Optimal Response to Burstable Billing under Demand Uncertainty

Transcription

1 1 Opimal Response o Bursable Billing under Demand Uncerainy Yong Zhan, Suden Member, IEEE, Mahdi Ghamkhari, Suden Member, IEEE, Hossein Akhavan-Hejazi, Suden Member, IEEE, Du Xu, Member, IEEE, and Hamed Mohsenian-Rad, Senior Member, IEEE arxiv: v1 [cs.ni] 18 Mar 2016 Absrac Bursable billing is widely adoped in pracice, e.g., by colocaion daa cener providers, o charge for heir users, e.g. daa ceners, for ransferring daa. However, here is sill a lack of research on wha he bes way is for a user o manage is workload in response o bursable billing. To overcome his shorcoming, we propose a novel mehod o opimally respond o bursable billing under demand uncerainy. Firs, we develop a racable mahemaical expression o calculae he 95h percenile usage of a user, who is charged by a provider via bursable billing for bandwidh usage. This model is hen used o formulae a new bandwidh allocaion problem o imize he user s surplus, i.e., is ne uiliy minus cos. Addiionally, we examine differen non-convex soluion mehods for he formulaed sochasic opimizaion problem. We also exend our design o he case where a user can receive service from muliple providers, who all employ bursable billing. Using real-world workload races, we show ha our proposed mehod can reduce user s bandwidh cos by 26% and increase is oal surplus by 23%, compared o he curren pracice of allocaing bandwidh on-demand. Index Terms Bursable billing, bandwidh, demand uncerainy, nonlinear mixed-ineger programming, surplus imizaion. 1 INTRODUCTION BURSTABLE billing, is a smar daa pricing (SDP) mehod ha is used in pracice, e.g., by Inerne service providers, o charge for ransferring daa [1], [2], [3], [4]. Recenly, bursable billing is also widely adoped by Colocaion Daa Cener (CDC) providers, e.g., Creaive Daa Conceps [5], NeSource Communicaions [6] and Co-Locaion.com [7], as a means o charge heir users for bandwidh usage. According o Colocaion America, bandwidh billing has become he second larges aspec of CDC users overall coss, second o energy billing [8]. Under bursable billing, he provider, who provides is users wih links for daa ransferring, will measure Y. Zhan and D. Xu are wih he Key Laboraory of Opical Fiber Sensing and Communicaions, Universiy of Elecronic Science and Technology of China, Chengdu, China. yzhan.china, xudu.uesc}@gmail.com. M. Ghamkhari, H. Akhavan-Hejazi and H. Mohsenian-Rad are wih he Deparmen of Elecrical Engineering, Universiy of California, Riverside, CA, USA. ghamkhari, shejazi, hamed}@ece.ucr.edu. This work was done when Y. Zhan was a Visiing Suden a he Smar Grid Research Lab, Universiy of California a Riverside. The corresponding auhor is H. Mohsenian-Rad. Cliens Workloads $ Our Focus Daa Cener (User) Bandwidh $ Via Bursable Billing CDC Provider Fig. 1. An example seup for he applicaion of bursable billing: a daa cener, i.e., user of a CDC provider, uilizes bandwidh provided by he CDC provider o serve ouside cliens wih uncerain demands. each of is user s usage of bandwidh based on he user s peak usage a a cerain percenile, ofen a he 95h percenile usage. By consrucion, bursable billing neglecs he user s usage of bandwidh during any ime oher han period of peak use. Hence, bursable billing allows users o exceed heir usage hresholds for a shor period wihou facing financial penaly [3]. In general, bursable billing can be sudied from wo differen viewpoins: provider s and user s. For sudies ha address bursable billing from he provider s viewpoin [2], [3], [9], [10], a common sraegy is for he provider o move differen users workloads across space and ime o avoid coinciding heir peak usages, hus, reducing he overall peak demand for bandwidh [10]. However, wheher or no users are willing o modulae heir workloads is ofen overlooked. The sudies ha address bursable billing from he user s perspecive have emerged only recenly. So far, due o he lack of a racable mahemaical expression o calculae he 95h percenile usage of bandwidh, a common approach has been o use experimenal and/or heurisic mehods, e.g., as in [11], [12], [13], [14], [15], [16]. There are also few sudies ha are analyical; however, hey assume ha he workload has a specified disribuion, e.g., Gaussian disribuion [17], or hey focus on peak pricing, i.e., he 100 percenile billing insead of 95 percenile billing [18], or hey assume ha he cos of bandwidh is volume-based [19], [20]. In his paper, as illusraed by Fig. 1, we are ineresed

2 Megabi Per Second 2 in sudying bursable billing from he user s viewpoin by aking ino consideraion he rade-off beween cos and performance based on user s preferences. Specifically, we seek o answer his fundamenal quesion: Wha is he bes way for an individual user, such as a daa cener in a CDC, who is charged via bursable billing, o manage is operaion and he use of bandwidh? Our approach o answer his quesion is based on formulaing and solving an opimizaion problem for bandwidh usage which aims a imizing he user s surplus, i.e., is ne uiliy minus cos. We ake ino consideraion he fac ha, in pracice, neiher he user nor he provider have perfec knowledge abou he workload, and hus he demand for bandwidh in he fuure. For example, when i comes o a user in a CDC as in Fig. 1, he workload is iniiaed by he user s cliens, no he user iself. Therefore, in our analysis, we address demand uncerainy wihin a sochasic opimizaion framework. The main conribuions of his paper are as follows: 1) To he bes of our knowledge, his is he firs paper o sudy he problem of opimal responding o bursable billing from a single user s viewpoin under demand uncerainy wih arbirary probabiliy disribuions. 2) To faciliae he use of sysemaic opimizaion, we develop a racable mahemaical expression o calculae he 95h percenile usage of bandwidh. This model is hen used o formulae a novel bandwidh allocaion problem o imize he user s surplus. Addiionally, we examine differen soluion mehods o find he exac and near-opimal soluions of he formulaed problem. 3) We exend our design as well o anoher emerging pracical scenario where a user can receive service from muliple providers, e.g., when a user can reques conen i needs from muliple providers ha all employ bursable billing. Accordingly, our problem formulaion also addresses workload disribuion in addiion o bandwidh allocaion. 4) We evaluae our design based on a real-world workload race: Wikipedia Page View daa [21]. Wih a ypical workload forecasing mehod, we show ha he use of our design is paricularly rewarding if a user is charged by high bandwidh price and/or i is more sensiive o price han o performance. Finally, we also show he advanage of uilizing services from muliple providers, where we can furher increase he user s surplus by disribuing is workload o muliple providers ha employ bursable billing. 2 PROBLEM FORMULATION In his secion, we formulae a mahemaical expression for a user s 95h percenile usage, which is a key concep in bursable billing. This model is hen used o obain Samples 95h percenile usage 13 rd Jun rd Jun nd Jul nd Jul 1998 Fig. 2. An example for calculaing he 95h percenile usage: a oal of 8640 samples are colleced for a user during one billing cycle. Afer hrowing away he op 5%, i.e., /100 = 432 samples, he 95h percenile usage is obained as Mbps, which is equal o he highes recorded bandwidh usage of he remaining /100 = 8208 samples. The 95h percenile usage is shown by he red line. Here, he user is allowed o have a oal of 432 burss above he red line wihou facing financial penaly. he user s expeced bandwidh cos and surplus prior o a billing cycle h Percenile Usage In order o apply bursable billing, a provider firs divides a billing cycle ino τ ime inervals of equal lengh T. The lengh of ime inervals could be as low as 30 seconds, hough ypically he ime inervals of T = 5 minues are considered [2]. Nex, o obain a user s 95h percenile usage, he provider akes samples of he user s usage of bandwidh, e.g., once every five minues during ha billing cycle. Then, he op 5% of he samples gahered wihin he billing cycle are hrown away and he highes elemen of he remaining 95% samples is aken as he user s 95h percenile usage. An example for calculaing he 95h percenile usage is shown in Fig. 2. Similarly, he user can obain is own 95h percenile usage, denoed by µ 95 (x[]), given he usage samples x[1],, x[τ] from he mahemaical expression provided in he following heorem: Theorem 1: Given x[], = 1,..., τ as he τ samples of he bandwidh usage for a user during a billing cycle, we can model he 95h percenile usage for ha user as µ 95 (x[]) = min ρ ρ[]x[] s.. ρ[] 0, 1},, ρ[] = 0.95τ, where he variables in he above minimizaion are ρ[] for all = 1,..., τ, and denoes he ceiling funcion. Proof: Le us define ρ[1],..., ρ[τ] such ha ρ[] = 0 for each ime slo a which x[] is wihin he op 5% of he values in array x[1],..., x[τ], and ρ[] = 1 oherwise. (1)

3 3 Clearly, we have µ 95 (x[]) = x[] ρ[]}, (2) which in his case, Theorem 1 holds. Nex, we noe ha ρ[1],..., ρ[τ] is a feasible soluion o problem (1). To complee he proof, we show ha ρ[1],..., ρ[τ] is in fac he opimal soluion of he minimizaion problem in (1). We prove his by conradicion. Suppose ρ[1],..., ρ[τ] is he opimal soluion of problem (1), where for a leas one ime slo, we have ρ[] ρ[]. Due o he equaliy consrain in (1), 95% of he variables in ρ[1],..., ρ[τ] are equal o one. Therefore, ρ[1],..., ρ[τ] could be differen from ρ[1],..., ρ[τ] only if here exiss a ime slo for which ρ[] = 1 even hough x[] is wihin he op 5% of he values in array x[1],..., x[τ]. In ha case, we mus have x[] ρ[]} x[] ρ[]}. (3) Also, since ρ[1],..., ρ[τ] is assumed o be he opimal soluion of problem (1), by definiion of opimaliy, we mus have x[] ρ[]} x[] ρ[]}. (4) From (3) and (4), we can conclude ha x[] ρ[]} = x[] ρ[]}. (5) However, his conradics he assumpion ha ρ[1],..., ρ[τ] is no opimal. Therefore, ρ[1],..., ρ[τ] is he opimal soluion. ρ[] in problem (1) is an auxiliary variable. For each ime slo, if ρ[] = 0, i indicaes ha is corresponding usage, x[], is wihin he op 5% of he values in x[1],..., x[τ], hus, x[] has no impac on he 95h percenile usage µ 95 (x[]), i.e., he user can uilize bandwidh on-demand wihou exra cos. On he conrary, if ρ[] = 1, he user may resric is usage a his ime slo o reduce is 95h percenile usage. 2.2 Cos of Bandwidh under Bursable Billing Nex, we formulae he user s bandwidh cos given he bandwidh usage samples x[1],..., x[τ] based on bursable billing as C 95 (x[]) = δ µ 95 (x[]), (6) where δ ($/Mbps) denoes he price of bandwidh under bursable billing. Noe, he price of bandwidh δ can vary wih he lengh of billing cycle. However, in his paper, we assume ha he lengh of each billing cycle is fixed, i.e., he price of bandwidh δ is consan. 2.3 Expeced Surplus Prior o a Billing Cycle Consider a user ha aims o plan for is bandwidh usage prior o a billing cycle. A key quesion is how o model he expeced surplus, i.e., ne uiliy minus cos, under uncerain demand. Therefore, in his secion we formulae he user s expeced ne uiliy and surplus prior o a billing cycle. Le D[] (Mbps) be he user s demand for bandwidh a ime slo, which is he amoun of bandwidh user needs o fully saisfy is cliens, i.e., o obain he highes ne uiliy. Noe ha, he user may no know is exac demand in he fuure, raher has a disribuion for is demand, i.e., D[] is a random variable. Nex, we noe ha he user may no always choose o serve is full demand for bandwidh a a given ime. Le X[] (Mbps) be he planned usage of bandwidh during ime inerval = 1,..., τ for he user prior o he billing cycle. Here, he planned usage of bandwidh X[] is decided based on he demand D[]. We assume a general ne uiliy funcion in his paper ha depends only on user s bandwidh usage. A each ime slo, he uiliy funcion U( ) is a concave and nondecreasing funcion of he oal bandwidh, as in [22], [23]. However, he user canno gain any exra uiliy by using more bandwidh han is demand. Therefore, for a billing cycle, we formulae he user s expeced ne uiliy as R = E(U(T minx[], D[]})) (7) corresponding o is planned usage samples X[1],..., X[τ]. Here, E(.) denoes mahemaical expecaion. From he opimizaion-based model in (1), one can calculae he 95h percenile usage for each billing cycle, which is denoed by µ 95 (X[]), as a funcion of planned usage samples X[1],..., X[τ]. Furher, he corresponding bandwidh cos a each billing cycle can be calculaed via (6). From (6) and (7), he user s expeced surplus, i.e., is expeced ne uiliy minus is bandwidh cos during a billing cycle, is obained as S = E (U(T minx[], D[]})) δ µ 95 (X[]). (8) 3 SURPLUS MAXIMIZATION In his secion, we aim o opimally plan he user s bandwidh usage prior o a billing cycle o achieve he highes surplus. In oher words, we formulae he problem o obain he opimal planned usage so as o imize he expeced surplus. Typically, neiher he user nor he provider have perfec knowledge abou he user s demand for bandwidh in an upcoming billing cycle, i.e., D[1],..., D[τ] are ofen uncerain. Here, we assume ha he predicions of user s demand D[] are given, which could be eiher deerminisic values or sochasic probabiliy funcions. Accordingly, we formulae he opimizaion problems of imizing he user s surplus prior o a billing cycle under deerminisic and sochasic predicion of D[].

4 4 3.1 Surplus Maximizaion wih Deerminisic Predicion If he predicion of demand for bandwidh is deerminisic, i.e., parameers D[1],..., D[τ] are deerminisic, from (1) and (8), we formulae he opimizaion problem o imize he user s surplus over a billing cycle as: X[],ρ[] U(T minx[], D[]}) δ ρ[]x[] s.. X[] 0,, ρ[] 0, 1}, ρ[] = 0.95τ. Here, X[] is he principal variable while ρ[] is he auxiliary variable ha is used o calculae he expeced 95h percenile usage as explained in Theorem 1. Noe ha, since he ne uiliy funcion does no depend on he auxiliary variable ρ[], and also because price parameer δ is nonnegaive, if he principal variable X[] is se o be fixed, hen he imizaion in (9) over X[] and ρ[] reduces o he minimizaion in (1) over ρ[]. Therefore, i is guaraneed ha once we solve he problem in (9), he choice of auxiliary variable ρ[] is auomaically seleced in a way ha µ 95 (X[]) is calculaed as in (1). 3.2 Surplus Maximizaion wih Sochasic Predicion Anoher common approach in addressing uncerainy is o obain a probabiliy mass funcion [24] for each random parameer using hisorical workload daa. This can be done in various levels of deails and accuracy, e.g., see [25]. In such case, we assume ha each D[] shall be expressed by K possible realizaions: D 1 [],..., D K [], where each realizaion D k [] may occur wih probabiliy π k,. We have K π k, = 1,. (10) k=1 Once we use he above modeling mehod, we can hen formulae he sochasic opimizaion problem o imize he user s expeced surplus over a billing cycle as: X[],ρ[] K k=1, (9) π k, U(T minx[], D k []}) δ ρ[]x[] s.. X[] 0,, ρ[] 0, 1}, ρ[] = 0.95τ. 4 SOLUTION METHOD, (11) Boh problems (9) and (11) are nonlinear, mixed-ineger programmings, which are generally considered o be hard problems o solve. Neverheless, in his secion, we explain how hese problems can be solved wih reasonable compuaional complexiies. 4.1 Deerminisic Problem For he deerminisic problem (9), we can inuiively obain he opimal soluion for variables ρ[1],..., ρ[τ] wihou numerically solving he problem. This propery can be expressed mahemaically in he following heorem. Theorem 2: Le ϑ denoe he se of all ime slos a which D[] is wihin he op 5% of he values in D[1],..., D[τ]. There exiss an opimal soluion for he deerminisic problem (9) in which he values of auxiliary variables ρ[1],..., ρ[τ] are as follows: ρ 0, ϑ; [] = (12) 1, oherwise. Once he opimal values of ρ in he deerminisic problem (9) are replaced from (12), he soluion for he principal variables X[1],..., X[τ] of he deerminisic problem (9) are obained from he following convex opimizaion problem: X[] s.. U(T X[]) δ ρ []X[] 0 X[] D[],, (13) where ρ [] is given by (12). Proof: Firs, one can easily find ha he objecive funcion in he deerminisic problem (9) is a non-increasing funcion of X[], when X[] D[]. Therefore, he opimizaion problem (9) can be reformulaed as X[],ρ[] U(T X[]) δ ρ[]x[] s.. 0 X[] D[],, ρ[] 0, 1}, ρ[] = 0.95τ., (14) Nex, we noe ha ρ [] in (12) is a feasible soluion for he problem (14). Le ϑ denoe he complemen se of ϑ, i.e., ϑ = 1,..., τ} ϑ. Le ρ c [] denoe he rue opimal soluion of ρ[] for he problem (14). The soluion of usage X [], obained from (14) by seing ρ[] = ρ [], is as follows: X D[], ϑ; [] = minµ, D[]}, ϑ, (15) where µ is he opimal 95h percenile usage of bandwidh corresponding o X []. To complee he proof of heorem we only need o show ha ρ c [] = ρ []. Nex, We prove by conradicion ha his argumen indeed holds. In oher words, if we assume ha ρ c [] ρ [], hen he user s oal surplus wih ρ c [] will be less han he user s surplus wih ρ []. Le ρ c [] ρ [] so ha:

5 5 ρ c [] = 0, ν; 1. ν, (16) where ν is some se so ha ν ϑ and ν is he complemen se of ν. This assumpion implies ha, for a leas one ime slo ν, D[] is no wihin he op 5% of he values in array D[1],..., D[τ]. The opimal usage of bandwidh X c [] in his case becomes X c D[], ν [] = minµ c (17), D[]}, ν, where µ c is he opimal 95h percenile usage of bandwidh corresponding o X c []. We prove ha µ c ϑ D[]. Considering a scenario where he user plans o uilize bandwidh on-demand, i.e., 1,..., τ}, X[] = D[]. In his case, he 95h percenile usage of bandwidh becomes ϑ D[], which is obviously he highes feasible 95h percenile usage of bandwidh of problem (14). Thus, µ c ϑ D[]. Then, we prove ha (X [], ρ []) is he opimal soluions of problem (14). Le X D[], ϑ; [] = minµ c, D[]}, ϑ. (18) Since µ c ϑ D[], X []ρ [] = µ c. Namely, wih (X [], ρ []), he 95h percenile usage of bandwidh equals µ c. In his case, we have C 95 (X []) = C 95 (X c []) = δ µ c. (19) Also, from (7), we have R(X []) = ϑ and R(X c []) = ν U(T D[]) + ϑ U(T minµ c, D[]}) (20) U(T D[]) + ν U(T minµ c, D[]}). (21) Le f(d[]) = U(T D[]) U(T minµ c, D[]}). From (20) and (21), we calculae R(X []) R(X c []), which equals o f(d[]) f(d[]). (22) ϑ ν ν ϑ Nex, noe ha f(d[]) is in fac equal o: U(T D[]) U(T µ c ), if D[] µ c ; f(d[]) = 0, oherwise. (23) Since U( ) is nondecreasing and T 0, from (23), f(d[]) is nondecreasing, oo. Then, we can find ha D[ 1 ] D[ 2 ] 1 ϑ ν, 2 ν ϑ. (24) From (22) and (24) and since f(d[]) is nondecreasing over D[] and ϑ ν = ν ϑ, we have R(X []) R(X c []). (25) From (19) and (25), he obained surplus wih (X [], ρ []) is no less han he one wih (X c [], ρ c []). Since X [] is he opimal soluion of X[] of problem (14) corresponding o ρ [], he obained surplus wih (X [], ρ []) is no less han he one wih (X [], ρ []). Therefore, he obained surplus wih (X [], ρ []) is no less han he one wih (X c [], ρ c []). Since (X c [], ρ c []) was assumed o be opimal, (X [], ρ []) is an opimal soluion of problem (14). From Theorem 2, one can conver he non-convex problem (9) ono a convex program (13), which can be effecively solved using convex programming echniques [26]. 4.2 Sochasic Problem If parameers D[1],..., D[τ] are random, hen we do no know a wha ime slos he burs will occur in he demand for bandwidh. Accordingly, we canno separaely figure ou he opimal values of ρ[1],..., ρ[τ]. Therefore, we have no choice bu solving he original sochasic problem (11). A key difficuly in solving he sochasic problem (11) is ha even if we relax he binary consrains, i.e., even if we choose ρ[] o be a coninuous number beween 0 and 1, he relaxed problem is sill difficul o solve due o he non-convex erm ρ[]x[] in he objecive funcion. Ineresingly, we can ackle his undesirable propery as i is explained in a heorem below. Theorem 3: The sochasic problem (11) is equivalen o: K π k, U(T minx[], D k []}) δ φ X[],ρ[],φ k=1 s.. X[] φ + L(1 ρ[]),, X[] 0, ρ[] 0, 1}, ρ[] = 0.95τ, (26) where L is a large number compared o he available bandwidh, and φ is anoher auxiliary variable. Proof: A each ime slo, if ρ[] = 0, hen he firs consrain in problem (26) reduces o X[] φ + L,, which always holds regardless of he values of X[] and φ. If ρ[] = 1, hen he firs consrain in (26) reduces o X[] φ,. In ha case, since he objecive funcion in (26) is o minimize φ, we necessarily obain ha φ = ρ[]x[] a any opimal soluion of problem (26). This is clearly an oucome ha we inended. Given he equivalence of he sochasic problem (11) and (26), we can solve problem (26) insead of (11). Nex, we noice ha from (26), once we relax he binary consrains, he relaxed problem is convex. Therefore, we can find he exac opimal soluion of problem (26) using branch-and-bound mehod [27], where a each branching,,

6 6 sep we need o solve a convex opimizaion problem. We refer o his approach as he convex branch-andbound (CBB) mehod. While he CBB mehod is effecive o obain he exac opimal soluion of he sochasic problem (11), solving a nonlinear (alhough convex) problem a each ieraion of he branch-and-bound algorihm could be ime consuming. Since he nonlineariy in problem (26) is due o he nonlinear uiliy funcion U( ), one way o make problem (26) linear is o replace U( ) wih is piece-wise linear approximaion. This is explained in he following heorem. Theorem 4: Le N denoe he number of angen lines in he piece-wise linear approximaion of he uiliy funcion U( ). If N, hen he problem in (26) is equivalen o he mixed-ineger linear opimizaion problem: X[],ρ[],φ, Q k [],h k [] K π k, h k [] δ φ k=1 s.. X[] φ + L(1 ρ[]),, X[] 0, ρ[] 0, 1}, ρ[] = 0.95τ,,, Q k [] X[],, k, Q k [] D k [],, k, h k [] U(n [])+ U (n [])(T Q k [] n []),, k, n, (27) where n = 1,..., N. Here, Q k [] and h k [] are auxiliary variables for angen line k. Proof: As i can be seen from (27), we firs replace minx[], D k []} in he objecive funcion of (26) wih an auxiliary variable Q k []. Here, Q k [] is upper bounded by X[] and D k [], which is exacly he ype of consrain ha we need o model he min funcion minx[], D k []}. Nex, he concave funcion U(T Q k []) is replaced by a new variable h k []. Also, as in he las consrain in (27), h k [] is upper bounded by N number of angens lines o he concave curve U(T Q k []). Therefore, if N, h k [] is equivalen o U(T Q k []). Accordingly, he problem formulaion in (27) becomes equivalen o he one in (26). The usefulness of problem (27) depends on he choice of parameer N. However, as we will see in Secion 6.2, we can obain he near exac opimal soluion of he sochasic surplus imizaion problem even if N = 3. There exis effecive solvers o solve mixedineger linear programming (MILP), such as CPLEX [28]. We will see in Secion 6.2 ha solving he MILP in (27) is compuaionally more racable han he CBB mehod. Before we end his secion, we mus poin ou ha one can obain an approximae soluion for problem (27) by erminaing he opimizaion solver a cerain guaraneed opimaliy bounds in order o significanly lower compuaional complexiy. We will furher discuss his opion in Secion EXTENSIONS AND REMARKS In his secion, we discuss wo ineresing analysis wih regards o he proposed design. Firs, we exend our design o a scenario where a user has he opion o receive service from muliple providers. An example is a user can download specified conen over differen ransi links ha is owned by differen ISPs, who charge he user via bursable billing. Second, we show in his secion ha a user can furher improve is surplus by updaing he usage of bandwidh in real-ime, i.e. during he billing cycle, based on he newly exposed acual demand informaion. 5.1 Exension o Muliple Providers Le X i [] denoe he planned usage of bandwidh a provider i a ime slo decided based on he user s demand D[]. Le δ i ($/Mbps) denoe he price of bandwidh a provider i. In his case, in each billing cycle, he expeced surplus of he user wih muliple providers is obained as ( ) I S msp = E U(T min X i [], D[]}) I δ i µ 95 (X i []). i=1 i=1 (28) We can also formulae a user s opimizaion of he usage a muliple providers o achieve imum expeced surplus under sochasic predicion of he demand D[] by: X i[],ρ i[] K I π k, U(T min X i [], D k []}) k=1 I i=1 δ i ρ i []X i [] i=1 s.. X i [] 0,, i, ρ i [] 0, 1},, i, ρ i [] = 0.95τ, i. (29) A special case of problem in (29) is where, K = 1 and π k, = 1, i.e., he case where he predicion of user s demand is deerminisic and problem (29) reduces o a deerminisic opimizaion. Noe ha, for he case of muliple providers, he exac soluion of he opimizaion in (18), even for he deerminisic opimizaion, canno be obained from he mehod discussed in Theorem 2. Therefore, for he soluion of he problem in (18), as in Secion 4.2, we ransform he nonlinear mixed-ineger

7 Worload (Mbps) Worload (Mbps) Wikipedia English 0 13 rd Jan h Jun h Dec s Jun Wikipedia English Mobile 13 rd Jan h Jun h Dec s Jun 2015 Fig. 3. Examples for he real-world workload races used in his paper from [21]; a) daa race of Wikipedia English, b) daa race of Wikipedia English Mobile. programming (29) ino an equivalen mixed-ineger convex programming (as shown in Theorem 3) or MILP (as shown in Theorem 4), where he mixed-ineger convex programming and he MILP can be solved via CBB and MILP solvers, respecively. 5.2 Updaing Usage of Bandwidh During a Cycle Nex, we show ha he user can furher improve is surplus during a billing cycle, by updaing is planned usage of bandwidh a each ime slo based on he newly exposed demand. We also show ha he user s final surplus afer a billing cycle will be no less han he expeced surplus. Here, we assume ha, a he beginning of each ime slo, he user s demand for bandwidh is exposed o he user. We denoe he exposed demand value a ime slo by D[]. Generally, he demand D[] may no be he same as he exposed value D[]. Therefore, a user can updae is planned usage of bandwidh in real-ime based on he newly learned exposed demand informaion, i.e., D[], o furher improve is surplus while keeping is bandwidh cos unchanged. For example, if X[] < D[] and X[] < µ 95 (X[]), he user can increase is usage from X[] o min D[], µ 95 (X[])}. In his way, he user s ne uiliy can be enhanced while remaining is bandwidh cos unchanged. In pracice, he expeced 95h percenile usage µ 95 (X[]) is reaed as a rae limier. According o Theorem 1, when ρ[] = 1, he user resrics is usage a his imes slo o reduce is 95h percenile usage. Specifically, when ρ[] = 1, if D[] µ 95 (X[]), he user can uilize bandwidh ondemand, and if D[] > µ95 (X[]), he user needs o resric is uilizaion of bandwidh o ensure ha is 95h percenile usage equals o µ 95 (X[]). On he conrary, he user can always uilize bandwidh on-demand when ρ[] = 0 since he usage a his ime slo has no impac on he 95h percenile usage. Therefore, we formulae he user s updaed usage of bandwidh a each ime slo during a cycle, which is denoed by X[], as D[], if ρ[] = 0 or D[] µ95 (X[]); X[] = µ 95 (X[]), oherwise. (30) From (30), we ensure ha, X[] D[]. Similar o (7) and (8), afer a billing cycle, he ne uiliy wih updaed usage values X[1],..., X[τ] can be calculaed as R = U(T X[]). (31) Furher, from (1), (6) and (31), we formulae he user s surplus wih updaed usage values X[1],..., X[τ] via S = U(T X[]) δ µ 95 ( X[]). (32) We can show ha a user s surplus wih updaed usage values X[1],..., X[τ] is always no less han is surplus wih planned usage values X[1],..., X[τ]. From (30), we ensure ha µ 95 ( X[]) µ 95 (X[]). Therefore, he bandwidh cos over a billing cycle wih X[] is always no higher han he bandwidh cos wih X[]. Nex, we noice ha he ne uiliy over a billing cycle wih updaed usage values, X[], is always no less han he bandwidh cos wih planned usage values, X[], i.e., U(T min X[], D[]}) U(T minx[], D[]}),. (33) To verify ha (33) indeed holds, consider hree cases: Case 1: If ρ[] = 0, X[] = D[]. Since he ne uiliy funcion U( ) is nondecreasing and T > 0, (33) is saisfied. Case 2: If ρ[] = 1 and D[] µ 95 (X[]), X[] = D[]. Same as case 1, in his case, (33) is saisfied. Case 3: If ρ[] = 1 and D[] > µ 95 (X[]), X[] = µ95 (X[]) and X[] µ 95 (X[]). In his case, (33) is also saisfied. Accordingly, as he surplus of a user over a cycle equals is ne uiliy minus is bandwidh cos over ha cycle, We can see ha a user s surplus wih updaed usage values is always no less han is surplus wih planned usage values. Idenically, if he user can receive service from muliple providers, we can also updae is planned usage of bandwidh a provider i, i.e., X i [], in real-ime based on he newly learned informaion of he exposed demand D[]. Le X i [] denoe he updaed usage of bandwidh a

8 Opimaliy (%) Compuaion TIme (Sec) 8 Opimaliy (%) Parameer N Wikien Wikimw Fig. 4. The impac of he number of angens lines on he opimaliy of he soluion for MILP-based problem (27). 1.E+05 1.E+03 1.E+01 1.E Parameer τ CBB MILP Near provider i a ime slo and i is defined as D[], if ρi [] = 0 or X i [] = D[] µ 95 (X i []); µ 95 (X i []), oherwise, (34) where ρ i [] is he auxiliary variable as used in Theorem 1. Then, we formulae he user s surplus over a cycle via S msp = U(T I i=1 X i []) I δ i µ 95 ( X i []). (35) i=1 Similarly, a user can also furher improve is surplus via updaing is planned usage according o (34) if i can receive service from muliple providers. Noe ha, since he final surplus a user can achieve in our design is obained from (32) and (35), we use hese values as he user s surplus, in he res of his paper, o evaluae he performance of our design. 6 CASE STUDIES In his secion, wih real-world daa races, we firs sudy he compuaion ime and performance of our proposed soluion mehods for solving he sochasic problem (26). Second, we evaluae our design wih a simple mehod o forecas he demand for bandwidh. Third, we discuss he impac of price and uiliy facor on he performance of our design. Forh, we show ha, wih muliple providers, he user can furher improve is surplus wih our design. 6.1 Seup We use wo daa ses in our case sudies: 1) Wikien: he page view daa of Wikipedia English from January 2014 o May 2015 [21], 2) Wikimw: he page view daa of Wikipedia English Mobile from January 2014 o May 2015 [21], Example races of hese daa ses are shown in Fig. 3. Each ime slo akes one hour and he billing cycle akes 28 days for Wikien and Wikimw daa ses. The uiliy funcions are seleced as follows: A(1 a) 1 x 1 a, if a (0, 1); U(x) = (36) Alog(x), if a = 1, which is commonly used in economics [29], [30]. Here, A > 0 is he uiliy facor decided by he user and Parameer τ CBB MILP Near Fig. 5. Comparing differen soluion mehods in solving problem (26): Compuaion ime, Opimaliy. a (0, 1] measures he concaviy of he user s uiliy. Namely, as a increases, he user s uiliy becomes more concave. Specifically, we assume ha a = 0.1, A = 0.08 and he impac of he uiliy facor A on he surplus of user will be discussed in Secion 6.4. We use a very simple workload forecasing mehod. Le D 1 [] and D 2 [] denoe he workload a ime slo in he las wo billing cycles, respecively. Suppose ha π 1, = π 2, = 0.5, = 1,..., τ. Specifically, for deerminisic surplus imizaion, we assume ha D[] = π 1, D 1 [] + π 2, D 2 [], for any = 1,..., τ. 6.2 Compuaion Complexiy of Proposed Soluion Mehods Recall from Secion 4.2 ha here are muliple opions o solve he sochasic problem (26). Specifically, he proposed CBB mehod leads o he exac opimal soluion. The efficiency of he MILP mehod, however, depends on he number of angen lines N. Suppose we choose [] = T D k []/N. Fig. 4 shows he opimaliy in percenage in applying he MILP mehod versus he number of angen lines N for differen daases. We can see ha he resuls are accurae when N 3. Therefore, for he res of his paper, we assume ha N = 3. Nex, we evaluae he compuaion ime for each soluion mehod. We use a personal compuer wih Inel Xeon CPU The resuls are shown in Fig. 5. We can see ha he compuaion ime of CBB is much longer han MILP. Even for he MILP approach, i may ake several hours o find he global opimal soluion of problem (27) as he size of he problem increases.

9 9 As we poined ou in Secion 4.2, one can obain an approximae soluion for problem (27) by erminaing he opimizaion solver a cerain guaraneed opimaliy bounds. This can be done by seing up a sopping condiion for he MILP mehod based on he raio beween he upper-bound and he lower-bound soluions. The upperbound soluion is he surplus ha can be achieved if we relax he remaining binary variables a he curren branching sage. The lower-bound soluion is he surplus a he bes binary soluion ha has been obained so far a he curren branching sage. Clearly, his raio indicaes a guaraneed opimaliy in he soluion of MILP ha has already been reached a he curren branching sage. In his paper, we obain an approximae soluion by sopping he MILP mehod in CPLEX once he above menioned raio reaches 5%, which guaranees a leas 95% opimaliy. We refer o his approximae soluion approach as he Near mehod. As we can see in Fig. 5, he Near mehod is significanly less complex in erms of required compuaion, compared o he CBB and MILP mehods. Specifically, he compuaional ime for he Near mehod grows only linearly wih respec o he number of ime slos. Ineresingly, we can see in Fig. 5 ha he acual achieved opimaliy is around 99% or more, i.e., much beer han he guaraneed 95% wors case opimaliy value. Therefore, for he res of his paper, we use he Near mehod a 95% guaraneed opimaliy. 6.3 Performance Evaluaion As a Baseline for performance comparison, we consider he case where he bandwidh is allocaed on-demand, i.e., X[] = X[] = D[], for any = 1,..., τ. Noe ha, his approach resembles how he bandwidh is currenly allocaed in pracice. Nex, we also assume an Ideal case where he usage of bandwidh is opimized based on rue knowledge of demand, i.e.,, D[] = D[]. While he Baseline shows how well we can perform compared o he exising pracice, he Ideal case shows he bes performance ha we can ever ge, assuming ha we can perfecly predic he upcoming workload. Nex, we compare he Baseline and Ideal cases wih our proposed Deerminisic and Sochasic mehods. The Deerminisic mehod refers o he case where he bandwidh usage is scheduled based on he opimal soluion of he deerminisic surplus imizaion problem in (13). The Sochasic mehod refers o he case where he bandwidh usage is scheduled based on he opimal soluion of he sochasic surplus imizaion problem in (27) using he Near mehod wih 95% guaraneed opimaliy. The mehod of forecasing he workload in each case was already explained in Secion 6.1. The resuls on performance comparison are shown in Fig. 6, Fig. 7 and Fig. 8, where he resuls for all mehods are normalized wih respec o he resuls of he Ideal case. Here, he price of bandwidh is se o be $15 per Mbps. We can make he following observaions based on hese resuls: Fig. 6. Comparing normalized bandwidh cos under differen mehods and differen workloads: a) Wikien, b) Wikimw. Fig. 7. Comparing normalized surplus under differen mehods and differen workloads: a) Wikien, b) Wikimw. As shown in Fig. 6 and Fig. 7, even hough we use

10 10 Fig. 8. Comparing average bandwidh cos and surplus under differen mehods and differen workloads: a) Wikien, b) Wikimw. Fig. 9. The impac of he price of bandwidh on average surplus under differen workloads: a) Wikien, b) Wikimw. a very simple mehod o forecas he demand for bandwidh, he Deerminisic and Sochasic soluions ouperform he Baseline in boh bandwidh cos reducion and surplus improvemen. Thus, our mehod is robus o he error of predicion of user s demand. Meanwhile, Deerminisic and Sochasic have similar oucomes. As shown in Fig. 8, on average, our proposed opimizaion-based approach o respond o bursable billing can grealy reduce he user s bandwidh cos while improving is surplus when comparing agains Baseline. For example,wih daa race of Wikien, boh Deerminisic and Sochasic surplus imizaion can reduce he user s bandwidh cos by 26% while increasing is oal surplus by 23%, respecively. 6.4 Impac of Price and Uiliy Facor Inuiively, increasing he price for bandwidh would increase he user s cos. Accordingly, he surplus ha he user may gain decreases as we increase price parameer δ. However, he rae of such decrease is no he same for differen mehods. The resuls are shown in Fig. 9. We can see ha he rae of decrease in surplus is higher for he Baseline compared o he Deerminisic and Sochasic mehods. As a resuls, he surplus improvemens wih our proposed opimizaion-based approaches are higher when he price of bandwidh is high. Fig. 10. The impac of he uiliy facor on average surplus under differen workloads: a) Wikien, b) Wikimw.

11 11 Fig. 11. Comparing normalized bandwidh cos wih muliple providers under differen workloads: a) Wikien, b) Wikimw. Fig. 12. Comparing normalized surplus wih muliple providers under differen workloads: a) Wikien, b) Wikimw. Nex, we analyze he impac of uiliy facor A. Clearly, increasing A resuls in higher surplus for he same usage of bandwidh. By analysing Fig. 10, we find ha he disance beween Baseline and Deerminisic/Sochasic is slighly larger when A is small. Namely, users wih smaller uiliy facors, who are more sensiive o price han performance, are more likely o response o he bursable billing o improve heir surpluses. We can also see ha he Deerminisic and Sochasic mehods ouperform he Baseline a all choices of A. 6.5 Impac of Muliple Providers Suppose he user can receive service from wo providers, who are referred o as providers 1 and 2. Boh of hem offer bandwidh a $15 per Mbps. To evaluae our proposed approach o response o bursable billing wih muliple providers, we simulae six differen cases: Ideal-MSP: I is defined as he oucome of imizing surplus, under he assumpion ha he demand for bandwidh is known wih muliple providers. Baseline-SSP: In his case, he user uilizes bandwidh from provider 1 on-demand. Deerminisic-SSP: In his case, he user uilizes bandwidh from provider 1 and makes is decisions based on our design wih deerminisic predicion abou is demand. Sochasic-SSP: In his case, he user uilizes bandwidh from provider 1 and makes is decisions based on our design wih sochasic predicion abou is demand. Deerminisic-MSP: In his case, he user uilizes bandwidh from boh provider 1 and 2 and makes is decisions based on our design wih deerminisic predicion abou is demand. Sochasic-MSP: In his case, he user uilizes bandwidh from boh provider 1 and 2 and makes is decisions based on our design wih sochasic predicion abou is demand. Figures 11 and 12 show he normalized bandwidh cos and surplus, obained in six differen cases, where he base for normalizaion is he surplus under he Ideal-MSP case. We can see ha Deerminisic-MSP and Sochasic-MSP mehods always ouperform Baseline-SSP in boh bandwidh cos reducion and surplus improvemen. Finally, we also find ha Deerminisic-MSP and Sochasic-MSP are always beer han Deerminisic-SSP and Sochasic-SSP. We may infer ha he availabiliy of muliple providers furher reduce he user s bandwidh cos and improves is surplus under opimal response mechanism o bursable billing. 7 CONCLUSION AND FUTURE WORK A novel opimizaion-based approach was proposed o selec he usage of bandwidh for a user, such as a user of

12 12 a colocaion daa cener, who is charged for bandwidh usage under bursable billing. Our proposed approach considers workload demand uncerainy, and is general in he sense ha i does no make any assumpion abou he saisical characerisics of workload. Numerical resuls based on empirical case sudies confirm ha even wih a simply workload forecasing mehod, he user can obain significanly higher surplus under he proposed opimal mehod for responding o bursable billing, compared o he curren pracice of allocaing bandwidh on-demand. We also exended our design o anoher emerging pracical scenario where a user can receive service from muliple providers. Accordingly, besides bandwidh allocaion, our problem formulaion also addresses workload disribuion. This paper can be exended in several direcions. Firs, one can adop a more advanced workload forecasing mehod o beer model probabiliy disribuion funcions for he demand for bandwidh. 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