Better Bounds for Online Load Balancing on Unrelated Machines


 Erica Golden
 3 years ago
 Views:
Transcription
1 Beer Bound for Online Load Balancing on Unrelaed Machine Ioanni Caragianni Abrac We udy he roblem of cheduling ermanen ob on unrelaed machine when he obecive i o minimize he L norm of he machine load. he roblem i known a load balancing under he L norm. We reen an imroved uer bound for he greedy algorihm hrough imle analyi; hi bound i alo hown o be be oible wihin he cla of deerminiic online algorihm for he roblem. We alo addre he queion wheher randomizaion hel online load balancing under L norm on unrelaed machine; hi i a challenging queion which i oen for more han a decade even for he L 2 norm. We rovide a oiive anwer o hi queion by reening he fir randomized online algorihm which ouerform deerminiic one under any inegral L norm for 2,..., 37. Our algorihm eenially comue in an online manner a fracional oluion o he roblem and ue he fracional value o make random choice. he local oimizaion crierion ued a each e i novel and raher counerinuiive: he value of he fracional variable for each ob correond o flow a an aroximae Wardro equilibrium for an aroriaely defined nonaomic congeion game. A corollarie of our analyi and by exloiing he relaion beween he L norm and he makean of machine load, we obain new comeiive algorihm for online makean minimizaion, making rogre in anoher longanding oen roblem. Inroducion We udy he following claical cheduling roblem. We have m arallel machine and n indeenden ob, where ob i induce a oibly infinie oiive inegral load w i when roceed by machine. he load of a machine i he oal load of he ob aigned o i. he co of an aignmen or chedule of a e of ob i defined a he L norm of he load vecor, / i.e., l m l for finie and l max m { l } for he load vecor l l,..., l m. he goal of a load balancing algorihm i o aign all ob Reearch Academic Comuer echnology Iniue & Dearmen of Comuer Engineering and Informaic, Univeriy of Para, Rio, Greece. E mail: hi work wa arially uored by he Euroean Union under IS FE Inegraed Proec FP AEOLUS and by a Caraheodory gran from he Univeriy of Para. o machine o ha he co i minimized. hi i he mo general verion of he roblem known a load balancing or cheduling on unrelaed arallel machine. Imoran ecial cae are hoe wih relaed machine and rericed aignmen. In he relaed machine model, each machine ha a oiive eed and each ob i ha a weigh w i and he load of ob i when i i aigned o machine i w i /. he rericed aignmen model i imilar; he main difference i ha each ob ha a ube of ermiible machine where i can be cheduled. he load vecor of ob i in he rericed aignmen model can be hough of a w i / for hoe machine which are ermiible for ob i, and for all oher machine. Secial cae where all machine have he ame eed idenical machine or all ob have he ame weigh are alo inereing. Almo all verion of he roblem are known o be NPhard [22]; hence, mo of he work ha focued on efficien aroximaion algorihm. A cenario which beer reflec racical iuaion i he one when he informaion abou he ob i no known in advance and i only revealed gradually. Comuaion i hen erformed in e; when a ob aear ogeher wih i load vecor, an online algorihm ha o make an irrevocable deciion and aign i o a machine. A naural online algorihm i he greedy algorihm for he L norm which aign each ob o ha machine ha minimize he increae of he h ower of he co. he erformance of online algorihm i aeed hrough he noion of comeiive raio defined a he maximum raio over all equence of ob of he execed co of he algorihm over he oimal co. hi definiion i quie general o caure randomized algorihm and aee he erformance of he online algorihm again obliviou adverarie, i.e., adverarie which may have acce o he robabiliy diribuion ued by he algorihm o make random choice bu no o he random choice hemelve. Scheduling o minimize he L norm of he load vecor alo called he makean ha received much aenion. Lenra e al. [29] and Shmoy and ardo [34] rovided 2aroximaion algorihm for unrelaed machine while he roblem ha been roved o be inaroximable wihin a raio beer han 3/2 [29].
2 Horowiz and Sahni [26] reened a olynomialime aroximaion cheme PAS when he number of machine i conan. For relaed machine, Hochbaum and Shmoy [25] ee alo [24] reened a PAS. In he online cae, Azar e al. [9] howed ha no randomized online algorihm can be beer han ln mcomeiive again obliviou adverarie even on he rericed aignmen model, where m i he number of machine. heir lower bound for deerminiic algorihm i log m. hey alo howed ha he greedy algorihm ha aign each ob o ha machine o ha he increae in he makean i minimized ha oimal comeiivene in he ame model. Ane e al. [3] oberved ha hi greedy algorihm i only Ωmcomeiive on m unrelaed machine. hey obained a deerminiic 4 log mcomeiive algorihm by combining everal ohiicaed echnique uch a exonenial weighing and doubling of eimae of he oimal makean. A randomized e log mcomeiive algorihm wa obained by inroducing randomized doubling o he algorihm of Ane e al.; hi reul i aribued o Indyk in [2]. he inereed reader may look in andard exbook on online comuaion uch a [3] for a coverage of hee reul and he relaed echnique. Conan comeiivene i oible for he relaed machine model [3, 2]. An imoran ecial cae i cheduling o minimize he makean on idenical machine. Graham [23] howed ha he greedy algorihm i 2 m comeiive in hi cae. he greedy algorihm i oimal only for m 3; for any m > 3, beer algorihm exi [2, 33]. Baral e al. [0] were he fir o how an algorihm whoe comeiive raio i below 2 δ for ome conan δ > 0 and arbirary m. See [] for he be known bound for he roblem. he L norm obecive ha been mainly inroduced ince, in many alicaion, he makean i no a uiable way o meaure how well he ob are balanced. Chandra and Wong [5] and Cody and Coffman [8] were he fir o conider he obecive of minimizing he um of quare of he machine load i.e., he quare of he L 2 norm. For unrelaed machine, Awerbuch e al. [6] howed ha he greedy algorihm ha comeiivene a mo + 2 under he L 2 norm and a mo c + Olog for higher norm, where c.7632 i he roo of he equaion c ln c. Furhermore, hey roved a lower bound of aroximaely for any deerminiic online algorihm. Among oher reul, Caragianni e al. [4] howed ha he uer bound of + 2 for he greedy algorihm under he L 2 norm i igh even for rericed aignmen. Beer comeiivene bound under he L 2 norm are known for rericed aignmen and ob wih equal weigh [4, 7, 35]. For he idenical machine cae, Avidor e al. [4] roved comeiivene bound of 4/3 under he L 2 norm and 2 O for higher norm. ln We oin ou ha he greedy algorihm rovided he be known aroximaion guaranee for he offline cae a well unil he recen work of Azar and Eein [7] who ued convex rogramming o comue fracional oluion o he roblem and rounding cheme o obain efficien inegral oluion. hee echnique yielded a randomized 2aroximaion algorihm for he L 2 norm and deerminiic 2aroximaion algorihm for higher norm. Beer aroximaion guaranee eecially for mall norm were reened in [28]. Prior o [7] and [28], 2aroximaion algorihm for all norm imulaneouly had been reened in [8] for rericed aignmen while, in he ame aer, he roblem wa roved o be APXhard for any L norm wih 2. Alon e al. [2] roved ha PASe are oible for rericed aignmen on idenical machine or for a conan number of unrelaed machine. Eein and Sgall [9] reened a PAS for relaed machine. Before reening our reul, we oen a arenhei o menion ha he recenly emerging field of Algorihmic Game heory [3] ha exenively conidered load balancing from a differen erecive. An aumion made here i ha he aignmen of ob i no cenrally conrolled bu, inead, each ob i owned by a elfih agen. Each agen favor aignmen of her ob o a machine o ha he laency he exerience i minimum given he aignmen of he ob conrolled by he oher agen. Following he eminal work of Kououia and Paadimiriou [27], a va amoun of he recen relaed lieraure udie how much he overall erformance of load balancing under everal obecive deeriorae a Nah equilibria, i.e., a aignmen where no agen ha an incenive o change he aignmen of her ob. Load balancing game are u ecial cae of aomic congeion game [5, 6] which model elfih behavior in nework rouing. Acually, load balancing can be hough of a rouing demand beween wo node hrough m arallel link. A lighly differen model which ha received much aenion recenly and whoe udy ha ared a lea half a cenury ago in he Economic and ranoraion lieraure ee [32] and he reference herein i ha of nonaomic congeion game in nework. We decribe he imle cae of uch game here; hee are he only gameheoreic definiion which we acually ue in he curren aer. In a nonaomic congeion game on m arallel link connecing a ource node o a deinaion node, here are infiniely many agen, each conrolling a negligible amoun of raffic flow from he ource o he deinaion o ha he oal amoun of raffic flow i uniy. Each link ha a nonnegaive laency funcion f which indi
3 cae ha he laency exerienced by all agen rouing heir raffic flow hrough i f x, where x i he oal amoun of flow roued hrough. Again, agen favor aignmen of heir flow o ha he laency exerienced i minimum given he aignmen of he oher agen. Aignmen from where no agen ha an incenive o deviae are called Wardro equilibria [36] and aify he following condiion: a flow vecor x i a a Wardro equilibrium if for any wo link, wih x > 0, i hold ha f x f x. Beckmann e al. [] have reened a convex oenial funcion whoe local minima correond o Wardro equilibria; hence, Wardro equilibria for nonaomic congeion game in nework can be arbirarily aroximaed uing convex rogramming echnique e.g., [30]. See alo [20] and he reference herein for recen diribued aroximaion of uch equilibria. Much eaier and faer aroximaion of Wardro equilibria are oible for game on arallel link rovided ha he laency funcion have ome reaonable form e.g., hey are olynomial. In hi aer, we udy he online oimizaion verion of load balancing in a coordinaed way. Alhough everal recen aer have conidered imoran ecial cae of online load balancing, no rogre on he general roblem wih he L norm obecive ha been made ince he work of Awerbuch e al. [6] in 995. We make uch a rogre and imrove mo of he reul of [6]. We renghen and ignificanly imlify he analyi of he greedy algorihm and how ha i i a mo 2 /  comeiive under he L norm. We alo how ha hi bound i igh for any, i.e., we how ha no deerminiic online algorihm can have beer comeiive ne han greedy. Our bound aroache ln from below a increae. A a corollary of our analyi and by exloiing he relaion of he L norm and he makean of machine load, we obain ha he greedy algorihm for he L ln m norm i e log mcomeiive for makean minimizaion on m unrelaed machine. hi i a raher urriing reul ince i mean ha he ame comeiivene wih he randomized verion of he algorihm of Ane e al. [3] can be obained by a much imler deerminiic greedylike algorihm. hee reul are reened in Secion 3. In Secion 4, we reen he mo inereing and echnically involved reul of he aer. We demonrae ha we can bea he lower bound on he comeiivene of deerminiic algorihm for load balancing under L norm by uing randomizaion. We reen a general algorihm called Balance which i arameerized by and a arameer vecor α. he algorihm i waching he momen of he random variable denoing he load on each machine and, a each e, i ue hi informaion ogeher wih he ob load in order o comue he robabiliy diribuion according o which i will make he deciion for he ob conidered. hi can be hough of a comuing a fracional oluion o he roblem which i rounded in a randomized way in order o obain an inegral aignmen. A each e, he crierion ued in order o comue he robabiliy diribuion of he random choice ugge a nice inerlay wih game. he algorihm conider a nonaomic congeion game on arallel link correonding o he machine wih aroriaely defined laency funcion, comue an aroximae Wardro equilibrium, and e he robabiliie of he random choice equal o he flow in hi equilibrium. Noe ha, unlike he aroach in Algorihmic Game heory menioned above, he game i ued a a ool by our algorihm in order o coordinae he aignmen of ob efficienly. Our analyi lead o ufficien condiion for he elecion of he arameer vecor α. Uing hee condiion, we are able o comue aroriae arameer o ha algorihm Balance i a mo.222comeiive again obliviou adverarie for inegral value of u o 37. For he L 2 norm, our algorihm i 5comeiive. By exloiing he relaion of L norm o he makean, we alo obain he fir comeiivene uer bound for makean minimizaion which bea he reul of Ane e al. [3] when he number of machine i u o an aronomically large conan i.e., e 37. We begin wih ome reliminary definiion in Secion 2 and conclude wih exenion and oen roblem in Secion 5. Due o lack of ace, ome roof have been omied. 2 Preliminarie We briefly reen he noaion ued in he analyi of our algorihm for he L norm. For an ineger k, we uually ue [k] o denoe he e {, 2,..., k}. When we conider an inance of he roblem, we denoe by l he load of machine in he oimal oluion. We ue he binary variable y i {0, } o denoe wheher ob i i aigned o machine in he oimal oluion y i or no y i 0. Clearly, i y iw i l and i y iwi i y iw i l for any. We alo denoe by Λ i he load of machine afer he aignmen of ob i. Hence, in order o rove uer bound on he comeiive [ raio, i uffice o bound he raio ] / / of IE Λ n or imly Λ n when he /. greedy algorihm i concerned over l In he analyi we ue he Minkowki inequaliy or he riangle inequaliy for he L norm aing ha k / k / + a + b / k a b
4 for any and a, b 0, a well a Hölder inequaliy aing ha IE[Z] IE[Z ] / for any nonnegaive random variable Z and ee wikiedia.org for a deailed reenaion of hee inequaliie. Our reul for makean minimizaion follow by he analyi of he algorihm for he L norm and he following obervaion. Lemma 2.. Le A be a ccomeiive online algorihm for load balancing on unrelaed machine under he L norm. hen, algorihm A i cm / comeiive for makean minimizaion on m unrelaed machine. 3 Deerminiic online algorihm In hi ecion we comue he exac comeiivene bound of he greedy algorihm for he L norm and we how ha hi i he be oible among deerminiic online algorihm. heorem 3.. For any, he greedy algorihm ha comeiive raio a mo for load balancing 2 / on unrelaed machine under he L norm. Proof. A he e aociaed wih ob i, we have Λ i 3. Λ i, Λi, + y i w i Λ i, Λn + y i w i Λ n where he fir inequaliy follow by he greedy naure of he algorihm and he econd inequaliy follow ince Λ n Λ i for any i, and he funcion fz z +a z i nondecreaing in [0, for a 0 and. Nex we will need he following echnical lemma. Lemma 3.. Le, 0 and a i 0, for i,..., k. hen, k k + a i + a i. Uing 3. and umming over all e of he algorihm, we obain ha 3.2 i i Λ n Λ i Λ i, Λn + y i w i Λ n Λn + y i w i Λ n i Λ n + i Λ n + l Λ n y i w i Λ n + Λ n l Λ n he econd inequaliy follow by Lemma 3. while he la inequaliy i derived by alying Minkowki inequaliy. We divide all ide of 3.2 by l and e c n Λ / l. Now 3.2 become 2c c+ which yield he deired reul c 2 /. Uing he inequaliy e z + z, we have ha 2 / ln hi imrove he reviou bound of Olog of [6]. We alo how ha greedy i oimal wihin he cla of deerminiic online algorihm for any L norm; hi reul alo imrove he reviouly be known lower bound of of [6]. heorem 3.2. For any ɛ > 0 and any, no deerminiic online load balancing algorihm on unrelaed machine can be beer han 2 / ɛcomeiive under he L norm. Proof. We ue a imilar conrucion wih [6] i.e., rericed aignmen on idenical machine bu we alo allow ob have differen load. hi alo generalize a conrucion ued in [4] o how a lower bound of + 2 on he comeiivene of he greedy algorihm under he L 2 norm. Le δ > 0 and ineger k be arameer o be defined laer and le be uch ha +δ /. Conider he execuion of a deerminiic online algorihm on m 2 k machine and an adverary ha reveal ob in k hae. Iniially, all 2 k machine are acive. In each hae, he adverary mache he acive machine ino air. In he hae i i 0,..., k, for each air of machine a, b, he adverary reen one ob wih load 2 i/ on machine a and b, and infinie load on any oher machine. he machine ha are aigned ob by he algorihm remain acive for he nex hae; machine ha are no aigned ob become inacive. Denoe by o and de he h ower of he co of he oimal aignmen and he co of he aignmen comued by he deerminiic algorihm, reecively. We oberve ha a machine ha become inacive immediaely afer hae i for i 0,..., k i no aigned any ob a hae i and laer hae while i i aigned one ob in each hae before hae i if
5 any. he machine ha i aigned he ob a hae k i alo aigned one ob in each hae before hae k. Since 2 k i machine become inacive immediaely afer hae i, we have ha 3.3 de k 2 / 0 k 2 k i k + 2 k i i 2 / 0 i 2 / 0 k 2 k i 2 i/ 2 / k 2 k 2 i/ 2 / k 2 k 2 i/ 2 / 2 k k 2 i/ 2 / 2 k k + 2 / 2 / 2 k k / he econd inequaliy follow ince a a for any a [0, ] and while he la inequaliy follow ince 2 / i increaing for and, hence, 2 / /2 which yield 2. 2 / In order o bound o from above, conider he aignmen in which each ob i aigned in ooiion o he algorihm. Given a ob wih finie load on a air of machine a, b, he ob i aigned o a if he algorihm aign i o b, and vice vera. hen, each machine ha a mo one ob and uing he relaion beween and δ, we obain k 3.4 o 2 k i 2 i/ i0 k 2 k i 2 i/ + i0 + δk2 k. By comaring 3.3 and 3.4, we obain ha for any ɛ > 0 here are ufficienly large k and ufficienly mall δ o ha he raio of he co of he aignmen comued by he deerminiic algorihm over he oi mal co i de o / 2 / ɛ. We remark ha no rericion on he number of machine aear in he aemen of heorem 3.2. Our lower bound conrucion eenially how ha when o log m here exi an inance wih m machine on which any deerminiic online algorihm ha comeiive raio arbirarily cloe o 2 /. By ighening he inequaliie ued in order o obain 3.3, we can how ha he ame hold when o. log m log log m he lower bound of heorem 3.2 definiely doe no hold for ωlog m ince, in hi cae, he Olog m comeiive algorihm for makean minimizaion can be eaily roved o be Olog mcomeiive under he L norm. We conclude hi ecion by reening a reul for makean minimizaion. I follow a a corollary of our analyi of he greedy algorihm heorem 3. and Lemma 2.. heorem 3.3. he greedy algorihm for he L ln m norm i e log mcomeiive for makean minimizaion, where m i he number of unrelaed machine. he greedy algorihm for he L ln m norm ha everal advanage over he e log mcomeiive randomized algorihm obained by he echnique of Ane e al. [3] exended wih randomized doubling. Fir, i i deerminiic. Second, i doe no ue exonenial weighing and, hence, olynomial ace i alway ufficien in order o erform comuaion. Acually, he greedy algorihm for he L ln m norm can be hough of a uing ubexonenial weighing. hird, i doe no ue eimae of he oimal makean or doubling. 4 Randomized online algorihm In hi ecion we reen he fir randomized online algorihm for load balancing on unrelaed machine ha bea he lower bound on he comeiivene of deerminiic one under L norm. Our algorihm i called Balance, ue a vecor α of oiive value, and work a follow. When a new ob i arrive, Balance comue a robabiliy diribuion on he machine, i.e., robabiliie x i ha ob i i aigned o machine. hen, i imly ca a die ha ha one face for each machine wih x i > 0 wih x i being he robabiliy ha he face correonding o machine i he oucome of he die caing and aign ob i o he machine correonding o he oucome of die caing. Moivaed by he greedy algorihm, a naive way o comue he robabiliie in each e could be o [ ] minimize he increae in IE Λ n IE[Λ n ] due o he deciion a he e. Unforunaely, i i no hard o lighly modify he roof of heorem 3.2 and conruc inance where uch an algorihm mimic he greedy algorihm and canno achieve a
6 beer comeiive raio. Inead, algorihm Balance i waching all he inegral momen of he machine load and, a each e, i make i deciion rying o balance he increae in each of hem. In order o do o, i define aroriae game a each e, comue equilibria for hem, and make i deciion according o hee equilibria. A each e i, he algorihm conider a aricular nonaomic congeion game wih a uni amoun of flow ha ha o be roued from a ource node o a deinaion node which are conneced wih m arallel link. Each link correond o machine of he load balancing inance and ha an aroriaely defined nondecreaing laency funcion f i. Algorihm Balance comue an ɛaroximae Wardro equilibrium wih ɛ α 2 m, i.e., a flow vecor x i uch ha for any wo link, [m] wih x i > 0, i hold ha f i x i f i x i ɛ. hi alo yield 4.5 x i f i x i ɛ f i x i for any link [m]. In order o define he laency funcion on he link, he algorihm kee rack of auxiliary funcion g i for i [n], [m], and 0,,..., defined a follow. A he fir e, he algorihm e g z zw i if > 0, and g 0 z for any [m] and z 0. For i >, during e i, he algorihm define he auxiliary funcion g i for [m] and 0,..., uing he auxiliary funcion defined in he reviou e, he vecor of robabiliie x i comued a he reviou e, and he load of ob i on he machine, a follow: g i z z 0 +g i, x i, g i, x i, w i Alhough comlicaed a fir glance, he auxiliary funcion have been defined in uch a way ha g i x i IE[Λ i ]. hi i aed in he following lemma ogeher wih oher roerie of he auxiliary funcion g i ha will be ueful in our analyi. Lemma 4.. For any i > 0, [m], and 0,...,, he following roerie hold:. g i z zg i g i 0+g i 0, for any z g i 0 g i 0w i. 3. g i x i IE[Λ i ] IE[Λ n ]. 4. g i z u g i 0 u uz g i z u g i g i 0, for any u and z 0. Proof. Proerie and 2 follow rivially by he definiion of g i. he inequaliy in Proery 3 i obviou. In order o rove he equaliy in Proery 3, we will ue inducion on i. For i, if 0, we have IE[Λ 0 ] g 0x by definiion. Alo, if > 0, hen Λ i w wih robabiliy x and 0 wih robabiliy x i, i.e., IE[Λ ] x w g x by definiion. Now aume ha IE[Λ i ] g i x i for any i < i and 0,...,. hen, he random variable Λ i equal Λ i, + w i wih robabiliy x i and Λ i, wih robabiliy x i. Hence, uing lineariy of execaion, he inducive hyohei, and he definiion of auxiliary funcion g i, we have [ ] IE[Λ i] IE x i Λ i, + w i + x i Λ i, [ ] IE x i Λ i,w i + Λ i, 0 x i 0 x i 0 +g i, x i, g i x i. IE[Λ i,]w i + IE[Λ i,] g i, x i, w i Proery 4 clear hold if z 0 or if g i i a conan funcion. In order o rove i for z > 0 and when i i no conan, conider he funcion hz z + c u for ome nonnegaive conan c. Since h i convex in [0, + for u, i derivaive a oin z 2 > 0 i no maller han he loe of he line croing oin 0, h0 and z 2, hz 2, i.e., hz 2 h0 z 2 uz 2 + c u which yield hz 2 h0 uz 2 + c u z 2. Proery 4 hen follow by eing z 2 zg i g i 0 and c g i 0 ince by Proery hz 2 zg i g i 0 + g i 0 g i z. he laency funcion aociaed wih link i defined a f i z α g i z / 0 g i zw i. hi comlee he decriion of he algorihm. So far, i hould have become clear ha he algorihm doe no ue he oucome of reviou random choice in order o make deciion a any e. Inead, i ue he momen of he random variable denoing he load on each machine which in urn deend only on he robabiliy diribuion of reviou random choice. hi nice roery make he analyi of he algorihm
7 racable. We are now ready o rove he following aemen which characerize he comeiivene of algorihm Balance in erm of and he arameer vecor α. Lemma 4.2. Le 2 be an ineger. If here exi oiive number ξ > 0 for 0,..., uch ha he value of vecor α aify α ξ 4.6 ξ 0 κ + α κ ξ κ κ ξ α κ+ for,...,, hen algorihm α, Balance i β/α / comeiive again obliviou adverarie for load balancing on unrelaed machine under he L norm, where β α ξ 0 ξ. Proof. We wih o how ha he execaion of / he random variable X Λ n i a mo β /. α l By Hölder inequaliy on execaion of nonnegaive random variable, we have ha IE[X] IE[X ] / and, hence, i uffice o how ha [ ] / IE Λ n β /. α l By lineariy of execaion, we have α IE Λ n α IE [ Λ ] 4.7 n α IE [ Λ ] / n α IE [ Λ /. n] We will work on he fir um a he righhand ide of 4.7. Uing he roerie of funcion g i, we obain 4.8 n n α IE [ Λ ] / n α IE [ Λ ] / [ ] i IE Λ / i, α g i x i / g i 0 /. Nex we ue a echnical lemma which follow by he definiion of funcion f i and he roerie of funcion g i. Lemma 4.3. A each e i [n] and for each machine [m], i hold ha α g i x i / g i 0 / x i f i x i x 2 iα 2. Proof. We ue he roerie, 2, and 4 of funcion g i from Lemma 4. o obain x i α g i x i / g i 0 / x i x i x i 0 α g ix i / g i g i 0 α g ix i / α g ix i / 0 0 α g ix i / x i f i x i x 2 i 0 x i f i x i x 2 i 0 g i x i g i 0 w i α g ix i / w i g i g i 0 u0 α g ix i / u x i f i x i x 2 iα 0 x i f i x i x 2 iα 2. g iu 0w u i g i0 0w i g i 0w i g i x i w i he la inequaliy follow ince g i0 0 and w i. Now, working wih 4.8 and uing Lemma 4.3 and he inequaliy x2 i /m ince x i, we obain 4.9 α IE [ Λ ] / n
8 n n x i f i x i α 2 x i f i x i α 2. m Since he flow vecor x i i an α 2 m aroximae Wardro equilibrium for he congeion game conidered a e i, we ue 4.5 and 4.9 o obain a relaion of he lefhand ide of 4.9 o he deciion in he oimal aignmen, i.e., α IE [ Λ / n] x 2 i n y i f i x i. By ubiuing f i, uing he roery g i x i IE[Λ n ] for any i [n], [m], and 0,...,, and ince n y iwi l for any [m] and [], we obain a more clear relaion o he oimal aignmen: 4.0 n 0 α IE [ Λ ] / n α g i x i y i w i n α 0 y i w i n 0 y i w i 0 α α g i x i / IE [ Λ ] / [ ] n IE Λ n IE [ Λ ] / [ ] n IE Λ n IE [ Λ ] / [ ] n IE Λ n l. In order o uerbound he righhand ide of 4.0, we will ue he following echnical lemma. Lemma 4.4. Le γ, δ be nonnegaive ineger uch ha γ + δ and ζ, ζ 2 > 0. hen, 4. z γ zδ 2z γ δ 3 γ ζγ ζ δ 2z + δ ζγ ζδ 2 z 2 for any z, z 2, z γ + δ ζ γ ζδ 2z 3 Given ζ and ζ 2, wha Lemma 4.4 eenially doe i o bound he lefhand ide of 4. wih a olynomial of he form c z + c 2z 2 + c 3z 3 o ha he bound i igh when z ζ and z 2 ζ 2. We aly Lemma 4.4 o each erm a he righhand ide of 4.0. In aricular, alying Lemma 4.4 wih z IE [ /, Λn] z 2 IE [ Λ n] /, z3 l, ζ ξ, ζ 2 ξ, γ, and δ, we obain IE [ Λ ] / [ ] n IE Λ n l and, hence, 4.0 yield α IE [ Λ ] / n 0 +ξ + ξ κ+ + α IE [ Λ n α ξ α κ κ α ξ 0 + ξ + ξ ξie [ Λ ] / n ξ ξ ξl IE [ Λ ] / n ξ ξie [ Λ ] / n ] / + ξ ξ 0 ξκ κ ξ α IE [ Λ ] / n + β ξ l l ξ l IE [ Λ ] / n where he la inequaliy follow by 4.6[ and by he ] definiion of β. Hence, 4.7 yield IE Λ n β α l and Lemma 4.2 follow. By alying Lemma 4.2, we can eaily obain he following reul for he L 2 norm. Corollary 4.. Algorihm α, 2Balance wih α α 2 i 5comeiive again obliviou adverarie for load balancing on unrelaed machine under he L 2 norm. Proof. Alying Lemma 4.2 wih ξ 0 ξ ξ 2 2, we have ha inequaliy 4.6 i aified and β 5. For larger norm, comuing he be oible value for vecor α o ha here exi oiive ξ ha
9 aify he condiion 4.6 of Lemma 4.2 and imulaneouly minimize β i no eay ince hee condiion are quie comlicaed. So, in order o comue good value for he arameer of algorihm Balance, we reaon a follow. We will denoe by ˆα he correonding arameer vecor. Fir, we normalize vecor ˆα by eing ˆα. We gue aroriae value ˆξ for ξ and require ha he condiion 4.6 of Lemma 4.2 are aified wih equaliy. hi yield a yem of linear equaion in which he ˆα for,..., are he unknown and which can be rereened a Aˆα b where A i an uer riangular marix wih A, ˆξ ˆξ 0 for,...,, A,κ 0 for,..., and κ,...,, and κ A,κ ˆξ κ κ ˆξ for,..., and κ +,...,, and b ˆξ for,...,. Of coure, we mu gue he ˆξ o ha hi yem of linear equaion yield oiive value for he arameer ˆα. he following lemma i roof i omied rovide a ufficien condiion for hi. Lemma 4.5. If ˆξ i increaing in, and ˆξ, hen he yem of linear equaion Aˆα b ha a unique oiive oluion. We have ued ˆξ and have olved he correonding yem of linear equaion for inegral value of u o 37 by imlemening a imle back ubiuion rouine in C. he limiaion on come from he range of number he double daa ye of C can rereen. Our reul ugge ha, for he aricular elecion of he arameer vecor ˆα, he comeiive raio of algorihm ˆα, Balance i alway beer han he lower bound for deerminiic online algorihm and a mo.222. A a corollary, uing Lemma 2., we obain imroved comeiivene for online makean minimizaion a well when he number of machine i u o an aronomically large conan. he nex aemen ummarize he dicuion of hi ecion. heorem 4.. Algorihm ˆα, Balance ha comeiive raio a mo.222 again obliviou adverarie for load balancing under he L norm for inegral 37. Algorihm ˆα, ln m Balance ha comeiive raio a mo.222e ln m log m again obliviou adverarie for makean minimizaion on m e 37 unrelaed machine. 5 Exenion and oen roblem Our load balancing algorihm for L norm can be adaed o work wih imilar comeiivene by making he analyi lighly more comlicaed in nework rouing when he obecive i o minimize he oal laency or he maximum laency er link and he delay funcion on he nework link are olynomial; he adaaion include aroximaing he Wardro equilibrium of more general nonaomic congeion game in nework. Concerning oen roblem, here are ill many. Fir, i would be inereing o heoreically rove uer bound on he comeiivene of algorihm Balance for any value of. We conecure ha hee bound will be only marginally larger han hoe comued in Secion 4. Furhermore, he queion abou he limi of randomizaion for online load balancing under he L norm i very inereing even for he L 2 norm. Finally, alhough we have made ome rogre on online makean minimizaion on unrelaed machine, he ga beween our uer bound and he lower bound of [9] remain. For hi roblem, i i even widely oen wheher randomizaion i really neceary in order o obain he be oible comeiivene. Our reul in Secion 4 ugge ha game equilibria can be ueful in online combinaorial oimizaion. We lan o inveigae hi relaion more yemaically in he fuure by conidering differen online roblem. Reference [] S. Alber. On randomized online cheduling. In Proceeding of he 34h Annual ACM Symoium on heory of Comuing SOC 02, , [2] N. Alon, Y. Azar, G. J. Woeginger and. Yadid. Aroximaion cheme for cheduling. In Proceeding of he 8h Annual ACMSIAM Symoium on Dicree Algorihm SODA 97, , 997. [3] J. Ane, Y. Azar, A. Fia, S. Plokin, and O. Waar. Online rouing of virual circui wih alicaion o load balancing and machine cheduling. Journal of he ACM, 443, , 997. [4] A. Avidor, Y. Azar, and J. Sgall. Ancien and new algorihm for load balancing in he L norm. Algorihmica, 29, , 200. [5] B. Awerbuch, Y. Azar, and A. Eein. he rice of rouing unliable flow. In Proceeding of he 37h Annual ACM Symoium on heory of Comuing SOC 05, , [6] B. Awerbuch, Y. Azar, E. F. Grove, M.Y. Kao, P. Krihnan, and J. S. Vier. Load balancing in he L norm. In Proceeding of he 36h Annual IEEE Symoium on Foundaion of Comuer Science FOCS 95, , 995. [7] Y. Azar and A. Eein. Convex rogramming for cheduling unrelaed arallel machine. In Proceeding
10 of he 37h Annual ACM Symoium on heory of Comuing SOC 05, , [8] Y. Azar, L. Eein, Y. Richer, and G. Woeginger. Allnorm aroximaion algorihm. Journal of Algorihm, 522, , [9] Y. Azar, J. Naor, and R. Rom. he comeiivene of online aignmen. Journal of Algorihm, 82, , 995. [0] Y. Baral, A. Fia, H. J. Karloff and R. Vohra. New algorihm for an ancien cheduling roblem. Journal of Comuer and Syem Science, 53, , 995. [] M. Beckmann, C. B. McGuire, and C. B. Winen. Sudie in he Economic of ranoraion, Yale Univeriy Pre, 956. [2] P. Berman, M. Charikar, and M. Karinki. Online load balancing for relaed machine. Journal of Algorihm, 35,. 082, [3] A. Borodin and R. ElYaniv. Online comuaion and comeiive analyi. Cambridge Univeriy Pre, 998. [4] I. Caragianni, M. Flammini, C. Kaklamani, P. Kanellooulo, and L. Mocardelli. igh bound for elfih and greedy load balancing. In Proceeding of he 33rd Inernaional Colloquium on Auomaa, Language, and Programming ICALP 06, LNCS 405, Sringer, Par I, , [5] A. K. Chandra and C. K. Wong. Worcae analyi of a lacemen algorihm relaed o orage allocaion. SIAM Journal on Comuing, 43, , 975. [6] G. Chriodoulou and E. Kououia. he rice of anarchy of finie congeion game. In Proceeding of he 37h Annual ACM Symoium on heory of Comuing SOC 05, , [7] G. Chriodoulou, V. S. Mirrokni, and A. Sidirooulo. Convergence and aroximaion in oenial game. In Proceeding of he 23rd Annual Symoium on heoreical Aec of Comuer Science SACS 06, LNCS 3884, Sringer, , [8] R. A. Cody and E. G. Coffman, Jr. Record allocaion for minimizing execed rerieval co on crumlike orage device. Journal of he ACM, 23,. 035, 976. [9] L. Eein and J. Sgall. Aroximaion cheme for cheduling on uniformly relaed and idenical arallel machine. Algorihmica, 39, , [20] S. Ficher, H. Räcke, and B. Vöcking. Fa convergence o Wardro equilibria by adaive amling mehod. In Proceeding of he 38h Annual ACM Symoium on heory of Comuing SOC 06, , [2] G. Galambo and G. Woeginger. An online cheduling heuriic wih beer wor cae raio han graham li cheduling. SIAM Journal on Comuing, 222, , 993. [22] M. R. Garey and D. S. Johnon. Comuer and Inracabiliy, W. H. Freeman and Comany, 979. [23] R. L. Graham. Bound for cerain muliroceor anomalie. Bell Syem echnical Journal, 45, , 966. [24] D. S. Hochbaum and D. B. Shmoy. Uing dual aroximaion algorihm for cheduling roblem heoreical and racical reul. Journal of he ACM, 34, , 987. [25] D. S. Hochbaum and D. B. Shmoy. A olynomial aroximaion cheme for cheduling on uniform roceor: uing he dual aroximaion aroach. SIAM Journal on Comuing, 73, , 988. [26] E. Horowiz and S. Sahni. Exac and aroximae algorihm for cheduling nonidenical roceor. Journal of he ACM, 23, , 976. [27] E. Kououia and C. Paadimiriou. Worcae equilibria. In Proceeding of he 6h Inernaional Symoium on heoreical Aec of Comuer Science SACS 99, LNCS 563, Sringer, , 999. [28] V. S. Anil Kumar, M. V. Marahe, S. Parhaarahy, and A. Srinivaan. Aroximaion algorihm for cheduling on mulile machine. In Proceeding of he 46h Annual IEEE Symoium on Foundaion of Comuer Science FOCS 05, , [29] J. K. Lenra, D. B. Shmoy, and E. ardo. Aroximaion algorihm for cheduling unrelaed arallel machine. Mahemaical Programming, 46, , 990. [30] Y. Neerov and A. Nemirokii. Inerioroin olynomial algorihm in convex rogramming. SIAM Sudie in Alied Mahemaic, SIAM, 994. [3] N. Nian,. Roughgarden, E. ardo, and V. V. Vazirani. Algorihmic game heory, Cambridge Univeriy Pre, 2007, o aear. [32]. Roughgarden and E. ardo. How bad i elfih rouing? Journal of he ACM, 492: , [33] S. S. Seiden. Online randomized muliroceor cheduling. Algorihmica, 282, 7326, [34] D. B. Shmoy and E. ardo. An aroximaion algorihm for he generalized aignmen roblem. Mahemaical Programming, 62, , 993. [35] S. Suri, C. óh and Y. Zhou. Selfih load balancing and aomic congeion game. Algorihmica, 47, , [36] J. G. Wardro. Some heoreical aec of road raffic reearch. In Proceeding of he Iniue of Civil Engineer, P. II, , 952.
How Much Can Taxes Help Selfish Routing?
How Much Can Taxe Help Selfih Rouing? Tim Roughgarden (Cornell) Join wih Richard Cole (NYU) and Yevgeniy Dodi (NYU) Selfih Rouing a direced graph G = (V,E) a ource and a deinaion one uni of raffic from
More informationTopic: Applications of Network Flow Date: 9/14/2007
CS787: Advanced Algorihm Scribe: Daniel Wong and Priyananda Shenoy Lecurer: Shuchi Chawla Topic: Applicaion of Nework Flow Dae: 9/4/2007 5. Inroducion and Recap In he la lecure, we analyzed he problem
More information2.4 Network flows. Many direct and indirect applications telecommunication transportation (public, freight, railway, air, ) logistics
.4 Nework flow Problem involving he diribuion of a given produc (e.g., waer, ga, daa, ) from a e of producion locaion o a e of uer o a o opimize a given objecive funcion (e.g., amoun of produc, co,...).
More informationChapter 13. Network Flow III Applications. 13.1 Edge disjoint paths. 13.1.1 Edgedisjoint paths in a directed graphs
Chaper 13 Nework Flow III Applicaion CS 573: Algorihm, Fall 014 Ocober 9, 014 13.1 Edge dijoin pah 13.1.1 Edgedijoin pah in a direced graph 13.1.1.1 Edge dijoin pah queiong: graph (dir/undir)., : verice.
More information6.003 Homework #4 Solutions
6.3 Homewk #4 Soluion Problem. Laplace Tranfm Deermine he Laplace ranfm (including he region of convergence) of each of he following ignal: a. x () = e 2(3) u( 3) X = e 3 2 ROC: Re() > 2 X () = x ()e d
More informationOPTIMIZING PRODUCTION POLICIES FOR FLEXIBLE MANUFACTURING SYSTEM WITH NONLINEAR HOLDING COST
OPIMIZING PRODUCION POLICIE FOR FLEXIBLE MANUFACURING YEM WIH NONLINEAR HOLDING CO ABRAC Leena Praher, Reearch cholar, Banahali Vidayaeeh (Raj.) Dr. hivraj Pundir, Reader, D. N. College, Meeru (UP) hi
More informationCircle Geometry (Part 3)
Eam aer 3 ircle Geomery (ar 3) emen andard:.4.(c) yclic uadrilaeral La week we covered u otheorem 3, he idea of a convere and we alied our heory o ome roblem called IE. Okay, o now ono he ne chunk of heory
More information#A81 INTEGERS 13 (2013) THE AVERAGE LARGEST PRIME FACTOR
#A8 INTEGERS 3 (03) THE AVERAGE LARGEST PRIME FACTOR Eric Naslund Dearmen of Mahemaics, Princeon Universiy, Princeon, New Jersey naslund@mahrinceonedu Received: /8/3, Revised: 7/7/3, Acceed:/5/3, Published:
More informationTHE PRESSURE DERIVATIVE
Tom Aage Jelmer NTNU Dearmen of Peroleum Engineering and Alied Geohysics THE PRESSURE DERIVATIVE The ressure derivaive has imoran diagnosic roeries. I is also imoran for making ye curve analysis more reliable.
More informationOn the Connection Between MultipleUnicast Network Coding and SingleSource SingleSink Network Error Correction
On he Connecion Beween MulipleUnica ework Coding and SingleSource SingleSink ework Error Correcion Jörg Kliewer JIT Join work wih Wenao Huang and Michael Langberg ework Error Correcion Problem: Adverary
More informationCommon Virtual Path and Its Expedience for VBR Video Traffic
RADIOENGINEERING, VOL. 7, NO., APRIL 28 73 Coon Virual Pah and I Exedience for VBR Video Traffic Erik CHROMÝ, Ivan BAROŇÁK De. of Telecounicaion, Faculy of Elecrical Engineering and Inforaion Technology
More informationAn Approach for Project Scheduling Using PERT/CPM and Petri Nets (PNs) Tools
Inernaional Journal of Modern Engineering Research (IJMER) Vol., Issue. 5, Se  Oc. 222 ISSN: 2295 n roach for Projec Scheduling Using PERT/CPM and Peri Nes (PNs) Tools mer. M. oushaala (Dearmen of
More informationFortified financial forecasting models: nonlinear searching approaches
0 Inernaional Conference on Economic and inance Reearch IPEDR vol.4 (0 (0 IACSIT Pre, Singapore orified financial forecaing model: nonlinear earching approache Mohammad R. Hamidizadeh, Ph.D. Profeor,
More informationCHAPTER 11 NONPARAMETRIC REGRESSION WITH COMPLEX SURVEY DATA. R. L. Chambers Department of Social Statistics University of Southampton
CHAPTER 11 NONPARAMETRIC REGRESSION WITH COMPLEX SURVEY DATA R. L. Chamber Deparmen of Social Saiic Univeriy of Souhampon A.H. Dorfman Office of Survey Mehod Reearch Bureau of Labor Saiic M.Yu. Sverchkov
More informationHow has globalisation affected inflation dynamics in the United Kingdom?
292 Quarerly Bullein 2008 Q3 How ha globaliaion affeced inflaion dynamic in he Unied Kingdom? By Jennifer Greenlade and Sephen Millard of he Bank Srucural Economic Analyi Diviion and Chri Peacock of he
More informationLecture 2: Telegrapher Equations For Transmission Lines. Power Flow.
Whies, EE 481 Lecure 2 Page 1 of 13 Lecure 2: Telegraher Equaions For Transmission Lines. Power Flow. Microsri is one mehod for making elecrical connecions in a microwae circui. I is consruced wih a ground
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationWHAT ARE OPTION CONTRACTS?
WHAT ARE OTION CONTRACTS? By rof. Ashok anekar An oion conrac is a derivaive which gives he righ o he holder of he conrac o do 'Somehing' bu wihou he obligaion o do ha 'Somehing'. The 'Somehing' can be
More informationThe Chase Problem (Part 2) David C. Arney
The Chae Problem Par David C. Arne Inroducion In he previou ecion, eniled The Chae Problem Par, we dicued a dicree model for a chaing cenario where one hing chae anoher. Some of he applicaion of hi kind
More informationSAMPLE LESSON PLAN with Commentary from ReadingQuest.org
Lesson Plan: Energy Resources ubject: Earth cience Grade: 9 Purpose: students will learn about the energy resources, explore the differences between renewable and nonrenewable resources, evaluate the environmental
More informationON A FAIR VALUE MODEL FOR PARTICIPATING LIFE INSURANCE POLICIES
Invemen Managemen and Financial Innovaion, Volume 3, Iue 2, 2006 05 ON A FAIR VALUE MODEL FOR PARTICIPATING LIFE INSURANCE POLICIES Fabio Baione, Paolo De Angeli, Andrea Forunai Abrac The aim of hi aer
More informationANALYTIC PROOF OF THE PRIME NUMBER THEOREM
ANALYTIC PROOF OF THE PRIME NUMBER THEOREM RYAN SMITH, YUAN TIAN Conens Arihmeical Funcions Equivalen Forms of he Prime Number Theorem 3 3 The Relaionshi Beween Two Asymoic Relaions 6 4 Dirichle Series
More informationPhysical Topology Discovery for Large MultiSubnet Networks
Phyical Topology Dicovery for Large MuliSubne Nework Yigal Bejerano, Yuri Breibar, Mino Garofalaki, Rajeev Raogi Bell Lab, Lucen Technologie 600 Mounain Ave., Murray Hill, NJ 07974. {bej,mino,raogi}@reearch.belllab.com
More informationSensor Network with Multiple Mobile Access Points
Sensor Newor wih Mulile Mobile Access Poins Parvahinahan Veniasubramaniam, Qing Zhao and Lang Tong School of Elecrical and Comuer Engineering Cornell Universiy, Ihaca, NY 4853, USA Email: v45@cornell.edu,{qzhao,long}@ece.cornell.edu
More informationSinglemachine Scheduling with Periodic Maintenance and both Preemptive and. Nonpreemptive jobs in Remanufacturing System 1
Absrac number: 050407 Singlemachine Scheduling wih Periodic Mainenance and boh Preempive and Nonpreempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationState Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University
Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween
More informationRobust Bandwidth Allocation Strategies
Robu Bandwidh Allocaion Sraegie Oliver Heckmann, Jen Schmi, Ralf Seinmez Mulimedia Communicaion Lab (KOM), Darmad Univeriy of Technology Merckr. 25 D64283 Darmad Germany {Heckmann, Schmi, Seinmez}@kom.udarmad.de
More informationCALCULATION OF OMX TALLINN
CALCULATION OF OMX TALLINN CALCULATION OF OMX TALLINN 1. OMX Tallinn index...3 2. Terms in use...3 3. Comuaion rules of OMX Tallinn...3 3.1. Oening, realime and closing value of he Index...3 3.2. Index
More informationHeat demand forecasting for concrete district heating system
Hea demand forecaing for concree diric heaing yem Bronilav Chramcov Abrac Thi paper preen he reul of an inveigaion of a model for horerm hea demand forecaing. Foreca of hi hea demand coure i ignifican
More informationFormulating CyberSecurity as Convex Optimization Problems Æ
Formulaing CyberSecuriy a Convex Opimizaion Problem Æ Kyriako G. Vamvoudaki,João P. Hepanha, Richard A. Kemmerer 2, and Giovanni Vigna 2 Cener for Conrol, Dynamicalyem and Compuaion (CCDC), Univeriy
More informationOptimal Path Routing in Single and Multiple Clock Domain Systems
IEEE TRANSACTIONS ON COMPUTERAIDED DESIGN, TO APPEAR. 1 Opimal Pah Rouing in Single and Muliple Clock Domain Syem Soha Haoun, Senior Member, IEEE, Charle J. Alper, Senior Member, IEEE ) Abrac Shrinking
More informationFormulating CyberSecurity as Convex Optimization Problems
Formulaing CyberSecuriy a Convex Opimizaion Problem Kyriako G. Vamvoudaki, João P. Hepanha, Richard A. Kemmerer, and Giovanni Vigna Univeriy of California, Sana Barbara Abrac. Miioncenric cyberecuriy
More informationAn Optimized Resolution for Software Project Planning with Improved Max Min Ant System Algorithm
Inernaional Journal of Muliedia and Ubiquiou Engineering Vol No6 (5) 538 h://dxdoiorg/457/iue564 An Oiized Reoluion for Sofware Proec Planning wih Iroved Max Min An Sye Algorih WanJiang Han HeYang Jiang
More informationTrading Strategies for Sliding, Rollinghorizon, and Consol Bonds
Trading Sraegie for Sliding, Rollinghorizon, and Conol Bond MAREK RUTKOWSKI Iniue of Mahemaic, Poliechnika Warzawka, 661 Warzawa, Poland Abrac The ime evoluion of a liding bond i udied in dicree and
More informationAnalysis of the development trend of China s business administration based on time series
SHS Web of Conference 4, 000 7 (06) DOI: 0.05/ hconf/0640007 C Owned by he auhor, ublihed by EDP Science, 06 Analyi of he develomen rend of China buine adminiraion baed on ime erie Rui Jiang Buine School,
More informationDIFFERENTIAL EQUATIONS with TI89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations
DIFFERENTIAL EQUATIONS wih TI89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationCrosssectional and longitudinal weighting in a rotational household panel: applications to EUSILC. Vijay Verma, Gianni Betti, Giulio Ghellini
Croecional and longiudinal eighing in a roaional houehold panel: applicaion o EUSILC Viay Verma, Gianni Bei, Giulio Ghellini Working Paper n. 67, December 006 CROSSSECTIONAL AND LONGITUDINAL WEIGHTING
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationEquity Valuation Using Multiples. Jing Liu. Anderson Graduate School of Management. University of California at Los Angeles (310) 2065861
Equiy Valuaion Uing Muliple Jing Liu Anderon Graduae School of Managemen Univeriy of California a Lo Angele (310) 2065861 jing.liu@anderon.ucla.edu Doron Niim Columbia Univeriy Graduae School of Buine
More informationTight Bounds for Selfish and Greedy Load Balancing
Tight Bounds for Selfish and Greedy Load Balancing Ioannis Caragiannis 1, Michele Flammini, Christos Kaklamanis 1, Panagiotis Kanellopoulos 1, and Luca Moscardelli 1 Research Academic Computer Technology
More informationFourier series. Learning outcomes
Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Halfrange series 6. The complex form 7. Applicaion of Fourier series
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationMath 201 Lecture 12: CauchyEuler Equations
Mah 20 Lecure 2: CauchyEuler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem
More informationYTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.
. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationExplore the Application of Financial Engineering in the Management of Exchange Rate Risk
SHS Web o Conerence 17, 01006 (015) DOI: 10.1051/ hcon/01517 01006 C Owned by he auhor, publihed by EDP Science, 015 Explore he Applicaion o Financial Engineering in he Managemen o Exchange Rae Rik Liu
More informationFourier Series Solution of the Heat Equation
Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,
More informationCircuit Types. () i( t) ( )
Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All
More informationNew Evidence on Mutual Fund Performance: A Comparison of Alternative Bootstrap Methods. David Blake* Tristan Caulfield** Christos Ioannidis*** and
New Evidence on Muual Fund Performance: A Comparion of Alernaive Boorap Mehod David Blake* Trian Caulfield** Chrio Ioannidi*** and Ian Tonk**** June 2014 Abrac Thi paper compare he wo boorap mehod of Koowki
More informationDifferential Equations and Linear Superposition
Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y
More informationA Reexamination of the Joint Mortality Functions
Norh merican cuarial Journal Volume 6, Number 1, p.166170 (2002) Reeaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali
More informationA Comparative Study of Linear and Nonlinear Models for Aggregate Retail Sales Forecasting
A Comparaive Sudy of Linear and Nonlinear Model for Aggregae Reail Sale Forecaing G. Peer Zhang Deparmen of Managemen Georgia Sae Univeriy Alana GA 30066 (404) 6514065 Abrac: The purpoe of hi paper i
More informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationWhy Do Real and Nominal. InventorySales Ratios Have Different Trends?
Why Do Real and Nominal InvenorySales Raios Have Differen Trends? By Valerie A. Ramey Professor of Economics Deparmen of Economics Universiy of California, San Diego and Research Associae Naional Bureau
More information1 HALFLIFE EQUATIONS
R.L. Hanna Page HALFLIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of halflives, and / log / o calculae he age (# ears): age (halflife)
More informationOptimal RealTime Scheduling for Hybrid Energy Storage Systems and Wind Farms Based on Model Predictive Control
Energies 2015, 8, 80208051; doi:10.3390/en8088020 Aricle OPEN ACCESS energies ISSN 19961073 www.mdi.com/journal/energies Oimal RealTime Scheduling for Hybrid Energy Sorage Sysems and Wind Farms Based
More information3 RungeKutta Methods
3 RungeKua Mehods In conras o he mulisep mehods of he previous secion, RungeKua mehods are singlesep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he
More informationIlliquidity and Pricing Biases in the Real Estate Market
Illiquidiy and ricing Biases in he Real Esae arke Zhenguo Lin Fannie ae 39 Wisconsin Avenue Washingon DC 16 Kerry D. Vandell School of Business Universiy of Wisconsin adison 975 Universiy Avenue adison,
More informationLongevity 11 Lyon 79 September 2015
Longeviy 11 Lyon 79 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univlyon1.fr
More informationReputation and Social Network Analysis in MultiAgent Systems
Repuaion and Social Neork Analyi in MuliAgen Syem Jordi Sabaer IIIA  Arificial Inelligence Reearch Iniue CSIC  Spanih Scienific Reearch Council Bellaerra, Caalonia, Spain jabaer@iiia.cic.e Carle Sierra
More informationUsefulness of the Forward Curve in Forecasting Oil Prices
Usefulness of he Forward Curve in Forecasing Oil Prices Akira Yanagisawa Leader Energy Demand, Supply and Forecas Analysis Group The Energy Daa and Modelling Cener Summary When people analyse oil prices,
More informationOPTIMAL BATCH QUANTITY MODELS FOR A LEAN PRODUCTION SYSTEM WITH REWORK AND SCRAP. A Thesis
OTIMAL BATH UANTITY MOELS FOR A LEAN ROUTION SYSTEM WITH REWORK AN SRA A Thei Submied o he Graduae Faculy of he Louiiana Sae Univeriy and Agriculural and Mechanical ollege in parial fulfillmen of he requiremen
More informationRepresenting Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationChapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE.
Chaper 6 Firs Order PDEs 6.1 Characerisics 6.1.1 The Simples Case Suppose u(, ) saisfies he PDE where b, c are consan. au + bu = 0 If a = 0, he PDE is rivial (i says ha u = 0 and so u = f(). If a = 0,
More information2.5 Life tables, force of mortality and standard life insurance products
Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationSubsistence Consumption and Rising Saving Rate
Subience Conumpion and Riing Saving Rae Kenneh S. Lin a, HiuYun Lee b * a Deparmen of Economic, Naional Taiwan Univeriy, Taipei, 00, Taiwan. b Deparmen of Economic, Naional Chung Cheng Univeriy, ChiaYi,
More informationAPPLICATION OF QMEASURE IN A REAL TIME FUZZY SYSTEM FOR MANAGING FINANCIAL ASSETS
Inernaional Journal on Sof Comuing (IJSC) Vol.3, No.4, November 202 APPLICATION OF QMEASURE IN A REAL TIME FUZZY SYSTEM FOR MANAGING FINANCIAL ASSETS Penka Georgieva and Ivan Pochev 2 Burgas Free Universiy,
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationInfrastructure and Evolution in Division of Labour
Infrarucure and Evoluion in Diviion of Labour Mei Wen Monah Univery (Thi paper ha been publihed in RDE. (), 906) April 997 Abrac Thi paper udie he relaionhip beween infrarucure ependure and endogenou
More informationLectures # 5 and 6: The Prime Number Theorem.
Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζfuncion o skech an argumen which would give an acual formula for π( and sugges
More informationMultiresource Allocation Scheduling in Dynamic Environments
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol. 00, No. 0, Xxxxx 0000, pp. 000 000 in 15234614 ein 15265498 00 0000 0001 INFORMS doi 10.1287/xxxx.0000.0000 c 0000 INFORMS Mulireource Allocaion Scheduling
More information1 The basic circulation problem
2WO08: Graphs and Algorihms Lecure 4 Dae: 26/2/2012 Insrucor: Nikhil Bansal The Circulaion Problem Scribe: Tom Slenders 1 The basic circulaion problem We will consider he maxflow problem again, bu his
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationINVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS
INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,
More informationAn empirical analysis about forecasting Tmall airconditioning sales using time series model Yan Xia
An empirical analysis abou forecasing Tmall aircondiioning sales using ime series model Yan Xia Deparmen of Mahemaics, Ocean Universiy of China, China Absrac Time series model is a hospo in he research
More informationRC, RL and RLC circuits
Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.
More informationSc i e n c e a n d t e a c h i n g:
Dikuionpapierreihe Working Paper Serie Sc i e n c e a n d e a c h i n g: Tw o d i m e n i o n a l i g n a l l i n g in he academic job marke Andrea Schneider Nr./ No. 95 Augu 2009 Deparmen of Economic
More information4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay
324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find
More informationNanocubes for RealTime Exploration of Spatiotemporal Datasets
Nanocube for RealTime Exploraion of Spaioemporal Daae Lauro Lin, Jame T Kloowki, and arlo Scheidegger Fig 1 Example viualizaion of 210 million public geolocaed Twier po over he coure of a year The daa
More informationGraphing the Von Bertalanffy Growth Equation
file: d:\b1732013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationThe Experts In Actuarial Career Advancement. Product Preview. For More Information: email Support@ActexMadRiver.com or call 1(800) 2822839
P U B L I C A T I O N S The Eers In Acuarial Career Advancemen Produc Preview For More Informaion: email Suor@AceMadRiver.com or call (8) 8839 Preface P Conens Preface P7 Syllabus Reference P Flow
More informationand Decay Functions f (t) = C(1± r) t / K, for t 0, where
MATH 116 Exponenial Growh and Decay Funcions Dr. Neal, Fall 2008 A funcion ha grows or decays exponenially has he form f () = C(1± r) / K, for 0, where C is he iniial amoun a ime 0: f (0) = C r is he rae
More informationA NOTE ON THE ALMOST EVERYWHERE CONVERGENCE OF ALTERNATING SEQUENCES WITH DUNFORD SCHWARTZ OPERATORS
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXII 1991 FASC. I A OTE O THE ALMOST EVERYWHERE COVERGECE OF ALTERATIG SEQUECES WITH DUFORD SCHWARTZ OPERATORS BY RYOTARO S A T O (OKAYAMA) 1. Inroducion.
More informationModule 3 Design for Strength. Version 2 ME, IIT Kharagpur
Module 3 Design for Srengh Lesson 2 Sress Concenraion Insrucional Objecives A he end of his lesson, he sudens should be able o undersand Sress concenraion and he facors responsible. Deerminaion of sress
More informationData Analysis Toolkit #7: Hypothesis testing, significance, and power Page 1
Daa Analyi Toolki #7: Hypohei eing, ignificance, and power Page 1 The ame baic logic underlie all aiical hypohei eing. Thi oolki illurae he baic concep uing he mo common e,  e for difference beween mean.
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationA Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)
A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke
More informationSTABILITY OF LOAD BALANCING ALGORITHMS IN DYNAMIC ADVERSARIAL SYSTEMS
STABILITY OF LOAD BALANCING ALGORITHMS IN DYNAMIC ADVERSARIAL SYSTEMS ELLIOT ANSHELEVICH, DAVID KEMPE, AND JON KLEINBERG Absrac. In he dynamic load balancing problem, we seek o keep he job load roughly
More informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prinou should hae 1 quesions. Muliplechoice quesions may coninue on he ne column or page find all choices before making your selecion. The
More informationCHARGE AND DISCHARGE OF A CAPACITOR
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
More informationA closer look at Black Scholes option thetas
J Econ Finan (2008) 32:59 74 DOI 0.007/s29700790008 A closer look a Black Scholes oion heas Douglas R. Emery & Weiyu Guo & Tie Su Published online: Ocober 2007 # Sringer Science & Business Media, LLC
More informationRelative velocity in one dimension
Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies
More informationWeek #9  The Integral Section 5.1
Week #9  The Inegral Secion 5.1 From Calculus, Single Variable by HughesHalle, Gleason, McCallum e. al. Copyrigh 005 by John Wiley & Sons, Inc. This maerial is used by permission of John Wiley & Sons,
More information