Static Fairness Criteria in Telecommunications

Transcription

1 Teknillinen Korkeakoulu ERIKOISTYÖ Teknillisen fysiikan koulutusohjelma Mat-208 Sovelletun matematiikan erikoistyöt Stati Fairness Criteria in Teleommuniations Vesa Timonen, R

2 Stati Fairness Criteria in Teleommuniations Vesa Timonen 9th November 2002 Contents Introdution 2 Network model 2 2 Network setup 3 22 Network setup Fairness riteria 4 4 Basi fairness riteria 4 4 Maximum throughput 4 42 Max-min fairness 5 43 Proportional fairness 6 44 Potential delay minimization 8 5 Fairness riteria with parameters 9 5 Weighted max-min fairness 0 52 Weighted proportional fairness 0 53 Weighted potential delay minimization 6 Utility approah 2 6 General utility funtion 3 62 (p, α)-proportional fairness 4 7 Conlusions 7

3 Introdution Abstrat The main objetive of bandwidth sharing in teleommuniations network is to use all available bandwidth without disrupting the onstraints and maintain a ertain fairness The ahieved fairness depends losely on the used fairness riterion Different fairness riteria favor or disriminate single soures or whole traffi lasses on different basis In this paper the notion of fairness is introdued, essential stati fairness riteria and their relationship is studied A network model is presented and the behavior of different fairness riteria is examined via two simple network setups Introdution Basially any teleommuniation network an be onsidered as a set of links with finite apaities as resoures and a finite number of soures as network users The most simple traffi model is suh that a data flow between a sending soure and the reeiving point oupies a onstant amount of link apaity in all the links through whih it traverses When several sessions have to ompete for the finite resoures or apaity of a network, the rate alloated for soures has to be regulated by some ontrol mehanism to avoid ongestion and to redue paket losses in the network Beause of the finite resoures, the bandwidth share or equally the rate alloation is a ompromise that should be fair, whih leads to the onept of fairness Notion of fairness has no unique definition It may depend on several different session priorities and servie requirements, eg a session an require a minimum guaranteed rate for sending data or has a maximum on allowed network delay It is generally aepted that traffi with the same priority should be treated equally The most simplified definition is to alloate the same share to eah onnetion Different fairness riteria favor or disriminate soures or traffi lasses on different basis The objetive an be to use the network apaity as effetively as possible without onsidering a single soure (the throughput maximization), or on the ontrary, the goal an be to ensure as equal sharing of the resoures as possible (max-min fairness) As a mathematial notion fairness an alternatively be thought of as an optimization problem, where the objetive is to find a rate alloation that minimizes or maximizes a utility funtion speifi for the used fairness riterion In this approah eg a ost for ahieved rate alloation an be easily added to the examination Further, the optimization approah provides a generalization of the onept of fairness In this paper our objetive is to introdue the notion of fairness from the viewpoint of teleommuniations and study essential fairness riteria and their relationship as they are onsidered in some earlier papers, and further, onretize these results via two different example ases Some assumptions and simplifiations are made onerning the network and the traffi The network topology, onsisting of the links with onstant apaities and soures, is fixed and known All soures send elasti traffi with adjustable onstant rate to a single target via a fixed route Round trip times and proessing delays are negleted Thus, this study is limited to stati fairness riteria Organization of the paper is as follows: in Setio the network model of this study and two different example network setups are presented The notion

4 2 Network model 2 of fairness is briefly disussed in Setion 3 and basi fairness riteria are studied in Setion 4 Weighted fairness riteria ontaining some additional parameters are presented in Setion 5 and in Setion 6 fairness as a optimization problem is studied more losely Relation between the riteria presented in previous setions and the utility approah is stated The paper is onluded in Setion 7 2 Network model In this setion the basi network model is presented and two different network setups are introdued In the following network model some assumptions are made; we assume that there is a fixed number of flows and routes in the network Also eah soures sending rate is assumed to be onstant In a way, the used network model is statial, sending rates, number of flows or used routes do not vary in time A network an be onsidered as a set of links L, eah link l L has a finite apaity C l > 0 Let S denote the set of all soures in the network and let denote the sending rate of soure s S A route r is a non-empty subset of L Denote by R the set of all routes used by the soures s S Let l r denote that link l belongs to route r A 0 matrix A is defined as follows: A = (A ls ) () where A ls = l s 2, l L and s S Finite apaities of the links set some restritions to the rate alloations that an be arried out in the network A rate alloation that does not disrupt these limitations is alled feasible Definition (Feasible alloation) A rate alloation } s S is feasible if 0 s S and it satisfies the apaity onstraints A ls C l, l L (2) s S Let x = } s S denote a rate alloation vetor and C = C l } l L a vetor of the link apaities Thus, Definition an be presented in a matrix form as follows: Definitio (Feasible alloation) A rate alloation x is feasible if x 0 and it satisfies the apaity onstraints Ax C (3) Let F denote the set of all feasible rate alloations, ie F def = x x is feasible} The set F is onvex, whih is proved as follows: Proposition Set F is onvex Set R an ontain idential elements, ie several soures an use exatly the same route 2 More generally, we may assume that A ls [0, ], whih orresponds to splitting the traffi over parallel paths [5, 4]

5 2 Network setup 3 Proof Let x and x 2 be two rate alloations suh that x, x 2 F Now, A (tx + ( t)x 2 ) = t Ax }} C x F A (tx + ( t)x 2 ) C t [0, ] tx + ( t)x 2 F t [0, ] F is onvex [, Def 2] +( t) Ax }} 2 tc + ( t)c = C C x 2 F In Setions 2 and 22 two simple linear network setups are introdued These networks are used in the later setions to demonstrate the differenes between various fairness riteria 2 Network setup This network setup (Figure ) is a widely used example to illustrate the omplexity related to fairness (see eg [2]) There are L links and L + traffi lasses R 0,, R L, eah ontaining n r flows, r = 0,, L Flows of lass R 0 travel through all links and flows of lass R l only through link l, l =,, L Figure : The linear network In the following setions Example refers to this network setup with following parameter values: L = 2, C l and n 22 Network setup 2 This network setup (Figure 2) is also known as the parking lot senario There are L links and L traffi lasses R,, R L eah ontaining n r flows, r =,, L Flows of lass R l travel through links l, l+,, L, l =,, L In link l there are flows of lasses R,, R l ompeting of the links finite apaity Figure 2: The linear network (the parking lot senario) In the following setions Example 2 refers to this network setup with following parameter values: L = 2, C = and C 2 =

6 3 Fairness riteria 4 3 Fairness riteria The main objetive of bandwidth sharing is to use all available bandwidth without disrupting the onstraints and maintain a ertain fairness The ahieved fairness depends losely on the used fairness riterion Different fairness riteria favor or disriminate single soures or whole traffi lasses on different basis The objetive an be to use the network apaity as effetively as possible without onsidering a single soure (the throughput maximization), or on the ontrary, the goal an be to ensure as equal sharing of the resoures as possible (max-min fairness) In the following setions various definitions of fairness riteria are introdued It is assumed that all the soures use a fixed route and send with a onstant rate Also the whole network, its resoures and topology is known Examinations of fairness are restrited to the uniast ase, ie flows are traversing from one soure to a single destination Thus, the riteria are onsidered as stati, varying of any omponent in time is exluded from this study 4 Basi fairness riteria The most ommon fairness riteria have no additional parameters As basi fairness riteria are onerned throughput maximization, max-min fairness, proportional fairness and potential delay minimization 4 Maximum throughput A straightforward objetive for bandwidth alloation is to find a feasible rate alloation that maximizes the total throughput [2, 7, 9, 4] In a way, it would be the most effiient way to use the network resoures Formulated as an optimization problem: maximize u(x) = s S (4) subjet to x F Example Let as denote n i x i = s R i, that is, x i is the mean rate alloation for flows in the lass R i, i = 0,, 2 In a link l the alloated rate for lass R l has to satisfy equation n 0 x 0 + n l x l In fat, we learly have n l x l = n 0 x 0 The total throughput of the network is then s S = 2 l=0 n lx l = 2 2n 0 x 0, whih attains its maximum 2 when x 0 = 0 Thus, we have x 0 = 0 and x l = n l, l =, 2 Example 2 As before, all flows in the same lass R r have the same mean rate x r The maximum of the total throughput is = C 2 and it is obtained with all rate alloations x F that fulfill the ondition n x + x 2 = suh that n x This gives 0 x n and x 2 = ( n x ) Thus, there is an infinite number of feasible rate alloations that maximize the throughput, eg

7 42 Max-min fairness 5 rate alloation or x = 0,, 0, /n } } 2,, / } } } n elements elements x 2 = /n,, /n, ( )/,, ( )/ } } } } } n elements elements Still, it an be questioned whether the rate alloation x is as fair as x 2 42 Max-min fairness Max-min fairness [2, 4, 9, 7] is the most ommon definition for the onept of fairness Its objetive is to maximize the minimum of the given bandwidths, ie the rate of any soure an not be inreased without dereasing the rate of some other soure that already has a smaller rate Definition 3 (Max-min fairness) A rate alloation x is max-min fair if x F and s S l s suh that i l x i = C l and = max x i (5) i l The bandwidth alloation fulfilling the Definition 3 an be proved to be unique [9, 7] An alternative definition of max-min fairness is that every soure has a bottlenek link [2, 4] 3 A bottlenek link is defined as follows: Definition 4 (Bottlenek link) A link l L is alled a bottlenek link for soure s S if link l is saturated, ie A ls = C l, l L (6) s S and soure s that sends through link l has the maximum rate among all the other soures using the link l, ie x i x i suh that A li =, i S (7) where denotes the rate alloation of soure s S Example Let x be a max-min fair rate alloation All flows in the same lass R l have the same rate x l, l = 0,, 2 By definition the soures using routes of lass R 0 have a bottlenek link The bottlenek link is the most rowded link, that is, for the bottlenek link k it holds k = arg max l n l = 2 Thus, the following equation holds: n 0 x 0 + x 2 = Sine link 2 is also the bottlenek link of lass R 2, we have x 0 = x 2, from whih it follows that x 0 = n 0 + In 3 The equality of these definitions is proved eg in [2, 4]

8 43 Proportional fairness 6 the other link the exess apaity is used by the flows of lass R Thus, the max-min fair rate alloation is n x r = 0 + for r R 0, n l n 0 + for r R l, l, 2} In the ase n r we have x r 2 and the total throughput is 3 2 Maxmin fairness gives a smaller total throughput than ahieved with throughput maximization, but guarantees that all the lasses are given an equal proportion of the bandwidth available Thus, max-min fairness an be onsidered as a more fair bandwidth sharing objetive than pure maximizing of the total throughput Example 2 Let x be the max-min fair rate alloation All flows in the same lass R l have the same rate x l, l =, 2 By definition the soures using the routes of lass R 2 have a bottlenek link 2 Thus, the following equation holds: n x + x 2 = If the link 2 is also the bottlenek link for flows of lass R, then x = x 2 Thus, x = x 2 = n + Now, if n x >, then the bottlenek link for flows of lass R is link That is, if n < n + and further, if n >, then x = n and x 2 = In the ase n r we have x = x 2 = 2 if n < n2, that is, if < 2 Otherwise x = and x 2 = In both ases max-min fairness gives the maximum total throughput, but as a ontrast to throughput maximization, the rate alloation is unambiguous 43 Proportional fairness Proportional fairness was proposed in [6] In proportional fairness deviation from the fair alloation auses a negative average hange Definition 5 (Proportional fairness) A rate alloation x is proportionally fair if x F and x s 0 x F (8) s S The bandwidth alloation fulfilling the Definition 5 an be proved to be unique [9, 7] Example Let us assume that the proportionally fair alloation is n x r = 0 +n + for r R 0, n + n l n 0 +n + for r R l, l, 2} (9) whih is proved as follows: Proposition The rate alloation x proposed in (9) is proportionally fair

9 43 Proportional fairness 7 Proof Let y F In links and 2 it holds that n 0 y 0 + n l y l s S y s = 2 l=0 n l y l x l x l = n 0 y 0 x 0 x 0 + n y x x + y 2 x 2 x 2 n 0y 0 = n 0x 0 }} }} = n 0y 0 n 0 x 0 n y n x + + x 0 x ( n 0 (y 0 x 0 ) ) x 0 x x 2 (9) = n 0 (y 0 x 0 ) ( n 0 + n + n n ) 2 (n 0 + n + ) n + = 0 y F n 0y 0 }} y 2 = n 0x 0 }} x 2 x 2 n + (n 0 + n + ) Thus, based on Definition 5 rate alloation (9) is proportionally fair In the ase that n r we have x 0 = 3, x = x 2 = 2 3 and the total throughput is Proportional fairness gives a smaller total throughput than ahieved with throughput maximization, but greater than ahieved with max-min fairness Proportional fairness seems to penalize long routes more than max-min fairness with tendeny to ahieve greater total throughput Example 2 Let us assume that the proportionally fair alloation is whih is proved as follows: x = n + if n < n otherwise, and (0) x 2 = n + if n < otherwise,, Proposition The rate alloation x proposed in (0) is proportionally fair Proof Let y F It holds that n y and n y + y 2 s S y s = 2 l= n l y l x l x l = n y x x + y 2 x 2 x 2 n y = n x }} }} = n y n x y 2 x 2 + x x ( 2 (n y n x ) ) x x 2

10 44 Potential delay minimization 8 Now, if n <, it follows from (0) that x = x 2 and further x x 2 0 Otherwise x = n, from whih it follows that (n y }} n x ) 0 and = n } } x x 2 }} 0, = ie s S y s 0 y F Thus, based on Definition 5 rate alloation (0) is proportionally fair In the ase that n r we have x = x 2 = 2 if n < n2, that is, if < 2 Otherwise x = and x 2 = In both ases the total throughput is, that is, same as in all the previous ases 44 Potential delay minimization Objetive in potential delay minimization is to minimize the time delay needed to omplete transfers [7] The time delay is thought to be inversely proportional to the sending rate of the soure Thus, a fair rate alloation x F an be defined as solution to following optimization problem: minimize u(x) = s S subjet to x F () Example Routes inside the same lass R r get equal alloation of the bandwidth x r It is required that n 0 x 0 + n l x l =, from whih it follows that x l = n 0x 0 n l A fair rate alloation minimizes the utility funtion u(x), where u(x) = s S Differentiating gives = n 0 x 0 + n x + x 2 = n 0 x 0 + n2 + 2 n 0 x 0 u = n x 0 x 2 + n 0 0 ( n 0 x 0 ) 2 = 0 ( n 0x 0 ) 2 = x 2 ( 0 n 2 + 2) x 0 = n n2 2 and x l = n l n n n2 2 Thus x r = n 0+ +n n l n 0 + +n2 2 for r R 0, for r R l, l, 2}

11 5 Fairness riteria with parameters 9 In the ase that n r the rate alloation is x 0 = 2+, x r = 2 2+ for r and the total throughput is Thus, the bandwidth alloation given by this riterion is in between when ompared to max-min and proportional fairness Example 2 Routes inside the same lass R r get equal alloation of the bandwidth x r It is required that n x + x 2 =, from whih follows x 2 = n x A fair rate alloation minimizes the utility funtion u(x), where u(x) = s S Differentiating gives = n x + x 2 = n x + u = n 2 x x 2 + n x = 2 n x ( n x ) 2 = 0 ( n x ) 2 = x 2 2 n + and x 2 = n = n + n + if the link is not the bottlenek, ie n < Otherwise we have x = n and x 2 = That is, the solution redues to same as in the previous ases and x = x 2 = n + if n < n n +, if otherwise if n < n2, if otherwise In the ase that n r the total throughput is 5 Fairness riteria with parameters Conept of weighted shares provides generalizations for max-min and proportional fairness and potential delay minimization A weighting fator φ s is assoiated with soure s S Inrease in weight leads to inrease in the reeived share [7, 9] Thus, weights an be used eg to give a higher rate alloation to a single soure in a a speifi flow lass, or further, to all soures of a flow lass In the following examples it is assumed that in Example the weight φ 0 = for flows of lass R 0, φ and φ 2 are the weights for flows of lasses R and R 2, respetively It is also demanded that φ n φ 2 In Example 2 the weight φ = for flows of lass R and φ 2 is the weight for flows of lass R 2

12 5 Weighted max-min fairness 0 5 Weighted max-min fairness Max-min fairness defined with weights [7, 9]: Definition 6 A bandwidth alloation x is max-min fair with weights φ s x F and s S l s suh that x i = C l and x i = max φ s i l φ i i l if The definition leads to a rate alloation, in whih a single soure s is onsidered as φ s soures This is verified with following examples Example All flows in the same lass R l have the same rate x l, l = 0,, 2 Beause φ n φ 2, the link 2 is the bottlenek link By Definition 6 in the bottlenek link x 0 = x 2 φ 2 and n 0 x 0 + x 2 =, from whih it follows that φ x 0 = n 0 +φ 2 and x 2 = φ 2 x 0 = 2 n 0 +φ 2 In link the exess apaity is used by the flows of lass R Thus, the max-min fair rate alloation is n x r = 0 +φ 2 for r R 0, φ 2 n l n 0 +φ 2 for r R l, l, 2} Example 2 All flows in the same lass R l have the same rate x l, l =, 2 Flows of lass R 2 have a bottlenek link 2 Thus, the following equation holds: n x + x 2 = If the link 2 is also the bottlenek link for flows of lass R, then by Definition 6 we have x = x 2 φ 2 Thus, x = n +φ 2 and x 2 = φ 2 x = φ 2 n +φ 2 Now, if n x >, then the bottlenek link for flows of lass R is link That is, if n < n +φ 2 and further, if n > φ 2, then x = n and x 2 = In summary, x = n +φ 2 if n < φ 2 n otherwise, and x 2 = φ2 n +φ 2 if n < φ 2 n otherwise, 52 Weighted proportional fairness Weighted proportional fairness [7, 6] is a generalization of proportional fairness Definition 7 (Weighted proportional fairness) A bandwidth alloation x is proportionally fair with weights φ s if x F and s S φ s x s 0 x F (2) Similarly as in the ase of the weighted max-min fairness, the definition leads to a rate alloation, in whih a single soure s is onsidered as φ s soures This is verified with following results Example The weighted proportionally fair alloation is n x r = 0+φ n +φ 2 for r R 0, φ n +φ 2 n l n 0+φ n +φ 2 for r R l, l, 2} Derivation of this result is presented in Setion 62 (3)

13 53 Weighted potential delay minimization Example 2 The weighted proportionally fair alloation is x = n +φ 2 if n < φ 2 n otherwise, and (4) x 2 = φ2 n +φ 2 if n < φ 2 otherwise, Derivation of this result is presented in Setion Weighted potential delay minimization Potential delay minimization defined with weights: a fair rate alloation x F an be defined as solution to following optimization problem: minimize u(x) = s S subjet to x F φ s (5) Example All flows in the same lass R r get equal alloation of the bandwidth x r It is required that n 0 x 0 + n l x l =, from whih follows x l = n 0x 0 n l A fair rate alloation minimizes the utility funtion (5), where u(x) = s S Differentiating gives φ s = n 0 x 0 + φ n x + φ 2 x 2 = n 0 x 0 + φ + φ 2 2 n 0 x 0 u = n 0 φ + φ 2 2 x 0 x 2 + n 0 0 ( n 0 x 0 ) 2 = 0 ( n 0x 0 ) 2 = x 2 ( 0 φ + φ 2 2) x 0 = n 0 + φ + φ 2 2 and x l = n l φ + φ 2 2 n 0 + φ + φ 2 2 Thus x r = n 0 + φ +φ 2 2 φ +φ2n2 2 n l n 0 + φ +φ 2 2 for r R 0, for r R l, l, 2} Example 2 All flows in the same lass R r get equal alloation of the bandwidth x r It is required that n x + x 2 =, from whih follows x 2 = n x A fair rate alloation minimizes the utility funtion (5), where u(x) = s S φ s = n x + φ 2 x 2 = n x + φ 2 2 n x

14 6 Utility approah 2 Differentiating gives u = n φ 2 2 x x 2 + n ( n x ) 2 = 0 ( n x ) 2 = φ 2 2x 2 x = n + φ2 and x 2 = φ 2 n + φ 2 if the link 2 is the bottlenek, ie n x = Otherwise we have Thus, we have and x = n and x 2 = x = x 2 = n + φ 2 if n < φ 2 n l φ2 otherwise n n + φ 2 <, that is n < φ 2,, n + φ 2 if n < φ 2 otherwise As a differene to the ase of weighted max-min fairness and weighted proportional fairness, the effet of the weights is less signifiant; omparison between previous two examples and the orresponding non-weighted examples shows that with the weighted riterion a single soure s is onsidered as φ s soures 6 Utility approah Utility approah is a more general onept of fairness [4] Every soure s S has a utility funtion u s, where u s ( ) indiates the value to soure s of having the rate Every link l L has a ost funtion g l, where g l (f l ) indiates the ost to the network of supporting an amount of flow f l on link l A utility fair alloation x F is defined as solution to following optimization problem: where maximize H(x) = s S subjet to x F u s ( ) l L g l (f l ) (6) f l = s S A ls Different fairness riteria an be presented with riterion speifi utility and ost funtions, and the fair rate alloation is found by this onstrained optimization

15 6 General utility funtion 3 problem The fairness riteria presented in setions 4 and 5 share a ommon ost funtion g l, that guarantees the feasibility of a rate alloation: 0 forf l C l, g l (f l ) = forf l > C l Disrupting the feasibility onstrain auses an infinite ost, and thus fores the rate alloation into the feasible region 6 General utility funtion The following form of the objetive funtion was introdued in [3], based on the riterion introdued in [8]: u s φ s log x if α =, α(x) = (7) otherwise φ s x α α Loal optimal solution of (6) is unique global optimum, if funtion H : F R : H(x) is stritly onave for all x F and set F = is onvex [, Def 3, Th 342] Convexity of F is proved in Setio Now, let us proof the onavity of funtion H(x) = s S us α( ) in two parts Proposition Let us define funtion f α as follows: log x if α =, f α (x) def = x α α otherwise Funtion f α (x) is stritly onave for all x > 0 and α 0 Proof Funtion f α is stritly onave, if its seond derivative is negative for all x > 0 [, Def 3, Th 338] Let us first examine the ase α = : In ase α we have 2 x 2 log x = < 0 x > 0 x2 2 x α x 2 α = α x α } } < 0 x > 0 α R + } 0 x 0 Thus, funtion f α (x) is stritly onave for all x > 0 and α 0 We have u s α(x) = φ s f α (x), where φ s 0 Multiplying with a positive onstant maintains the onavity, thus also funtion u s α is stritly onave Proposition Let f i } i I be a finite set of stritly onave funtions, f i : F R, where F is a nonempty onveet Let us define funtion g as follows: g : F R : g(x) def = f i (x) i I Funtion g is stritly onave

16 62 (p, α)-proportional fairness 4 Proof Let x, x 2 F Now, g(tx + ( t)x 2 ) = i I > i I = t i I f i (tx + ( t)x 2 ) } } >tf i (x )+( t)f i (x 2 ) f i is stritly onave (tf i (x ) + ( t)f i (x 2 )) f i (x ) + ( t) i I f i (x 2 ) = tg(x ) + ( t)g(x 2 ) t [0, ] x, x 2 F g is stritly onave Now, funtion H is a sum of stritly onave funtions and thereby funtion H(x) is also stritly onave, when x F and F is onvex Thus, we have proved that the solution of (6) is unique global optimum We an define a general utility funtion as follows: u Gen : F R : u Gen (x) def = s S u s α( ) (8) With different values of α and φ s different fairness riteria are ahieved [3, 8] Let us onsider the ase φ s Now, when α 0, optimization problem (6) redues to (4) and the solution is the alloation that maximizes the throughput When α the solution of (6) is proportionally fair [8] In the ase that α 2 optimization problem (6) redues to () and the solution minimizes the potential delay In the ase that α the solution of () is equal with solution of (5) [8] Weighted versions of proportional fairness and potential minimization are ahieved when φ s 62 (p, α)-proportional fairness (p, α)-proportional fairness (presented in [8]) is a generalization of proportional fairness Definition 8 ((p, α)-proportional fairness) A bandwidth alloation x is (p, α)-proportionally fair if x F and where α R + s S p s x s α 0 x F (9) Note that if p s = φ s and α =, (9) redues to (2) Further, if p s and α =, (9) redues to (8) In [8] it is proved that (p, α)-proportionally fair bandwidth alloation is also solution of optimization problem (6), where H = u Gen, as defined in (8) All basi fairness riteria an be derived from (p, α)-proportional fairness and also the weighted potential delay minimization is ahieved when p s and α = 2 An interesting observation is that when α, the (p, α)-proportionally fair

17 62 (p, α)-proportional fairness 5 alloation is max-min fair regardless of the values of parameters p 4 s This is onfirmed in the following two examples Now, let us derive previous results by solving optimization problem (6), where H(x) = u Gen (x) Note, that u α = φ s x α s for all α Thus, it is not required to divide following examinations into two parts Example All flows in the same lass R r get equal alloation of the bandwidth x r It is required that n 0 x 0 + n l x l =, from whih follows x l = n0x0 n l (p, α)-proportionally fair rate alloation maximizes funtion u(x), where u(x) = s S u α ( ) = 2 n i u α (x i ) i=0 = n 0 u α (x 0 ) + n u α ( n0 x 0 n Differentiating gives u x 0 = n 0 x 0 α + n φ + φ 2 ( n0 x 0 ( n0 x 0 n ) α ( n 0 ) ( ) n0 x 0 + n u α ) α ( n ) 0 = n 0 ( x0 α (φ n α + φ 2 α ) ( n 0 x 0 ) α) = 0 ( n 0 x 0 ) α = (φ n α + φ 2 α ) x 0 α n 0 x 0 = α φ n α + φ 2 α x 0 x 0 = n 0 + α φ n α + φ 2 n and x α l = 2 n l ) n ( ) n 0 n 0 + α φ n α + φ 2 n α 2 Thus x r = n l n 0+ α φ n α +φ 2n α for r R 0, 2 α φ n α +φ 2n α 2 n 0 + α φ n α +φ 2 n α for r R l, l, 2} 2 (20) Now, different fairness riteria are ahieved with different values of parameters α and φ s : When α 0 and < φ + φ 2 <, solution (20) maximizes the throughput, ie solution redues to be a solution of (4) If 0 < φ + φ 2 <, letting α 0 leads to solution in whih all the apaity is alloated for lass 0 flows This atually minimizes the total throughput When α, solution (20) redues to (3) giving the weighted proportionally fair alloation Setting φ s gives the non-weighted solution (9) When α 2, solution minimizes the potential delay, ie solution redues to be a solution of (5) Setting φ s gives the solution of the non-weighted riteria () When α and 0 < φ s <, solution (20) is max-min fair 4 Weighted max-min fairness is ahieved as a solution of optimization problem (6), where H(x) =Ps S (φ s) α α and α

18 62 (p, α)-proportional fairness 6 Example 2 All flows in the same lass R r get equal alloation of the bandwidth x r It is required that n x + x 2 =, from whih follows x 2 = n x (p, α)-proportionally fair rate alloation maximizes funtion u(x), where u(x) = s S u α ( ) = n u α (x ) + u α (x 2 ) Differentiating gives ( ) n x = n u α (x ) + u α u x = n x α + φ 2 ( n x ) α ( n ) = n ( x α φ 2 α ( n x ) α) = 0 ( n x ) α = φ 2 α x α n x = α φ 2 x x = n + α and x 2 = φ 2 if the link 2 is the bottlenek, ie n x = Otherwise x = n and x 2 = ( n n + α φ 2 n n + α φ 2 <, that is n < α φ 2 ) Thus, x =, n + α φ 2 if n < α φ 2 n otherwise and (2) x 2 = α φ2, n + α φ 2 if n < α φ 2 otherwise Now, different fairness riteria are ahieved with different values of parameters α and φ s : When α 0, two different solutions are ahieved depending on the value of φ 2 ; if < φ 2 <, all the apaity is given to lass 2 flows, whereas ase 0 < φ 2 < leads to alloation in whih lass flows are given the maximum amount of apaity Nevertheless, solution (2) maximizes the throughput, ie solution redues to be a solution of (4) When α, solution (2) redues to (4) giving the weighted proportionally fair alloation Setting φ s gives the non-weighted solution (0) When α 2, solution minimizes the potential delay, ie solution redues to be a solution of (5) Setting φ s gives the solution of the non-weighted riteria () When α and 0 < φ s <, solution (2) is max-min fair

19 7 Conlusions 7 7 Conlusions In this paper we have presented all the most ommon stati fairness riteria and derived the rate alloations given by these riteria in two simple example network setups We have also examined the relations between different riteria and presented the equality of some fairness riteria and the utility approah, whih generalizes the onept of fairness The main objetive of bandwidth sharing is to use all available bandwidth without disrupting the onstraints and maintain a ertain fairness The ahieved fairness depends losely on the used fairness riterion Different fairness riteria favor or disriminate single soures or whole traffi lasses on different basis As it was verified by the examples, the throughput maximization gives the most effiient rate alloation at the expense of a single soures or some flow lasses The max-min fair alloation an be onsidered as the most fair riterion on the grounds that its objetive is to provide as equal rate alloation as possible to all soures This happens at the expense of the total throughput The proportional fairness penalizes long routes more than max-min fairness with tendeny to ahieve greater total throughput The objetive of potential delay minimization is to minimize the delay needed to omplete transfers and the provided rate alloation is between when ompared to max-min and proportional fairness In the ase of a single bottlenek, all basi riteria provide the same alloation The study in this paper was strongly bounded and a great number of simplifiations was made onerning the network model, traffi and the time sale Fairness in TCP traffi and the relation between different ongestion ontrol shemes and the realized fairness are topial themes onerned in reent researh Thus, this paper an be onsidered as preliminary study for further work

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