C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a

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1 C e n t r u m v o o r W s k u n d e e n I n f o r m a t c a Probablty, Networks and Algorthms Probablty, Networks and Algorthms Traffc-splttng networks operatng under alpha-far sharng polces and balanced farness P.M.D. Leshout REPORT PNA-R0707 JULY 2007

2 Centrum voor Wskunde en Informatca (CWI) s the natonal research nsttute for Mathematcs and Computer Scence. It s sponsored by the Netherlands Organsaton for Scentfc Research (NWO). CWI s a foundng member of ERCIM, the European Research Consortum for Informatcs and Mathematcs. CWI's research has a theme-orented structure and s grouped nto four clusters. Lsted below are the names of the clusters and n parentheses ther acronyms. Probablty, Networks and Algorthms (PNA) Software Engneerng (SEN) Modellng, Analyss and Smulaton (MAS) Informaton Systems (INS) Copyrght 2007, Stchtng Centrum voor Wskunde en Informatca P.O. Box 94079, 1090 GB Amsterdam (NL) Kruslaan 413, 1098 SJ Amsterdam (NL) Telephone Telefax ISSN

3 Traffc-splttng networks operatng under alpha-far sharng polces and balanced farness ABSTRACT We consder a data network n whch, besdes classes of users that use specfc routes, one class of users can splt ts traffc over several routes. We consder load balancng at the packetlevel, mplyng that traffc of ths class of users can be dvded among several routes at the same tme. Assumng that load balancng s based on an alpha-far sharng polcy, we show that the network has multple possble behavors. In partcular, we show that some classes of users, dependng on the state of the network, share capacty accordng to some Dscrmnatory Processor Sharng (DPS) model, whereas each of the remanng classes of users behaves as n a sngle-class sngle-node model. We compare the performance of ths network wth that of a smlar network, where packet-level load balancng s based on balanced farness. We derve explct expressons for the mean number of users under balanced farness, and show by conductng extensve smulaton experments that these provde accurate approxmatons for the ones under alpha-far sharng Mathematcs Subject Classfcaton: 60K25; 68M20 Keywords and Phrases: Traffc splttng; Alpha-far sharng; Balanced farness Note: Ths research has been funded by the Dutch Bsk/BRICKS (Basc Research n Informatcs for Creatng the Knowledge Socety) project.

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5 Traffc-Splttng Networks Operatng under Alpha-far Sharng Polces and Balanced Farness P. Leshout CWI P.O. Box 94079, 1090 GB Amsterdam, The Netherlands Emal: July 7, 2007 Abstract We consder a data network n whch, besdes classes of users that use specfc routes, one class of users can splt ts traffc over several routes. We consder load balancng at the packet-level, mplyng that traffc of ths class of users can be dvded among several routes at the same tme. Assumng that load balancng s based on an alpha-far sharng polcy, we show that the network has multple possble behavors. In partcular, we show that some classes of users, dependng on the state of the network, share capacty accordng to some Dscrmnatory Processor Sharng (DPS) model, whereas each of the remanng classes of users behaves as n a sngle-class sngle-node model. We compare the performance of ths network wth that of a smlar network, where packet-level load balancng s based on balanced farness. We derve explct expressons for the mean number of users under balanced farness, and show by conductng extensve smulaton experments that these provde accurate approxmatons for the ones under alphafar sharng. 1 1 Ths research has been funded by the Dutch BSIK/BRICKS (Basc Research n Informatcs for Creatng the Knowledge Socety) project. 1

6 1 Introducton The performance of communcaton networks can be mproved when the servce demands are effcently dvded among the avalable resources, so-called load balancng. One can apply ether statc or dynamc load balancng. In the former case the balancng s not affected by the state of the network, whereas n the latter case t does depend on the system state. It s clear that better performance can be acheved when usng dynamc load balancng, but t s often hard to fnd the optmal load balancng polcy. Even for smple systems such a dynamc load balancng problem has non-trval solutons [16]. In ths paper we analyze load balancng n data networks carryng elastc traffc, as consdered by [12]. Transfers n such networks can be represented by flows. We may dstngush between load balancng at the flow-level or the packet-level, dependng on whether an arrvng flow s entrely drected to a specfc route (that t uses untl the flow s fnshed) or a flow can be splt between several routes, respectvely. Ths paper deals wth packet-level load balancng,.e., we assume that packets of a flow can be dvded among several routes. Due to the dynamc nature of traffc, t s n general complcated to analyze the performance of such networks. Flows arrve accordng to some stochastc process and brng along a random amount of work. For each gven number of flows present n the system, the allocated servce rates are determned by some sharng polcy. As soon as the number of flows changes, t s assumed that these rates are adapted nstantly. We analyze a network n whch, besdes classes of users that use specfc routes, one class of users can splt ts traffc over several routes. We note that ths network s the smplest system to analyze the performance and potental gans of load balancng at the packet-level, and t s therefore of partcular nterest to gan nsght. In addton, ths system also accounts for rather explct results. We assume that packet-level load balancng s based on an alpha-far bandwdth sharng polcy as ntroduced n [13]. The famly of alpha-far polces covers several common notons of farness as specal cases, such as max-mn farness (α ), proportonal farness (α 1) and maxmum throughput (α 0). In [14] t has also been shown that the case α = 2, wth addtonal class weghts set nversely proportonal to the respectve round trp tmes, provdes a reasonable modelng abstracton for the bandwdth sharng realzed by TCP (Transmsson Control Protocol) n the Internet. We show that the above network has multple possble behavors. In partcular, we show that packet-level load balancng based on alpha-far sharng mples that some classes of users, dependng on the state of the network, share capacty accordng to some Dscrmnatory Processor Sharng (DPS) model, whereas each of the remanng classes of users behaves as n a sngle-class sngle-node model. The flow-level performance of the above network s compared wth that of a smlar network, where packet-level load balancng s based on balanced farness, so-called nsenstve load balancng at the packet-level. The term nsenstve refers to the fact that the correspondng steady-state dstrbuton depends on the traffc characterstcs through the traffc ntensty only. Insenstve load balancng at the flow-level was frst ntroduced n [4], and extended to nsenstve load balancng at the packet-level n [10]. Optmal nsenstve load balancng at the flow-level utlzng local state nformaton was addressed n [1]. In [8] t was shown that one can acheve 2

7 1 2 L Fgure 1: The bandwdth-sharng network. even better performance f capacty allocaton and load balancng are optmzed jontly. A comparson between packet-level and flow-level nsenstve load balancng was conducted n [11]. Assumng Posson arrvals and exponentally dstrbuted servce requrements, the dynamcs of the flow populaton may be descrbed by a Markov process under both packet-level load balancng polces. We derve closed-form expressons for the mean number of users of each class under nsenstve load balancng. Extensve smulaton experments show that these are also qute accurate approxmatons for the ones n a smlar network where load balancng s based on alpha-far sharng, for whch no explct expressons are avalable. The above results are n lne wth the fndngs of [3], n whch t was shown that the performance of networks operatng under unweghted max-mn farness, unweghted proportonal farness and balanced farness s nearly smlar. The results n ths paper suggest that balanced farness s n fact a reasonable approxmaton for all unweghted alpha-far sharng polces. The remander of ths paper s organzed as follows. In Secton 2 we frst provde a detaled model descrpton, and ntroduce balanced farness and alpha-far sharng. In the next secton we consder the model for a fxed flow populaton, and we characterze how bandwdth s allocated under both polces. In Secton 4 we consder the model at large tme-scales, such that the state of the network vares, and we derve explct expressons for the mean number of users under balanced farness, and show by conductng extensve smulaton experments that these provde accurate approxmatons for the ones under alpha-far sharng. In the next secton we examne the gan than one can acheve for both polces by usng packet-level load balancng nstead of statc or flow-level load balancng. Secton 6 concludes wth some fnal observatons. 2 Model We consder the network as depcted n Fgure 1. The network conssts of L nodes, where node has servce rate C, = 1,...,L. There are L + 1 classes of users. Class requres servce at node, = 1,...,L, whereas class 0 can be served at all nodes at the same tme,.e., class-0 users can splt ther traffc. We assume that class- users arrve accordng to a Posson process of rate λ, and have exponentally dstrbuted servce requrements wth mean µ 1, = 0,...,L. The arrval processes are all ndependent. The traffc load of class s then ρ = λ µ 1. Let n = (n 0,...,n L ) denote the state of the network, wth n representng the number of class- users. 3

8 2.1 Balanced farness We frst assume that the bandwdth s shared accordng to balanced farness, as ntroduced n [4]. Let φ (n) denote the servce rate allocated to class, = 0,...,L, wth balanced farness, when the network s n state n (here φ 0 (n) = L =1 φ 0(n)). These servce rates have to satsfy the balance condtons φ (n e j ) φ (n) = φ j(n e ) φ j (n),j = 0,...,L, n,n j > 0, (1) where e denotes the ( + 1)th unt vector n R L+1. All balanced servce rates can be expressed n terms of a unque balance functon Φ( ), so that Φ(0) = 1 and φ (n) = Φ(n e ) Φ(n) n : n > 0, = 0,...,L. (2) Hence, characterzaton of Φ(n) mples that φ (n) s characterzed as well. Defne Φ(n) = 0 f n / N L+1 0. In order to obtan Φ(n), we need to solve the followng maxmzaton problem for each n N L+1 0 \{0}: (BF) max Φ(n) 1 s.t. L j=1 φ 0j (n) = Φ(n e 0) Φ(n) φ (n) = Φ(n e ), Φ(n) = 1,...,L φ 0 (n) + φ (n) C, = 1,...,L φ 0 (n),φ (n) 0, = 1,...,L. It s clear that Φ(n) can be obtaned recursvely: the Φ(n e )s are requred to determne Φ(n). Also note that (BF) s a smple LP-problem, whch can be solved usng standard LP algorthms. In Secton 3.1, however, we solve (BF) by rewrtng the LP-problem n terms of a related network. 2.2 Alpha-far sharng We next assume that the network operates under a so-called alpha-far sharng polcy, as ntroduced n [13]. When the network s n state n 0, the servce rate x allocated to each of the class- users s obtaned by solvng the followng optmzaton problem: (AF) max G(x) s.t. n 0 x 0 + n x C, = 1,...,L x 0,x 0, = 1,...,L, where the objectve functon G(x) s defned by { ( L =1 n G(x) := 0 κ x 0) 1 α 0 1 α + L =1 n x κ 1 α 1 α f α (0, )\{1}; n 0 κ 0 log( L =1 x 0) + L =1 n κ log(x ) f α = 1. 4

9 The κ s are non-negatve class weghts, and α (0, ) may be nterpreted as a farness coeffcent. The cases α 0, α 1 and α correspond to allocatons whch acheve maxmum throughput, proportonal farness, and max-mn farness, respectvely. The value of x 0 denotes how much capacty s assgned to path (that requres servce at node ) of class 0. Here x 0 = L =1 x 0 denotes how much capacty s assgned to a sngle class-0 user n the network. Let s (n) := x n denote the total servce rate allocated to class, = 0,...,L. 3 Statc settng In ths secton we consder the model for a fxed flow populaton,.e., the state n N L+1 0 \{0} s fxed, and we derve how bandwdth s shared between the varous classes n case of balanced farness and alpha-far sharng, respectvely. The dffculty n solvng problem (BF) and (AF), as presented n the prevous secton, s that no explct expressons are avalable for ther optmal solutons. We frst show that the network depcted n Fgure 1 s equvalent to another network. In order to do so, let us frst ntroduce the noton of the capacty set. The allocatons φ(n) = (φ 0 (n),...,φ L (n)) and s(n) = (s 0 (n),...,s L (n)) are clearly constraned by the capacty set C R L+1 C := x 0 : a 1,...,a L 0, + : L a j = 1, a x 0 + x C, = 1,...,L, j=1.e., φ(n) C and s(n) C for all n N L+1 0. It s straghtforward to show that the capacty set C can also be expressed as L C := x 0 : L x j C j, x C, = 1,...,L, j=0 j=1.e., C = C. Snce C s the capacty set correspondng to the tree network depcted n Fgure 2, t follows that the networks depcted n Fgures 1 and 2 are n fact equvalent. The tree has a common lnk wth capacty C C L, and L + 1 branches wth capactes, C 1,...,C L, respectvely. In ths network class- users requre servce at the node wth servce rate C and at the common lnk, = 1,...,L, whereas class-0 users only requre servce at the common lnk. Note that each class of users corresponds to a specfc route n the tree network. As a sde remark we menton that n general t s not true that a network (where some classes of users can splt ther traffc over several routes at the same tme) can be converted n a tree network. In fact, f we extend the model depcted n Fgure 1 by addng a class of users that requres servce at all L nodes smultaneously, then t s already not possble to represent the network as a tree network. However, we note that n general one may stll be able to convert a traffc-splttng network n some other network (wth dummy nodes) wthout traffc splttng. 3.1 Balanced farness In ths subsecton we derve the balanced farness allocaton by solvng problem (BF). Snce the models depcted n Fgures 1 and 2 are equvalent, t follows that the balance functon Φ( ) 5

10 class 0 C 1 class 1 L =1 C C L 1 classl 1 C L class L Fgure 2: Tree network correspondng to tree network concdes wth Φ( ),.e., Φ( ) = Φ( ), see [3]. In the followng lemma we present the soluton of the optmzaton problem (BF). Lemma 3.1 The balanced farness functon Φ(n) satsfes, wth Φ(0) = 1, { } Φ(n e 1 ) Φ(n) = max,..., Φ(n e L), C 1 C L L =0 Φ(n e ) L =1 C, n N L+1 0 \{0}. (3) Proof: From the above t follows that we can obtan Φ( ) by determnng Φ( ), as they are the same. Subsequently, Φ( ) s obtaned by usng Equaton (2) n [5]. We note that Lemma 3.1 s n agreement wth Equaton (19) n [10]. From Lemma 3.1 t follows that Φ(n) can be obtaned recursvely. The total servce rate allocated to class, = 0,...,L, n each state n N L+1 0 can be obtaned usng Lemma 3.1 and (2). 3.2 Alpha-far sharng In ths subsecton we focus on the alpha-far allocaton, that s obtaned by solvng problem (AF). Smlar to the prevous subsecton, we can obtan the alpha-far allocaton s(n) by determnng the alpha-far allocaton s(n) n the tree network, as both networks are the same, mplyng that s(n) = s(n). In order to obtan s(n) we need to solve the followng maxmzaton problem: (AF 2) max H(x) L s.t. n x =0 n x C, L C, =1 = 1,...,L, x 0, = 0,...,L, (4) 6

11 where the objectve functon H(x) s defned by H(x) := { L =0 n x κ 1 α 1 α f α (0, )\{1}; L =0 n κ log(x ) f α = 1. Below we show that (AF 2) s solvable, but the optmal soluton strongly depends on the state n 0. We present a smple algorthm for obtanng the alpha-far allocaton. Lemma 3.2 The alpha-far allocaton s(n) can be obtaned wth the followng algorthm: Set Stop:=False Set S := {0,...,L} WHILE Stop=False DO Determne the S -class DPS allocaton: s (n) := n IF s (n) C for all S\{0} THEN set Stop:=True ELSE Take any S\{0} such that s (n) > C Set S := S\{ } Set s (n) := C END END j S\{0} C j j S n j j, S Proof: Frst consder the Karush-Kuhn-Tucker (KKT) necessary condtons for problem (AF 2). If x s an optmal soluton to problem (AF2), then there exst constants p 0, = 0,...,L, such that, n 0 κ 0 x α 0 n κ x α n 0 p 0 ; (5) n (p 0 + p ), = 1,...,L; (6) ( L p 0 C =1 ) L n x = 0; (7) =0 p (C n x ) = 0, = 1,...,L. (8) Note that (5) and (6) hold for any α (0, ). Solvng (5)-(8) for (x 0,...,x L ) and (p 0,...,p L ) yelds L L! q=1 q!(l q)! = 2L 1 possble solutons, however, dependng on the state of the network n, only one of the 2 L 1 solutons, x, s such that p 0, = 0,...,L,.e., ths s the optmal soluton for (AF2). For each of the other solutons there exsts at least one Lagrange parameter that s negatve, mplyng that these solutons cannot be optmal. Note that the exstence of a unque optmal soluton x for (AF2) also follows as H(x) s strctly concave and the constrants are lnear. Straghtforward calculus shows that the correspondng alphafar allocaton s (n) = s (n) = n x, = 0,...,L, can be obtaned by the above algorthm. The algorthm reflects that 2 L 1 solutons exst for (5)-(8), but t also shows that only one of these solutons, x, s found after termnaton of the algorthm. The Lagrange parameters 7

12 correspondng to x are such that p = 0 f S\{0}, and p > 0 f / S\{0}, where S s the set obtaned after termnaton of the algorthm. Furthermore, p 0 = 0 f n 0 = 0 and f there exsts an such that n = 0, = 1,...,L, otherwse p 0 > 0. 4 Flow-level dynamcs In the prevous secton we consdered the model for a fxed flow populaton, and we derved expressons for the balanced farness and alpha-far allocatons n each state of the network. In ths secton we analyze the model at suffcently large tme scales. In ths case we also have to take the random nature of the traffc nto account,.e., the state of the network n vares at large tme scales. 4.1 Balanced farness Let N(t) = (N 0 (t),...,n L (t)) denote the state of the network at tme t. Snce we assumed Posson arrvals and exponentally dstrbuted servce requrements, N(t) s a Markov process wth transton rates: q(n,n + e ) = λ ; q(n,n e ) = µ φ (n), = 0,...,L, n case of balanced farness. In [3] t was shown that the process N(t) s stable f there exsts ( ρ 01,..., ρ 0L ) such that L =1 or equvalently, f ρ 0 = ρ 0 and ρ 0 + ρ < C, = 1,...,L, (9) L ρ < =0 L C j and ρ < C, = 1,...,L. (10) j=1 It may be verfed from (1) that the steady-state queue length dstrbuton s gven by π(n) = 1 L G(ρ) Φ(n) ρ n, n N L+1 0, (11) =0 where the normalzaton constant G(ρ) equals G(ρ) = G(ρ 0,...,ρ L ) =... n 0 =0 n L =0 Φ(n) As a sde remark we menton that (11) n fact holds for much more general traffc characterstcs, see [4] for a more detaled treatment. When applyng Lttle s formula we fnd that L =0 ρ n. EN BF = ρ G(ρ) ρ G(ρ) = ρ log G(ρ), = 0,...,L, (12) ρ 8

13 .e., characterzaton of G(ρ) mples that EN BF, = 0,...,L, s known as well. By explotng the results of [6] on tree networks we can determne G(ρ), and t can be verfed that ths results n G(ρ) = 1 1 L =0 ρ L =1 C L =1 ρ 1 L =1 C ( ). (13) L =1 1 ρ C Then by usng (12) we can obtan a closed-form expresson for EN BF, = 0,...,L. The expresson for EN BF, = 1,...,L, s n general qute complcated, n contrast to the expresson for the mean number of class-0 users, whch s gven by EN BF 0 = ρ 0 L =1 C L =0 ρ. From (13) t follows that EN BF, = 0,...,L, s fnte f the stablty condton (10) holds. 4.2 Alpha-far sharng As before, let N(t) = (N 0 (t),...,n L (t)) denote the state of the network at tme t. In case of alpha-far sharng N(t) s a Markov process wth transton rates: q(n,n + e ) = λ ; q(n,n e ) = µ s (n), = 0,...,L. Snce our network s equvalent to the tree network depcted n Fgure 2, t follows from Theorem 1 n [2] that the process N(t) s stable f (9) holds. Lemma 3.2 shows that, dependng on the state of the network n N L+1 0, the network has 2 L 1 possble behavors. Ths llustrates the complcaton of fndng closed-form expressons for the mean number of users of each class. In fact, so far no expressons for the mean number of users are avalable n case of alpha-far sharng. To gan some nsght, we derve n ths secton approxmatons for the mean number of users of each class,.e., EN AF, = 0,...,L. The approxmatons are valdated by means of smulaton experments. We consder the case where the network conssts of L = 2 nodes, but we note that the approxmatons can be extended to the case L > 2 n a smlar fashon. Usng Lemma 3.2 n Secton 3.2, t follows that the network, dependng on the state n, has three possble behavors: () f ( n 1 > C (κ2 ) 1/α ( ) ) 1/α 1 κ0 n 2 + n 0, C 2 κ 1 κ 1 then classes 0 and 2 behave as n a two-class DPS model wth capacty C 2 and relatve weghts, = 0,2, whereas class 1 behaves as an M/M/1 queue wth arrval rate λ 1 and servce rate µ 1 C 1 ; () If n 1 < C 1 C 2 ( κ2 κ 1 ) 1/α n 2 ( κ0 κ 1 ) 1/α n 0, then classes 0 and 1 behave as n a two-class DPS model wth capacty C 1 and relatve weghts, = 0,1, whereas class 2 behaves as an M/M/1 queue wth arrval rate λ 2 and servce 9

14 rate µ 2 C 2 ; () otherwse the network wll behave as n a three-class DPS model wth capacty C 1 + C 2 and relatve weghts, = 0,1,2. If the network were to behave as () all the tme and f ρ 1 < C 1 and ρ 0 + ρ 2 < C 2 (stablty condtons), then by explotng the results of [7] we would obtan ( ) µ 0 ρ EN () 0 = EN () 1 = EN () 2 = EN () 0 = ρ C 2 ρ 0 ρ 2 ρ 1 C 1 ρ 1 ; ρ C 2 ρ 0 ρ 2 ρ C 1 ρ 0 ρ 1 0 µ 0 (C 2 ρ 0 ) + 2 µ 2 (C 2 ρ 2 ) ( ) µ 2 ρ µ 0 (C 2 ρ 0 ) + 2 µ 2 (C 2 ρ 2 ) Lkewse, when the network behaves as () and f ρ 2 < C 2 and ρ 0 +ρ 1 < C 1 (stablty condtons), we fnd ( ) µ 0 ρ EN () 1 = EN () 2 = ρ C 1 ρ 0 ρ 1 ρ 2 C 2 ρ 2. 0 µ 0 (C 1 ρ 0 ) + 1 µ 1 (C 1 ρ 1 ) ( ) µ 1 ρ µ 0 (C 1 ρ 0 ) + 1 µ 1 (C 1 ρ 1 ) If the network behaves as a three-class DPS model,.e., as (), and f ρ 0 + ρ 1 + ρ 2 < C 1 + C 2 (stablty condton), then one can obtan the mean number of users of each class by solvng the followng set of lnear equatons for EN (), = 0,1,2: (C 1 + C 2 )EN () λ 2 j=0 j λ j λ EN() + λ λ EN() j ;. ; ; j µ j + µ = ρ, = 0,1,2, where λ := λ 0 +λ 1 +λ 2, see [7]. In ths case there also exsts a closed-form expresson for EN (), = 0,1,2, but t s complcated. We propose the followng approxmaton: EN AF EN AP, = 0,1,2, where EN0 AP := EN () 0 ; EN1 AP := max{en () 1, EN() 1 }; EN2 AP := max{en () 2, EN () 2 }. It can be verfed that EN0 AP s bounded f ρ 0 +ρ 1 +ρ 2 < C 1 +C 2, EN1 AP s bounded f ρ 1 < C 1 and ρ 0 + ρ 1 + ρ 2 < C 1 + C 2, and EN2 AP s bounded f ρ 2 < C 2 and ρ 0 + ρ 1 + ρ 2 < C 1 + C 2. Hence, the EN AP s are only all bounded f (9) holds,.e., f the process N(t) s also stable. In [3] t was argued that the performance of a network under proportonal farness (α = 1) and max-mn farness (α ) s closely approxmated by that under balanced farness. Therefore, we also propose the followng approxmaton: EN AF EN BF, = 0,1,2. The value of EN BF, = 0,1,2, can be obtaned usng (12), and s ndependent of the value of α. 10

15 γ EN0 AF EN1 AF EN2 AF EN0 AP EN1 AP EN2 AP EN0 BF EN1 BF EN2 BF Table 1: Smulaton results for scenaro I. γ α EN0 AF EN1 AF EN2 AF EN0 AP EN1 AP EN2 AP EN0 BF EN1 BF EN2 BF Table 2: Smulaton results for scenaro II. To examne the accuracy of the above approxmatons we have performed smulaton experments. We consder the settng wth C 1 = C 2 = 1, and we take λ = γ, µ = 1, = 0,1,2, such that ρ 0 = ρ 1 = ρ 2 = γ. We frst consder scenaro I, where κ = 1, = 0,1,2. Subsequently, we consder scenaro II, where κ 0 = 5, κ 1 = 1 and κ 2 = 2. In scenaro II we let the traffc load γ and the alpha-far coeffcent α vary, whereas n scenaro I we only let γ vary, as t can be verfed that the EN AF s and EN AP s are ndependent of the value of α n scenaro I. To ensure stablty we assume that γ < 2 3. The results are reported n Tables 1 and 2. Each reported smulaton value n these (and other) tables s measured over events,.e., arrvals or departures. Remark: We have also determned a 95% confdence nterval (CI) for each lsted smulaton value n ths paper, but these are not presented. We note, however, that the relatve effcency,.e., the rato of the half-length of the CI to the reported smulaton value, s less than 3% for all lsted cases n Tables 1, 2, 5 and 6, and less than 10% for all lsted cases n Tables 3 and 4. Table 1 compares the value of EN AF and EN BF, = 0,1,2, for scenaro I. The results show that EN AF obtaned by smulaton wth the approxmatons EN AP EN AP, = 0,1,2. Also, 11

16 the table shows that EN0 AF EN0 BF and EN AF EN BF, = 1,2. Overall we see that both approxmatons are accurate n case of equal class weghts, especally for low traffc load. Table 2 reports the results correspondng to scenaro II,.e., n case of unequal class weghts. In ths case EN AF and EN AP do depend on the value of α, as s shown n the table. Agan, we see that EN AF EN AP, = 0,1,2. For low traffc loads both approxmatons perform qute well, but for hgh traffc loads we see that the balanced farness approxmaton s less accurate than the other one. Tables 1 and 2 show that EN AF EN AP, = 0,1,2, whch may be explaned as follows. Frst note that the rate allocated to class 1 s smaller than or equal to C 1 at all moments n tme under alpha-far sharng, whereas rate C 1 s contnuously avalable to class 1 n (). Clearly, ths mples that EN1 AF EN () 1. Wth smlar reasonng, we fnd that ENAF 2 EN () 2. Snce class- users cannot be allocated more than C, = 1,2, under alpha-far sharng, whereas n the three-class DPS model the upper bound s C 1 + C 2 for both classes, one may expect that EN AF EN (), = 1,2. For any state n N 3 0 \{0} t can be verfed that the alpha-far allocaton to class 0 s larger or equal than the one obtaned n the three-class DPS model, so one would expect EN0 AF EN () 0 at frst sght. However, recall that we argued that the number of users of classes 1 and 2 n the model operatng under alpha-far sharng wll (on average) be larger than n the three-class DPS model, whch causes that the total servce allocated to class 0 n the model operatng under alpha-far sharng s less than or equal to that n the three-class DPS model,.e., we may also expect EN0 AF EN () 0. The above reasonng ndeed suggests that EN AF EN AP, = 0,1, Flud and quas-statonary regmes To test the performance of the two approxmatons n case of extreme parameter values, we now assume that the flow dynamcs of the varous classes occur on wdely separate tme scales,.e., n flud and quas-statonary regmes. Formally, let λ (r) := λ f (r) and µ (r) := µ f (r), where f (r) represents the tme scale assocated wth class as functon of r, = 0,...,L. Note that the traffc ntensty of class equals ρ (r) := λ (r) /µ (r) = ρ, = 0,...,L, so t s ndependent of r. Let N (r) be the number of class- flows n the r-th system. Before analyzng the qualty of the approxmatons, we frst present the followng useful proposton. Proposton 4.1 Assume that L+1 classes of users share C unts of capacty accordng to DPS, where class has relatve weght κ, = 0,...,L. If f (r)/f 1 (r) 0 as r, = 1,...,L,.e., hgher ndexed classes operate on faster tme scales, then EN ( ) = ρ C L j= ρ 1 + j=0 κ j κ ( C L r=j ρ r ρ ρ ) ( j C L r=j+1 ρ r ), = 0,...,L. Proof: In [9] the above result was already proved for L = 1. For L > 1 the authors showed that EN ( ) j, j = 1,...,L, could be obtaned by determnng EN ( ), = 0,...,j 1,.e., as a recurson. Straghtforward calculus, however, shows that ths recurson reduces to the above result. 12

17 γ EN0 AF EN1 AF EN2 AF EN AP( ) 0 EN AP( ) 1 EN AP( ) 2 EN0 BF EN1 BF EN2 BF Table 3: Results correspondng to the flud and quas-statonary regmes (scenaro I). Let us return to the settng wth L = 2 nodes and L+1 = 3 classes of users. Proposton 4.1 allows us to obtan smple closed-form expressons for E AP, = 0,1,2, when r. Assumng that hgher ndexed classes operate on faster tme scales and that the stablty condtons (10) hold, we fnd EN AP( ) 0 := EN AP( ) 1 := max ρ 0 ; C 1 + C 2 ρ 0 ρ 1 ρ 2 { ρ 1 ρ 1 C 1 + C 2 ρ 1 ρ ρ 0ρ 1, ; C 1 ρ 1 1 (C 1 + C 2 ρ 0 ρ 1 ρ 2)(C 1 + C 2 ρ 1 ρ 2) { } EN AP( ) ρ 2 ρ 2 1 j ρ jρ 2 2 := max, + C 2 ρ 2 C 1 + C 2 ρ 2 j=0 2 (C 1 + C 2 2 r=j ρr)(c1 + C2 2 r=j+1 ρr). In case of equal class weghts, κ = κ, = 0,1,2, t s not hard to see that EN AP( ) 0 = ρ 0 C 1 + C 2 ρ 0 ρ 1 ρ 2 ; { EN AP( ) ρ1 1 = max, C 1 ρ 1 { EN AP( ) ρ2 2 = max, C 2 ρ 2 ρ 1 C 1 + C 2 ρ 0 ρ 1 ρ 2 ρ 2 C 1 + C 2 ρ 0 ρ 1 ρ 2 Clearly, the EN AP( ) s strongly depend on the orderng of the classes wth respect to the tme scales. In case of other orderngs than the one mentoned above one can obtan expressons n a smlar fashon. The accuracy of the approxmatons n the flud and quas-statonary regmes s examned by performng smulaton experments. We take C 1 = C 2 = 1, λ 0 = γ, λ 1 = 10γ, λ 2 = 100γ, µ 0 = 1, µ 1 = 10, µ 2 = 100, so that ρ = γ, = 0,1,2, and assume that hgher ndexed classes operate on faster tme scales. Tables 3 and 4 report the results for scenaro I and II, respectvely. Recall that the EN AF s and EN AP( ) s are ndependent of the value of α n scenaro I, whereas they are senstve to the value of α n scenaro II. The tables show that also n the flud and quas-statonary regmes the approxmatons are promsng. } ; }. } 5 Comparson wth statc and flow-level load balancng In the prevous sectons we consdered load balancng at the packet-level. In ths secton we quantfy how much better packet-level load balancng s than statc and flow-level load balancng. 13

18 γ α EN0 AF EN1 AF EN2 AF EN AP( ) 0 EN AP( ) 1 EN AP( ) 2 EN0 BF EN1 BF EN2 BF Table 4: Results correspondng to the flud and quas-statonary regmes (scenaro II). We consder the same parameter values as n the prevous secton (wthout consderng flud and quas-statonary regmes), and calculate the mean number of users of each class under statc and flow-level load balancng, so that we can make a comparson wth packet-level load balancng. As before, we frst assume that load balancng s based on balanced farness, and subsequently on alpha-far sharng. 5.1 Balanced farness When statc or flow-level load balancng s used, that s based on balanced farness, we now need to keep track of the number of class-0 users at node, = 1,2. Let n 0 denote the number of class- users at node, = 1,2. Then the balance functon s gven by (see [1]) Φ(n) = and we obtan φ 0 (n) = ( n 01 + n 1 n 1 )( n 02 + n 2 n 2 C n 1+n 01 1 C n 2+n 02 2 n 0 n 0 + n C ; φ (n) = ), n n 0 + n C, = 1,2. Hence, at both nodes capacty s shared accordng to egaltaran Processor Sharng (PS). Let us frst consder statc load balancng. Clearly, consderng the symmetrc parameter settng of the prevous secton, the optmal statc polcy s to route class-0 arrvals to node, = 1,2, wth probablty 1 2. Usng the parameter values of the prevous secton, we thus fnd 14

19 γ ENBF st 0 ENBF st 1 ENBF st 2 BF fl EN0 BF fl EN1 BF fl EN2 EN0 BF EN1 BF EN2 BF Table 5: Results for statc, flow-level and packet-level load balancng n case of balanced farness. that class- (class-0) users arrve accordng to a Posson process of rate γ ( 1 2γ) at node, and both class-0 and class- users have exponentally dstrbuted servce requrements wth mean 1, = 1,2. Recallng that C = 1, = 1,2, and snce capacty s shared accordng to PS at both nodes, t s a straghtforward exercse to show that EN BFst := γ 1 3 = 0,1,2, 2γ, where EN0 BFst denotes the mean number of class-0 users n the network (at node 1 or node 2). In Table 5 we report the EN BFst s for dfferent values of the load γ. Usng the closed-form expressons for EN BFst and EN BF, = 0,1,2, t s straghtforward to derve that EN BFst 0 EN BF 0 = 2; EN BFst EN BF = 4 4γ 4 5γ 1, = 1,2, gven that the load γ of each class s smaller than 2 3. In case of flow-level load balancng t s optmal (under the current settng) to route class-0 users to node 1 f n 01 +n 1 < n 02 +n 2, and to node 2 f n 01 +n 1 > n 02 +n 2. If n 01 +n 1 = n 02 +n 2 then an arrvng class-0 user s sent to node wth probablty 1 2, = 1,2. In other words, an arrvng class-0 user should jon the shortest queue, see [15]. Snce no explct expressons are known for the mean number of users EN BFfl of class, = 0,1,2, under flow-level load balancng, we have performed smulaton experments to obtan these values. The results are also reported n Table 5. Table 5 shows that packet-level load balancng outperforms both statc and flow-level load balancng, and flow-level load balancng s better than statc load balancng, as was expected, EN BFst, = 0,1,2. For low values of γ (low loads), the results are qute smlar, but for hgher loads the dfferences become more sgnfcant. We note that these results are n lne wth the fndngs of [11]..e., EN BF EN BFfl 5.2 Alpha-far sharng In case statc or flow-level load balancng s executed through alpha-far sharng, we also need to be aware of the number of class-0 users at nodes 1 and 2. In case n class- users and n 0 class-0 users are present at node, the allocated servce rates are s (n) = n C 0 n 0 + n, s 0(n) = 0 n 0 C 0 n 0 + n, = 1,2. 15

20 γ α ENAF st 0 ENAF st 1 ENAF st 2 AF fl EN0 AF fl EN1 AF fl EN2 EN0 AF EN1 AF EN2 AF Table 6: Results for statc, flow-level and packet-level load balancng n case of alpha-far sharng (scenaro II). Hence, capacty s shared accordng to DPS wth relatve weghts 0 and at node, = 1,2. Agan, due to symmetrc parameter values, n case of statc load balancng t s optmal to route class-0 arrvals to node, = 1,2, wth probablty 1 2. Usng the parameter values of the prevous secton, we thus fnd that class- (class-0) users arrve accordng to a Posson process of rate γ ( 1 2γ) at node, and both class-0 and class- users have exponentally dstrbuted servce requrements wth mean 1, = 1,2. Usng that C = 1, = 1,2, and snce capacty s shared accordng to DPS at both nodes, the results of [7] mply that ( ) EN0 AFst 1 2 := γ γ γ (1 1 2γ) + κ1/α1 (1 γ) + EN AFst := γ γ 1 + ( 1 2 γ 0 0 (1 1 2γ) + κ1/α (1 γ) ) ( ) γ (1 1 2γ) + κ1/α 2 (1 γ), = 1,2. Note that EN AFst = EN BFst, = 0,1,2, n case of equal class weghts. Therefore, we only focus on scenaro II, and these results are shown n Table 6. The optmal flow-level load balancng polcy s as before to jon the shortest queue, see [15]. As no explct expressons for the mean number of users EN AFfl of class, = 0,1,2, are avalable under flow-level load balancng, we resort to smulaton experments to obtan these values. Note that EN AFfl = EN BFfl, = 0,1,2, n case of equal class weghts, so we only report the results correspondng to scenaro II, see Table ;

21 Tables 6 shows that packet-level load balancng performs better than both statc and flowlevel load balancng: EN AF EN AFfl EN AFst, = 0,1,2. Agan, the results seem to vary more n case of hgh values of γ. 6 Concluson We analyzed a network consstng of L nodes, wth L + 1 classes of users. Class- users requre servce at node only, = 1,...,L, whereas class-0 users can splt ther traffc over the L nodes. We consdered load balancng at the packet-level, mplyng that class-0 users can splt ther traffc over the L nodes at the same tme. We assumed that load balancng was based on balanced farness and an alpha-far bandwdth sharng polcy, respectvely. We characterzed how bandwdth s allocated n each state of the network under these two polces. Assumng Posson arrvals and exponentally dstrbuted servce requrements, we derved expressons (approxmatons) for the mean number of users of each class under these two polces. For both polces we also showed that one can acheve sgnfcant performance gans f one performs packet-level load balancng nstead of statc or flow-level load balancng, especally for hghly loaded systems. A topc for further research s extendng the results to a more general network, e.g., so-called lnear networks where some classes can splt ther traffc over multple nodes at the same tme. In ths case t s consderably harder, f possble at all, to derve expressons for the mean number of users of each class under the above-mentoned polces, as the network does not reduce to a tree network. Acknowledgments The author wshes to thank Sem Borst and Mchel Mandjes for valuable dscussons. References [1] T. Bonald, M. Jonckheere, A. Proutère (2004). Insenstve load balancng. In: Proceedngs of the ACM SIGMETRICS/Performance 2004 Conference, New York, USA, [2] T. Bonald, L. Massoulé (2001). Impact of farness on Internet performance. In: Proceedngs of the ACM SIGMETRICS/Performance 2001 Conference, Boston, USA, [3] T. Bonald, L. Massoulé, A. Proutère, J. Vrtamo (2006). A queueng analyss of max-mn farness, proportonal farness and balanced farness. Queueng Systems and Applcatons, 53: [4] T. Bonald, A. Proutère (2003). Insenstve bandwdth sharng n data networks. Queueng Systems, 44: [5] T. Bonald, A. Proutère, J.W. Roberts, J. Vrtamo (2003). Computatonal aspects of balanced farness. In: Proceedngs of the 18th Internatonal Teletraffc Congress, Berln, Germany,

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