# Maximizing the Bandwidth Multiplier Effect for Hybrid Cloud-P2P Content Distribution

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4 2 5 5 measurement data exponental fttng 5 5 S/l (KBps) (a) S vs. Fg. 3. Relatonsps between (a) peer swarm. D/l) (KBps) 3 2 measurement data exponental fttng measurement data exponental fttng 5 5 l/s (b) vs. and S, (b) and, as to a typcal D/l) (KBps) 3 2 measurement data exponental fttng Fg. 5. (a) measurement data 2 2 (c) only usng S/l 2 2 Modelng usng (b) = ( S ) l α f ) and (d) bot (.e., (.e., 2 (b) only usng l/s 2 (d) usng bot l/s and S/l 2 2 = ( ) β f ), (c) S (.e., = ( S ) α ( ) β f ), as to a typcal peer swarm. Clearly, te key mpact factors sould nclude bot S and so tat te model can matc te (a) measurement data well. 2 S/l (KBps) (a) log( S ) vs log( ). l/s (b) log( ) vs log( ). Fg.. Relatonsps between (a) log( ) and log( S ), (b) log( ) and log( ), as to a typcal peer swarm. key mpact factors sould nclude bot S we get te followng equaton: and l. Terefore, = ( S ) α ( ) β f, 2 () were < α <, β > and f >. Ten te aggregate download bandwdt of peer swarm s: = S α l α β β f. (2) Snce S s te only decson varable tat we can scedule, we also wrte as (S ). Fnally, te bandwdt multpler of peer swarm s: S = S α l α β β f. (3) To compute te constant parameters α, β and f, we frst transform Equaton () nto ts log form: log = log S α log β + log f, () so tat α, β and f can be computed by usng te measurements of,, S and, va te classcal lnear regresson metod. One tng to note s tat te abovementoned constant parameters (α, β and f ) can only be taken as constant durng a certan perod (typcally one day or several ours), so tey need to be perodcally updated usng te latest measurements. 2 If =, we just let = so tat ( ) s β gnored. TABLE I RELATIVE ERRORS OF THE THREE MODELS APPLIED TO ALL THE 57 PEER SWARMS, COMPARED WITH THEIR MEASUREMENTS DATA. Model Avg (relatve error) Mn Max (b) only usng (c) only usng S (d) usng bot S and B. OBAP and Its Optmal Soluton Tll now, te optmal bandwdt allocaton problem (OBAP) of CloudP2P can be formalzed as follows: OBAP Maxmze te overall bandwdt multpler ( D S ) subject to te followng condtons: D = m =, were m s te number of swarms; S = m = S, were S s taken as a constant durng an allocaton perod; S, {, 2,, m}; = S α l α β β f, {, 2,, m}; wt decson varables S, S 2,, S m. We can see tat OBAP s a constraned nonlnear optmzaton problem [8]. Gven tat S s taken as a constant durng an allocaton perod, maxmzng D S s equal to maxmzng D. Wen te optmal soluton of OBAP exsts, suppose te optmal soluton s S = (S, S2,, Sm) 3 and te correspondng aggregate download bandwdt of eac swarm s (D, D2,, Dm). Tus, accordng to te optmalty 3 Te bold font s used to represent a vector. It s possble tat OBAP as no optmal soluton wtn ts constraned set.

5 Ideal status 6 Maxmum D 2 8 Current status S o 8 P() P(2) P(3) P(5) P() P() Iteraton space S d Equal-effect surface Fg. 6. An exceptonal case n wc te peer swarm cannot be adjusted to tdeal status. condton of constraned nonlnear optmzaton [8], we ave: m (S ) m (S S ), S wt S = S. (5) = = Ten fxng an arbtrary and lettng j be any oter ndex, we construct a feasble soluton S to te constrants as: S =, S j = S + S j, S k = S k, k, j. Applyng S to Equaton (5), we get: ( D j(s j ) S j (S ) ) S,, j ( j). If S =, peer swarm gets no cloud bandwdt and tus we do not need to consder suc a swarm for cloud bandwdt allocaton. Consequently, we ave {, 2,, m}, S >, and ten D j (S j ) S j (S ),, j ( j). (6) Terefore, te optmal soluton S as te followng form: D j (S) = D 2(S2) = = D m(sm), (7) S S 2 S m wc means te margnal utlty of te cloud bandwdt allocated to eac peer swarm sould be equan te optmal soluton (f t exsts). In practce, tere s an exceptonal case n wc a peer swarm cannot be adjusted to ts deal status (.e., te margnal utlty of te cloud bandwdt allocated to peer swarm s equal to tat of te oter swarms), and ts exceptonal case wll cause OBAP to ave no optmal soluton n te form of Equaton (7). Allustrated n Fg. 6, for some reasons peer swarm as an upper bound of ts aggregate download bandwdt ( Maxmum ), wc prevents te bandwdt allocaton algortm from adjustng peer swarm to tdeal status. In ts stuaton, we just allocate te least cloud bandwdt to mprove ts aggregate download bandwdt to Maxmum so tat te relatve devaton of margnal utlty among all te peer swarms can be as lttle as possble. In concluson, we ave te followng teorem: Teorem. For CloudP2P content dstrbuton, te maxmum bandwdt multpler mples tat te margnal utlty of te cloud bandwdt allocated to eac peer swarm sould be equal. In practce, we want te relatve devaton of margnal utlty among all te peer swarms to be as lttle as possble,.e., larger bandwdt multpler mples more balanced margnal utltes among peer swarms. Fg. 7. A demo teraton process. Te equal-effect surface s te set of all te ponts P tat ave te same performance value f(p). IV. FAST-CONVERGENT ITERATIVE ALGORITHM In last secton we ave formulated te optmal bandwdt allocaton problem (OBAP) nto a constraned nonlnear optmzaton problem. Te optmal soluton of suc a problem s typcally obtaned va teratve operatonn multple steps untl te algortm converges [8]. Terefore, te convergence property of te teratve algortm s crtcan solvng OBAP. Te convergence property of an teratve algortm manly depends on two aspects: teraton drecton and teraton stepsze. For a d-dmenson constraned nonlnear optmzaton problem, alts feasble solutons compose a d-dmenson teraton space S d. Suppose te teratve algortm starts at an arbtrary pont P () = (P (), P () 2,, P () d ) Sd. Ten n eac subsequent teraton step, te algortm must determne an teraton drecton and an teraton stepsze to go furter to a new pont P (k) = (P (k), P (k) 2,, P (k) ) S d so tat P (k) s closer to te optmal pont P tan P (k ), as sown n Fg. 7. Specfcally, te teraton process can be formalzed as: P (k+) = P (k) + t (k) (P (k) P (k) ), untl f(p (k+) ) f(p (k) ) < ɛ. were f(.) s te performance functon, ɛ s a very small constant, (P (k) P (k) ) s te teraton drecton, and t (k) s te teraton stepsze n te k-t step. Te task of our fast-convergent teratve algortm (FA) s to determne approprate P (k) and t (k) n te k-t step so tat te teraton process can be as fast as possble. Iteraton drecton. For a nonlnear optmzaton problem, usually t mpossble to drectly fnd te ultmate drecton P P () (or P P (k) for a certan k) because ts bascally as dffcult as to drectly fnd P. Instead, FA utlzes te condtonal gradent metod [8] to determne te teraton drecton n eac step. For a functon f(p), t s well known tat f(p (k+) ) can be approxmated va te Taylor expanson: f(p (k+) ) = f(p (k) ) + f(p (k) )(P (k+) P (k) ) T + 2 (P(k+) P (k) ) 2 f(p (k) )(P (k+) P (k) ) T +. d-dmenson means te optmzaton problem deals wt d decson varables n total. As to OBAP, d s te total number of peer swarms. d (8) (9)

6 were f(x) = ( f(x) X, f(x) X 2,, f(x) X d ). Te condtonal gradent metod uses te frst-order Taylor expanson to approxmate f(p (k+) ): f(p (k+) ) f(p (k) ) + f(p (k) )(P (k+) P (k) ) T. () As to te OBAP problem, te dmenson (d) s just te number of peer swarms (m), so tat P (k) = S (k), f(p (k) ) = f(s (k) m ) = D(k) = S = D(S) S and (S ) = S α l α β β f. Ten we ave: f(s (k+) ) f(s (k) ) + f(s (k) )(S (k+) S (k) ) T. () Snce our goas to maxmze f(s) on condton tat m = S = S and S, {, 2,, m}, we need to (greedly) maxmze f(s (k+) ) n Equaton () n te k-t teraton step. Tus, we must fnd te specfc S tat satsfes te followng problem: Maxmze f(s (k) )(S S (k) ) T subject to m = S = S and S, {, 2,, m}. By expandng S, S (k) and f(s (k) ), we transform te above problem nto Maxmze m = (S (k) ) (S S (k) ) subject to m = S = S and S, {, 2,, m}. It s not dffcult to fnd tat te above problem s a lnear optmzaton problem and te optmal soluton S (k) s: S (k) j and = S, for te j = arg max {,2,,m} (S (k) ) ; S (k) =, {, 2,, j, j +,, m}. So we get te optmateraton drecton n te k-t step: (2) d (k) = S (k) S (k). (3) Iteraton stepsze. Tll now we ave got tat te k-t step of our FA teratve algortm proceeds as: S (k+) = S (k) + t (k) d (k) were d (k) s determned n Equaton (2) and (3). Ideally, te stepsze t (k) sould satsfy te followng condtons: Maxmze f(s (k) + t (k) d (k) ) subject to S (k) + t (k) d (k) s a feasble soluton. Unfortunately, te above problem s stll a nonlnear optmzaton problem and tut mpossble to drectly obtan ts optmal soluton. Instead, we utlze te Armjo rule [9] to adaptvely set te teraton stepsze t (k), n order to guarantee tat f(s (k+) ) s at least larger tan f(s (k) ) by a bound: f(s (k) + τ j d (k) ) f(s (k) ) στ j f(s (k) )d (k)t () were te two constant parameters τ, σ (, ), and j s tred successvely as,, 2,..., untl te above nequalty s satsfed for a certan j (wc s j (k) ). As a result, we get te adaptve teraton stepsze n te k-t step for FIFA: t (k) = τ j(k) (5) Summary of FA. Te fast-convergent teratve algortm (FA) effcently solves OBAP by fndng te optmal drecton and adaptvely settng te stepsze n eac teraton step. Frst, te convergence of FA s provable due to ts combnatory use of te condtonal gradent metod and te Armjo rule (refer to Proposton 2.2. n [8]). Second, FA s easy to use because all te related parameters, τ and σ, can be easly confgured. For example, we smply confgure τ =.5 and σ =. for FA, and ten t s well applcable to all te smulaton/mplementaton scenaron Secton V and VI. Fnally, altoug te accurate convergence speed of FA cannot be teoretcally proved, FA exbts nearly-lnear convergence speed n our performance evaluaton (refer to Secton V-C). Tat s to say, for a CloudP2P system consstng of m peer swarms, FA convergen nearly Θ(m) steps. Comparsons of WF, HC and FA. Te water-fllng algortm (WF) s a classcateratve algortm n solvng constraned nonlnear optmzaton problems (e.g., [8]) for ts smplcty and ntutve explanaton. In eac teratve step, WF only fnds two components of S (k),.e., S (k) and S (k) l satsfyng D te followng condtons: = arg max (S (k) ) {,2,,m} and l = arg mn {,2,,m} (S (k) constant porton δ from S (k) l S (k) S (k) l l ). Ten WF moves a to S (k) : S(k) S (k) + δ and δ. Ts movement looks lke fllng some water from one cup to te oter. In oter words, te teraton drecton and teraton stepsze of WF are set as follows: d (k) = (d (k), d(k) 2,, d(k) m ), were d =, d l =, d =,, l; and t (k) = δ. Obvously, WF uses a restrcted teraton drecton (only n two dmensons among te total m dmensons) and a fxed teraton stepsze (δ). Te fundamental problem of WF len te settng of δ. If δ s set too bg, WF wll not converge; f δ s set too small, WF wll converge slowly. Stll worse, on andlng a large number of gly dynamc peer swarms, settng an approprate (.e., neter too bg nor too small) teraton stepsze (δ) for WF becomes extremely dffcult. Consequently, te only practcal coce s to set an extremely small δ resultng n a uge number of teraton steps and slow convergence speed. On te contrary, te teraton stepsze of FA s adaptvely set so te number of teraton steps depends on te number of peer swarms. Fg. 8 s a comparson of te teratve operatons of WF and FA wen tere are only two peer swarms: S =.5 and S 2 =.85 (te total cloud bandwdt s normalzed as S = ). Addtonally, te restrcted teraton drecton furter slows down te teraton process of WF because WF always walks only n two dmensons among te total m dmensons. On te contrary, te teraton drecton of FA can be n all dmensons. Fg. 9 llustrates a demo comparson wen tere are tree peer swarms.

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