapaion co, again SCC and RER lead o robu reul while DWC, CC and DSU 0.4 canno be conidered robu. Inereingly he SCC 0.8 and SCC 0.85 raegie now perform beer han he RER raegie. Thi can be explained by looking a he objecive funcion of he RER model M3. The recoure co which are penalized are imilar o he penaly co in Table 2. If he demand of a ingle period in a cenario i high he objecive funcion aign raher low co o he rik of underfulfilling he demand in ha ingle period. If, however, laer hi cenario occur and a hor erm allocaion ha o be made, he co will be relaively high ince high fixed co are incurred for only a ingle period. The SCC raegy on he oher ide would bae i calculaion on he α quanile of he demand in ha period, running le rik of being forced o reallocae for a ingle period. Summarizing, he DWC raegy i no robu and in boh cae lead o very bad reul. Alhough he DWC raegy never lead o penaly or adapaion co, i baic plan, baed on he wor cae demand of all cenario, i ill much more expenive han he combinaion of penaly or adapaion and he planned co of he oher raegie. Only when he penaly co are e higher han 100 ime he variable co he DWC raegy perform accepably. Thu he DWC raegy canno be recommended for a wide range of parameer e of he bandwidh allocaion problem.

8 DED and DSU wih low urplu facor are alo no robu. Only if he urplu facor of DSU i e correcly i performance i accepable; i can hu no really be conidered robu. The chance-conrained raegy CC alo perform badly and i dominaed in performance and complexiy by SCC. SCC and RER can be conidered robu. SCC bae i calculaion on quanile of he demand diribuion and hu ue more informaion from he demand diribuion han he urplu raegie DSU which explain he beer performance. RER perform very good, obviouly he fine-grained conrol over he rik make i more robu han he deerminiic raegie. If unaified demand lead o penaly co he be reul are obviouly achieved if he recoure co are o equal he penaly co (ee RER 50 ). For horerm allocaion he influence of he recoure co i no ha ignifican, hey hould be e o lighly higher value (RER 75 ). 4.3 Evaluaing he General Performance So far only he robune of he raegie ha been evaluaed. In a econd more complex, bu alo more realiic erie of imulaion we ry o evaluae he general/average performance of he differen algorihm. The cenario creaion i modified o reflec a greaer uncerainy in he planning proce. The cenario are creaed and every raegy creae a plan baed on he cenario. Then one cenario i eleced o occur in realiy and he plan are evaluaed by heir performance wih he demand of ha cenario. The occuring cenario, however, i no par of he e of cenario, i i ju imilar o one of hoe cenario. We creae i by elecing one of he cenario and changing he demand of each period by +/- 2%. Thi reflec ha he cenario he deciion making inance bae i deciion on are kind of fuzzy, a hey would be in realiy. The average co, he average deviaion, and i andard deviaion over 20 problem inance for he aggregaed plan and penaly a well a plan and adapaion co can be found in Table 4. The ranking i baed on he average deviaion. The ranking in performance i quie imilar o he ranking regarding robune in Secion 4.2. The RER and SCC raegie perform be and can be recommended. RER again i beer uied for he cae wih penaly co (lo demand) while SCC i beer uied for hor-erm allocaion (reuling in adapaion co). DSU again only perform well if he urplu facor i e correcly. DED and DWC a well a CC perform relaively badly and canno be recommended. The concluion from he experimen are ha he RER Plan and Penaly Co Plan and Adapaion Co algorihm av. co av. dev x ddev rank av. co av. dev x ddev rank CERT 325' ' DED 431' % 10.66% ' % 4.11% 12 DSU ' % 7.90% ' % 3.53% 10 DSU ' % 5.55% 9 401' % 3.38% 9 DSU ' % 3.23% 6 394' % 2.37% 7 DSU ' % 3.44% 8 399' % 3.31% 8 DSU ' % 4.29% ' % 4.13% 11 DWC 431' % 6.17% ' % 6.17% 16 SCC ' % 3.17% 4 383' % 2.31% 2 SCC ' % 3.17% 5 383' % 2.94% 1 SCC ' % 3.32% 7 389' % 3.15% 5 CC ' % 4.73% ' % 4.55% 13 CC ' % 5.61% ' % 5.66% 14 CC ' % 5.91% ' % 5.95% 15 RER ' % 4.03% 3 390' % 2.98% 6 RER ' % 3.32% 1 387' % 3.02% 4 RER ' % 3.36% 2 385' % 2.62% 3 Table 4: Performance Evaluaion

9 raegy hould be ued if no hor-erm allocaion are made, i i robu and perform be for ha iuaion. The recoure co hould be e imilar o he eimaed (calculaory) penaly co of unaified demand for be performance. However, he raegy i robu again a wrong eing of he recoure co, i ill perform very good a long a he recoure co are in he ame order of magniude a he penaly co. For hor-erm allocaion SCC hould be preferred. However, i parameer α hould no be e oo high. If i i e oo high he raegy approache he DWC raegy which performed exremely bad. α=0.8 wa he be choice in our imulaion. The oher raegie are eiher no robu or perform oo badly o be recommended. In pracice one would inuiively ofen bae he calculaion on he expeced demand (DED raegy) or on he wor-cae demand (DWC). Boh approache lead o very bad reul. 4.4 Furher Reul In furher imulaion he fixed co were varied, he uncerainy increaed and he number of cenario varied. In all cae he general concluion from above and he general ranking of he raegie remained unalered in principle. For he problem inance of Secion 4.2 and Secion 4.3 allocaing reource once per week wihou renegoiaion lead o abou 3 ime higher co han hoe yielded by RER or SCC. Thi how again ha renegoiaion can ave a coniderable amoun of co. We have explained why reervaion in advance i vial o avoid he rik of no geing enough bandwidh in peak period. Even if ha i no he cae reervaion in advance can be beer ha hor-erm reervaion: Shor erm reervaion will be priced higher becaue hey leave he provider wih a much higher planning uncerainy and he rik of underuilizing hi reource. The reul how ha if hor-erm allocaion are priced even only 15 o 20% higher han long-erm reervaion he laer combined wih a robu algorihm are cheaper han he opimal hor-erm allocaion. 5. Summary and Oulook In hi paper, we have devied everal raegie for bandwidh allocaion under uncerainy. We have pu emphai on robu raegie which from a deciion-heoreic viewpoin are generally deirable. By imulaion we have examined our propoed raegie wih repec o robune a well a performance in erm of co minimizaion. Some of he more clever raegie howed excellen robune and performance characeriic wherea oher, mainly he mo imple and raighforward one bu alo a fairly ophiicaed one (CC), exhibied deficiencie. While we are aware ha our imulaion eing are quie arbirary (due o lack of empirical daa for uch ervice) we believe ha he principle leon from hee experimen are very general and ha cenario capure uncerainy in he bandwidh allocaion problem very well. A fuure work we perceive he inveigaion of more ophiicaed reource model han ju imple (one-dimenional) bandwidh capaciie, e.g., baed on conrolled burine a for example capured by imple oken bucke a ha been done for deerminiic demand in [15]. Furhermore, i would be inereing o exend he model oward muliple provider a ha been done for he deerminiic cae in [14], which, however, will cerainly be much more difficul for uncerain demand. Anoher iue which o u eem worhwhile furher inveigaion i a rigorou comparion of yem baed on reervaion in advance of variable capaciie v. yem baed on on-demand renegoiaion under differen demand iuaion which could quaniaively juify our aumpion ha for criical demand he laer yem bear oo many rik. 6. Reference [1] R. Bellmann and S. Dreyfu. Applied Dynamic Programming, Princeon Univeriy Pre, Princeon, N.J., [2] D. Black, S. Blake, M. Carlon, E. Davie, Z. Wang, and W. Wei. An Archiecure for Differeniaed Service. Informaional RFC 2475, December [3] R. Braden, D. Clark, and S. Shenker. Inegraed Service in he Inerne Archiecure: an Overview. Informaional RFC 1633, June [4] R. Callon, M. Suzuki, J. De Clercq, B. Gleeon, A. Mali, K. Muhukrihnan, E. Roen, C. Sargor, and J. J. Yu. A Framework for Provider Proviioned Virual Privae Nework. Inerne Draf, draf-ief-ppvpn-framework-03.x, January [5] G.B. Danzig. Linear programming under uncerainy. Managemen Science 1, pp , [6] N. G. Duffield, P. Goyal, A. Greenberg, K. K. Ramakrihnan, and J. E. van der Merwe. A flexible model for reource managemen in virual privae nework. Proceeding of SIG- COMM, Aug [7] S. Dye. The Sochaic Single Node Service Proviion Problem. hp://cieeer.nj.nec.com/26408.hml. [8] M. Falkner, M. Deveikioi, and I. Lambadari. Minimum Co Traffic Shaping: A Uer Perpecive on Connecion Admiion Conrol, IEEE Communicaion Leer, pp , Volume 3, Sepember [9] B. Gleeon, A. Lin, J. Heinanen, G. Armiage, and A. Mali. A Framework for IP Baed Virual Privae Nework. Informaional RFC draf-gleeon-vpn-framework-03.x. February [10] T.J. Gordon and H. Haywood. Iniial experimen wih he cro-impac marix mehod of forecaing. In: Fuure 1, pp , [11]M. Groglauer, S. Kehav, and D. Te. RCBR: A imple and efficien ervice for muliple ime-cale raffic. Proceeding of SIGCOMM 95, pp , Boon, MA, Sepember [12] O. Heckmann and J. Schmi. Muli-Period Reource Allocaion a Syem Edge (MPRASE). Technical Repor TR-KOM , Darmad Univeriy of Technology, fp:// fp.kom.e-echnik.u-darmad.de/pub/paper/hs00-2-paper.pdf, Ocober [13] O. Heckmann and J. Schmi. A Taxonomy for Muli-Period Reource Allocaion Problem a Syem Edge (MPRASE Taxonomy). Technical Repor TR-KOM , fp:// fp.kom.e-echnik.u-darmad.de/pub/paper/hs01-1-paper.pdf, Darmad Univeriy of Technology, Sepember 2001, currenly under ubmiion.

10 [14] O. Heckmann, J. Schmi, and R. Seinmez. Muli-Period Reource Allocaion a Syem Edge - Capaciy Managemen in a Muli-Provider Muli-Service Inerne. In Proceeding of IEEE Conference on Local Compuer Nework (LCN 2001), pp , November [15] O. Heckmann, F. Rohmer, and J. Schmi. The Token Bucke Allocaion and Reallocaion Problem (MPRASE Token Bucke). Technical Repor TR-KOM , Mulimedia Communicaion (KOM), fp://fp.kom.e-echnik.u-darmad.de/pub/paper/hrs01-1-paper.pdf, December 2001, currenly under ubmiion. [16]F. S. Hillier and G. J. Lieberman. Operaion Reearch. McGraw-Hill, [17] R. Iaac and I. Lelie. Suppor for Reource-Aured and Dynamic Virual Privae Nework. IEEE Journal on Seleced Area in Communicaion (JSAC) 19(3), Special Iue on Acive and Programmable Nework, [18] R. Iaac. Lighweigh, Dynamic and Programmable Virual Privae Nework. IEEE OPENARCH, pp. 3-12, March [19] P. Kall and S.W. Wallace. Sochaic Programming. Wiley, New York, [20] S. Karabuk and S.D. Wu. Sraegic Capaciy Planning in he Semiconducor Indury: A Sochaic Programming Approach. Lehigh Univeriy, Deparmen of Indurial and Manufacuring Syem Engineering, Repor No. 99T-12. hp://cieeer.nj.nec.com/karabuk99raegic.hml, [21] I. Khalil and T. Braun. Implemenaion of a Bandwidh Broker for Dynamic End-o-End Reource Reervaion in Ouourced Virual Privae Nework. Proceeding of IEEE Conference on Local Compuer Nework (LCN), November [22] E. Knighly and H. Zhang. Connecion Admiion Conrol for RED-VBR, a Renegoiaion-Baed Service. Proceeding of IWQoS 96, Pari, France, [23]D. Meadow and J. Rander, The Limi o Growh. Univere book, New York, [24]D. Mira, J. A. Morrion, and K. G. Ramakrihnan. VPN DE- SIGNER: a ool for deign of muliervice virual privae nework. Bell Lab Technical Journal Ocober-December 1998, pp [25]K. Nair and R.K. Sarin. Generaing Fuure Scenario - Their Ue in Sraegic Planning. In Long Range Planning (LRP) 12(3), pp , June [26]G. Procii, M. Gerla, J. Kim, S. S. Lee, and M. Y. Sanadidi. On Long Range Dependence and Token Bucke. Proceeding of SPECTS 2001, Orlando, Florida, pp , Jul [27] J. Schmi, O. Heckmann, M. Karen, and R. Seinmez. Decoupling Differen Time Scale of Nework QoS Syem. In Proceeding of he 2001 Inernaional Sympoium on Performance Evaluaion of Compuer and Telecommunicaion Syem, page Sociey for Modelling and Simulaion Inernaional, July [28] A. Scholl and O. Heckmann. Rollierende robue Planung von Produkionprogrammen. Schrifen zur Quaniaiven Beriebwirchaflehre, 02/00 (ISSN ), March [29] A. Scholl. Robue Planung und Opimierung: Grundlagen Konzepe und Mehoden Experimenelle Uneruchungen. Phyica, Heidelberg, [30] J. Rober. Traffic Theory and he Inerne. IEEE Communicaion, pp January [31] A. Tomagard, S. Dye, S.W. Wallace, J.A. Audead, L. Sougie, and M.H. van der Vlerk. Sochaic opimizaion model for diribued communicaion nework. Working paper #3-97, Deparmen of indurial economic and echnology managemen, Norwegian Univeriy of Science and Technology, N Trondheim, Norway, [32] Trace of he Inerne Traffic Archive. hp://ia.ee.lbl.gov/ hml/race.hml. [33] H. Vladimirou, S.A. Zenio and R.J-B. We (edior). Model for planning under uncerainy. Annal of Operaion Reearch, Vol. 59, J.C. Balzer AG Scienific Publiher, [34]W. K. Haneveld and M. H. van der Vlerk. Sochaic ineger programming: general model and algorihm. Annal of Operaion Reearch (85), pp , [35] H. Zhang and E. W. Knighly. RED-VBR: A renegoiaionbaed approach o uppor delay-eniive VBR video. Mulimedia Syem, 5(3): , 1997.

2.4 Network flows. Many direct and indirect applications telecommunication transportation (public, freight, railway, air, ) logistics

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