Comparison of the Discriminatory Processor Sharing Policies

apport   de recherche ISSN 0249-6399 ISRN INRIA/RR--6475--FR+ENG Thème COM INSTITUT N A TION AL DE RECHERCHE EN INFORMA TIQUE ET EN A UTOMA TIQUE Comparison of the Discriminatory Processor Sharing P olicies Natalia Osipov a N° 6475 June 2007 Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles, BP 93, 0690 2 Sophia Antipolis Cedex (France) Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65 Comparison of the Disriminatory Pro essor Sharing P oliies Natalia Osip o v a ∗ † Thème COM  Systèmes omm unian ts Pro jet MAESTR O Rapp ort de re her he n ° 6475  June 2007  17 pages Abstrat: Disriminatory Pro essor Sharing p oliy in tro dued b y Kleinro  k is of a great in- terest in man y appliation areas, inluding teleomm uniations, w eb appliations and TCP o w mo delling. Under the DPS p oliy the job priorit y is on trolled b y a v etor of w eigh ts. V arying the v etor of w eigh ts, it is p ossible to mo dify the servie rates of the jobs and optimize system  harateristis. In the presen t pap er w e presen t results onerning the omparison of t w o DPS p oliies with dieren t w eigh t v etors. W e sho w the monotoniit y of the exp eted so journ time of the system dep ending on the w eigh t v etor under ertain ondition on the system. Namely , the system has to onsist of lasses with means whi h are quite dieren t from ea h other. F or the lasses with similar means w e suggest to selet the same w eigh ts. Key-w ords: Disriminatory Pro essor Sharing, exp onen tial servie times, optimization. ∗ INRIA Sophia An tip olis, F rane, e-mail: Natalia.Osip o v asophia.inria.fr † The w ork w as supp orted b y F rane T eleom R&D Gran t Mo délisation et Gestion du T ra Réseaux In ternet no. 42937433. Comparaison des p olitiques DPS Résumé : L'ordre de servie DPS (Disriminatory Pro essor Sharing) qui était in tro duit par Kleinro  k est un problème très in téressan t et p eut être appliqué dans b eauoup de domaines omme les téléomm uniations, les appliations w eb et la mo délisation de ux TCP . A v e le DPS, les jobs qui viennen t dans le système son t on trlés par un v eteur de p oids. En mo dian t le v eteur de p oids, il est p ossible de on trler les taux de servie des jobs, donner la priorité à ertaines lasses de jobs et optimiser ertaines aratéristiques du système. Le problème du  hoix des p oids est don très imp ortan t et très diile en raison de la omplexité du système. Dans le présen t papier, nous omparons deux p olitiques DPS a v e les v eteurs de p oids diéren ts et nous présen tons des résultats sur la monotoniité du temps mo y en de servie du système en fontion du v eteur de p oids, sous ertaines onditions sur le système. Le système devrait onsister en plusieurs lasses a v e des mo y ennes très diéren tes. P our les lasses qui on t une mo y enne très pro  he il faut  hoisir les meme p oids. Mots-lés : Disriminatory Pro essor Sharing, le temp de servie exp onen tielle, optimisation. Comp arison of the Disriminatory Pr o  essor Sharing Poliies 3 1 In tro dution The Disriminatory Pro essor Sharing (DPS) p oliy w as in tro dued b y Kleinro  k [ 11 ℄. Under the DPS p oliy jobs are organized in lasses, whi h share a single serv er. The apait y that ea h lass obtains dep ends on the n um b er of jobs urren tly presen ted in all lasses. All jobs presen t in the system are serv ed sim ultaneously at rates on trolled b y the v etor of w eigh ts g k > 0 , k = 1 , . . . , M } , where M is the n um b er of lasses. If there are N j jobs in lass j , then ea h job of this lass is serv ed with the rate g j / P M k =1 g k N k . When all w eigh ts are equal, DPS system is equiv alen t to the standard PS p oliy . The DPS p oliy mo del has reen tly reeiv ed a lot of atten tion due to its wide range of applia- tion. F or example, DPS ould b e applied to mo del o w lev el sharing of TCP o ws with dieren t o w  harateristis su h as dieren t R TT s and pa k et loss probabilities. DPS also pro vides a nat- ural approa h to mo del the w eigh ted round-robin disipline, whi h is used in op erating systems for task s heduling. In the In ternet one an imagine the situation that serv ers pro vide dieren t servie aording to the pa ymen t rates. F or more appliations of DPS in omm uniation net w orks see [2℄, [4 ℄, [5 ℄, [7℄, [12 ℄. V arying DPS w eigh ts it is p ossible to giv e priorit y to dieren t lasses at the exp ense of others, on trol their instan taneous servie rates and optimize dieren t system  harateristis as mean so journ time and so on. So, the prop er w eigh t seletion is an imp ortan t task, whi h is not easy to solv e b eause of the mo del's omplexit y . The previously obtained results on DPS mo del are the follo wing. Kleinro  k in [ 11 ℄ w as rst studying DPS. Then the pap er of F a y olle et al. [ 6℄ pro vided results for the DPS mo del. F or the exp onen tially distributed required servie times the authors obtained the expression of the exp eted so journ time as a solution of a system of linear equations. The authors sho w that indep enden tly of the w eigh ts the slo wdo wn for the exp eted onditional resp onse time under the DPS p oliy tends to the onstan t slo wdo wn of the PS p oliy as the servie requiremen ts inreases to innit y . Rege and Sengupta in [13 ℄ pro v ed a deomp osition theorem for the onditional so journ time. F or exp onen tial servie time distributions in [14 ℄ they obtained higher momen ts of the queue length distribution as the solutions of linear equations system and also pro vided a theorem for the hea vy-tra regime. V an Kessel et al. in [8℄, [10 ℄ study the p erformane of DPS in an asymptoti regime using time saling. F or general distributions of the required servie times the appro ximation analysis w as arried out b y Guo and Matta in [7℄. Altman et al. [2 ℄ study the b eha vior of the DPS p oliy in o v erload. Most of the results obtained for the DPS queue w ere olleted together in the surv ey pap er of Altman et al. [ 1℄. A vra henk o v et al. in [3 ℄ pro v ed that the mean unonditional resp onse time of ea h lass is nite under the usual stabilit y ondition. They determine the asymptote of the onditional so journ time for ea h lass assuming nite servie time distribution with nite v ariane. The problem of w eigh ts seletion in the DPS p oliy when the job size distributions are ex- p onen tial w as studied b y A vra henk o v et al. in [3℄ and b y Kim and Kim in [10 ℄. In [10 ℄ it w as sho wn that the DPS p oliy redues the exp eted so journ time in omparison with PS p oliy when the w eigh ts inrease in the opp osite order with the means of job lasses. Also in [ 10 ℄ the authors form ulate a onjeture ab out the monotoniit y of the exp eted so journ time of the DPS p oliy . The idea of onjeture is that omparing t w o DPS p oliies, one whi h has a w eigh t v etor loser to the optimal p oliy pro vided b y cµ -rule, see [15 ℄, has smaller exp eted so journ time. Using the metho d desrib ed in [10 ℄ in the presen t pap er w e pro v e this onjeture with some restritions on the system parameters. The restritions on the system are su h that the result is true for systems for whi h the v alues of the job size distribution means are v ery dieren t from ea h other. The restrition an b e o v erome b y setting the same w eigh ts for the lasses, whi h ha v e similar means. The ondition on means is a suien t, but not a neessary ondition. It b eomes less strit when the system is less loaded. The pap er is organized as follo ws. In Setion 2 w e giv e general denitions of the DPS p oliy and form ulate the problem of exp eted so journ time minimization. In Setion 3 w e form ulate the RR n ° 6475 4 N. Osip ova main Theorem and pro v e it. In Setion 4 w e giv e the n umerial results. Some te hnial pro ofs an b e found in the App endix. 2 Previous results and problem form ulation W e onsider the Disriminatory Pro essor Sharing (DPS) mo del. All jobs are organized in M lasses and share a single serv er. Jobs of lass k = 1 , . . . , M arriv e with a P oisson pro ess with rate λ k and ha v e required servie-time distribution F k ( x ) = 1 − e − µ k x with mean 1 /µ k . The load of the system is ρ = P M k =1 ρ k and ρ k = λ k /µ k , k = 1 , . . . , M . W e onsider that the system is stable, ρ < 1 . Let us denote λ = P M k =1 λ k . The state of the system is on trolled b y a v etor of w eigh ts g = ( g 1 , . . . , g M ) , whi h denotes the priorit y for the job lasses. If in the lass k there are urren tly N k jobs, then ea h job of lass k is serv ed with the rate equal to g j / P M k =1 g k N k , whi h dep ends on the urren t system state, or on the n um b er of jobs in ea h lass. Let T DP S b e the exp eted so journ time of the DPS system. W e ha v e T DP S = M X k =1 λ k λ T k , where T k are exp eted so journ times for lass k . The expressions for the exp eted so journ times T k , k = 1 , . . . , M an b e found as a solution of the system of linear equations, see [ 6 ℄, T k   1 − M X j =1 λ j g j µ j g j + µ k g k   − M X j =1 λ j g j T j µ j g j + µ k g k = 1 µ k , k = 1 , . . . , M . (1) Let us notie that for the standard Pro essor Sharing system T P S = m 1 − ρ . One of the problems when studying DPS is to minimize the exp eted so journ time T DP S with some w eigh t seletion. Namely , nd g ∗ su h as T DP S ( g ∗ ) = min g T DP S ( g ) . This is a general problem and to simplify it the follo wing sub ase is onsidered. T o nd a set G su h that T DP S ( g ∗ ) ≤ T P S , ∀ g ∗ ∈ G. (2) F or the ase when job size distributions are exp onen tial the solution of (2) is giv en b y Kim and Kim in [10 ℄ and is as follo ws. If the means of the lasses are su h as µ 1 ≥ µ 2 ≥ . . . ≥ µ M , then G onsists of all su h v etors whi h satisfy G = { g | g 1 ≥ g 2 ≥ . . . ≥ g M } . Using the approa h of [10 ℄ w e solv e more general problem ab out the monotoniit y of the exp eted so journ time in the DPS system, whi h w e form ulate in the follo wing setion as Theorem 1. 3 Exp eted so journ time monotoniit y Let us form ulate and pro v e the follo wing Theorem. INRIA Comp arison of the Disriminatory Pr o  essor Sharing Poliies 5 Theorem 1. L et the job size distribution for every lass b e exp onential with me an µ i , i = 1 , . . . , M and we enumer ate them in the fol lowing way µ 1 ≥ µ 2 ≥ . . . ≥ µ M . L et us  onsider two dier ent weight p oliies for the DPS system, whih we denote as α and β . L et α, β ∈ G , or α 1 ≥ α 2 ≥ . . . ≥ α M , β 1 ≥ β 2 ≥ . . . ≥ β M . The exp e te d sojourn time of the DPS p oliies with weight ve tors α and β satises T DP S ( α ) ≤ T DP S ( β ) , (3) if the weights α and β ar e suh that: α i +1 α i ≤ β i +1 β i , i = 1 , . . . , M − 1 , (4) and the fol lowing r estrition is satise d: µ j +1 µ j ≤ 1 − ρ, (5) for every j = 1 , . . . , M . Remark 2. If for some lasses j and j + 1  ondition (5 ) is not satise d, then in pr ati e, by ho osing the weights of these lasses to b e e qual, we  an stil l use The or em 1. Namely, for lasses suh as µ j +1 µ j > 1 − ρ , we suggest to set α j +1 = α j and β j +1 = β j . Remark 3. The or em 1 shows that the exp e te d sojourn time T DP S ( g ) is monotonous a  or ding to the sele tion of weight ve tor g . The loser is the weight ve tor to the optimal p oliy, pr ovide d by cµ -rule, the smal ler is the exp e te d sojourn time. This is shown by the  ondition ( 4 ), whih shows that ve tor α is loser to the optimal cµ -rule p oliy then ve tor β . The or em 1 is pr ove d with r estrition (5). This r estrition is a suient and not a ne  essary  ondition on system p ar ameters. It shows that the me ans of the job lasses have to b e quite dier ent fr om e ah other. This r estrition  an b e over  ome, giving the same weights to the job lasses, whih me an values ar e similar. Condition ( 5) b e  omes less strit as the system b e  omes less lo ade d. T o pro v e Theorem 1 let us rst giv e some notations and pro v e additional Lemmas. Let us rewrite linear system (1) in the matrix form. Let T ( g ) = [ T ( g ) 1 , . . . , T ( g ) M ] T b e the v etor of T ( g ) k , k = 1 , . . . , M . Here b y [ ] T w e mean transp ose sign, so [ ] T is a v etor. By [ ] ( g ) w e note that this elemen t dep ends on the w eigh t v etor seletion g ∈ G . Let us onsider that later in the pap er v etors g , α, β ∈ G , if the opp osite is not notied. Let dene matries A ( g ) and D ( g ) in the follo wing w a y . A ( g ) =      λ 1 g 1 µ 1 g 1 + µ 1 g 1 λ 2 g 2 µ 1 g 1 + µ 2 g 2 . . . λ M g M µ 1 g 1 + µ M g M λ 1 g 1 µ 2 g 2 + µ 1 g 1 λ 2 g 2 µ 2 g 2 + µ 2 g 2 . . . λ M g M µ 2 g 2 + µ M g M . . . λ 1 g 1 µ M g M + µ 1 g 1 λ 2 g 2 µ M g M + µ 2 g 2 . . . λ M g M µ M g M + µ M g M      (6) D ( g ) =      P i λ i g i µ 1 g 1 + µ i g i 0 . . . 0 0 P i λ i g i µ 2 g 2 + µ i g i . . . 0 . . . 0 0 . . . P i λ i g i µ M g M + µ i g i      (7) RR n ° 6475 6 N. Osip ova Then (1) b eomes ( E − D ( g ) − A ( g ) ) T ( g ) =  1 µ 1 . . . . 1 µ M  T . (8) W e need to nd the exp eted so journ time of the DPS system T DP S ( g ) . A ording to the denition of T DP S ( g ) and equation (8) w e ha v e T DP S ( g ) = 1 λ [ λ 1 , . . . , λ M ] T ( g ) = 1 λ [ λ 1 , . . . , λ M ]( E − D ( g ) − A ( g ) ) − 1  1 µ 1 , . . . , 1 µ M  T . (9) Let us onsider the ase when λ i = 1 for i = 1 , . . . , M . This results an b e extended for the ase when λ i are dieren t, w e pro v e it follo wing the approa h of [10 ℄ in Prop osition 10 at the end of the urren t Setion. Equation (9) b eomes T DP S ( g ) = 1 ′ ( E − D ( g ) − A ( g ) ) − 1 [ ρ 1 , . . . , ρ M ] T λ − 1 . (10) Let us giv e the follo wing notations. σ ( g ) ij = g j µ i g i + µ j g j . Then σ ( g ) ij ha v e the follo wing prop erties. Lemma 4. σ ( g ) ij and σ ( g ) j i satisfy σ ( g ) ij g i = σ ( g ) j i g j , σ ( g ) ij µ i + σ ( g ) j i µ j = 1 µ i µ j . (11) Pr o of. F ollo ws from the denition of σ ( g ) ij . Then matries A ( g ) and D ( g ) giv en b y (6) and (7) an b e rewritten in the terms of σ ( g ) ij . A ( g ) i,j = σ ( g ) ij , i, j = 1 , . . . , M , D ( g ) i,i = X j σ ( g ) ij , i = 1 , . . . , M , D ( g ) i,j = 0 , i, j = 1 , . . . , M , i 6 = j. F or w eigh t v etors α, β the follo wing Lemma is true. Lemma 5. If α and β satisfy (4), then α j α i ≤ β j β i , i = 1 , . . . , M − 1 , ∀ j ≥ i. (12) Pr o of. Let us notie that if a < b and c < d , then ac < bd when a, b , c, d are p ositiv e. Also if j > i then there exist su h l > 0 that j = i + l . Then α i +1 α i ≤ β i +1 β i , α i +2 α i +1 ≤ β i +2 β i +1 , . . . α i + l α i + l − 1 ≤ β i + l β i + l − 1 , i = 1 , . . . , M − 2 . Multiplying left and righ t parts of the previous inequalities w e get the follo wing: α i + l α i ≤ β i + l β i , i = 1 , . . . , M − 2 , whi h pro v es Lemma 5. INRIA Comp arison of the Disriminatory Pr o  essor Sharing Poliies 7 Lemma 6. If α and β satisfy (12), then σ ( α ) ij ≤ σ ( β ) ij , i ≤ j, σ ( α ) ij ≥ σ ( β ) ij , i ≥ j. Pr o of. As (12 ) then α j α i ≤ β j β i , i ≤ j, α j µ i β i ≤ β j µ i α i , i ≤ j, α j ( µ i β i + µ j β j ) ≤ β j ( µ i α i + µ j α j ) , i ≤ j, α j µ i α i + µ j α j ≤ β j µ i β i + µ j β j , i ≤ j, σ ( α ) ij ≤ σ ( β ) ij , i ≤ j. W e pro v e the seond inequalit y of Lemma 6 in a similar w a y . Lemma 7. If α , β satisfy (4), then T DP S ( α ) ≤ T DP S ( β ) , when the elements of ve tor y = 1 ′ ( E − B ( α ) ) − 1 M ar e suh that y 1 ≥ y 2 ≥ . . . ≥ y M . Pr o of. Let us denote B ( g ) = A ( g ) + D ( g ) , g = α, β . Then as ( 10 ) T DP S ( g ) = λ − 1 1 ′ ( E − B ( g ) ) − 1 [ ρ 1 , . . . , ρ M ] T , g = α, β . F ollo wing the metho d desrib ed in [10℄ w e get the follo wing. T DP S ( α ) − T DP S ( β ) = λ − 1 1 ′ ( E − B ( α ) ) − 1 [ ρ 1 , . . . , ρ M ] T − λ − 1 1 ′ ( E − B ( β ) ) − 1 [ ρ 1 , . . . , ρ M ] T = = λ − 1 1 ′ (( E − B ( α ) ) − 1 − ( E − B ( β ) ) − 1 ) [ ρ 1 , . . . , ρ M ] T = = λ − 1 1 ′ (( E − B ( α ) ) − 1 ( B ( α ) − B ( β ) )( E − B ( β ) ) − 1 ) [ ρ 1 , . . . , ρ M ] T . Let us denote M as a diagonal matrix M = diag ( µ 1 , . . . , µ M ) and y = 1 ′ ( E − B ( α ) ) − 1 M . (13) Then T DP S ( α ) − T DP S ( β ) = 1 ′ ( E − B ( α ) ) − 1 M M − 1 ( B ( α ) − B ( β ) ) T ( β ) = = y M − 1 ( B ( α ) − B ( β ) ) T ( β ) = = X i,j  y j µ j σ ( α ) j i + y i µ i σ ( α ) ij −  y j µ j σ ( β ) j i + y i µ i σ ( β ) ij  T ( β ) j = = X i,j y j σ ( α ) j i µ j − σ ( β ) j i µ j ! + y i µ i ( σ ( α ) ij − σ ( β ) ij ) ! T ( β ) j . As (11 ): σ ( g ) j i µ j = 1 µ i µ j − σ ( g ) ij µ i , g = α, β , RR n ° 6475 8 N. Osip ova then T DP S ( α ) − T DP S ( β ) = X i,j − y j σ ( α ) ij µ i − σ ( β ) ij µ i ! + y i µ i ( σ ( α ) ij − σ ( β ) ij ) ! T ( β ) j = = X i,j  − y j µ i  σ ( α ) ij − σ ( β ) ij  + y i µ i ( σ ( α ) ij − σ ( β ) ij )  T ( β ) j = = X i,j   σ ( α ) ij − σ ( β ) ij  ( y i − y j ) 1 µ i  T ( β ) j . Using Lemma 6 w e get  σ (1) ij − σ (2) ij  ( y i − y j ) is negativ e for i, j = 1 , . . . , M when y 1 ≥ y 2 ≥ . . . ≥ y M . This pro v es the statemen t of Lemma 7 . Lemma 8. V e tor y given by (13 ) satises y 1 ≥ y 2 ≥ . . . ≥ y M , if the fol lowing is true: µ i +1 µ i ≤ 1 − ρ, for every i = 1 , . . . , M . Pr o of. The pro of ould b e found in the app endix. Remark 9. F or the job lasses suh as µ i +1 µ i > 1 − ρ we pr ove that to make y i ≥ y i +1 it is suient to set the weights of these lasses e qual, α i +1 = α i . Com bining the results of Lemmas 5 , 6 , 7 and 8 w e pro v e the statemen t of the Theorem 1. Remark 9 giv es the Remark 2 after Theorem 1 . Prop osition 10. The r esult of The or em 1 is extende d to the  ase when λ i 6 = 1 . Pr o of. Let us rst onsider the ase when all λ i = q , i = 1 , . . . , M . It an b e sho wn that for this ase the pro of of Theorem 1 is equiv alen t to the pro of of the same Theorem but for the new system with λ ∗ i = 1 , µ ∗ i = q µ i , i = 1 , . . . , M . F or this new system the results of Theorem 1 is eviden tly true and restrition (5) is not  hanged. Then, Theorem 1 is true for the initial system as w ell. If λ i are rational, then they ould b e written in λ i = p i q , where p i and q are p ositiv e in tegers. Then ea h lass an b e presen ted as p i lasses with equal means 1 /µ i and in tensit y 1 /q . So, the DPS system an b e onsidered as the DPS system with p 1 + . . . + p K lasses with the same arriv al rates 1 /q . The result of Theorem 1 is extended on this ase. If λ i , i = 1 , . . . , M are p ositiv e and real w e apply the previous ase of rational λ i and use on tin uit y . 4 Numerial results Let us onsider a DPS system with 3 lasses. Let us onsider the set of normalized w eigh ts v etors g ( x ) = ( g 1 ( x ) , g 2 ( x ) , g 3 ( x )) , P 3 i =1 g i ( x ) = 1 , g i ( x ) = x − i / ( P 3 i =1 x − i ) , x > 1 . Ev ery p oin t x > 1 denotes a w eigh t v etor. V etors g ( x ) , g ( y ) satisfy prop ert y (4) when 1 < y ≤ x , namely g i +1 ( x ) /g i ( x ) ≤ g i +1 ( y ) /g i ( y ) , i = 1 , 2 , 1 < y ≤ x . On Figures 1, 2 w e plot T DP S with w eigh ts v etors g ( x ) as a funtion of x , the exp eted so journ times T P S for the PS p oliy and T opt for the optimal cµ -rule p oliy . On Figure 1 w e plot the exp eted so journ time for the ase when ondition ( 5) is satised for three lasses. The parameters are: λ i = 1 , i = 1 , 2 , 3 , µ 1 = 160 , µ 2 = 14 , µ 3 = 1 . 2 , then ρ = 0 . 91 1 . INRIA Comp arison of the Disriminatory Pr o  essor Sharing Poliies 9 1 2 3 4 5 6 7 8 9 10 3.1 3.15 3.2 3.25 3.3 3.35 3.4 3.45 3.5 T DPS (g(x)) T PS T opt Figure 1: T DP S ( g ( x )) , T P S , T opt funtions, ondition satised. 1 2 3 4 5 6 7 8 9 10 3.7 3.72 3.74 3.76 3.78 3.8 3.82 3.84 3.86 3.88 3.9 T opt T PS T DPS (g(x)) Figure 2: T DP S ( g ( x )) , T P S , T opt funtions, ondition not satised On Figure 2 w e plot the exp eted so journ time for the ase when ondition ( 5 ) is not satised for three lasses. The parameters are: λ i = 1 , i = 1 , 2 , 3 , µ 1 = 3 . 5 , µ 2 = 3 . 2 , µ 3 = 3 . 1 , then ρ = 0 . 9 2 . One an see that T DP S ( g ( x )) ≤ T DP S ( g ( y )) , 1 < y ≤ x ev en when the restrition (5) is not satised. 5 Conlusion W e study the DPS p oliy with exp onen tial job size distributions. One of the main problems studying DPS is the exp eted so journ time minimization aording to the w eighs seletion. In the presen t pap er w e ompare t w o DPS p oliies with dieren t w eigh ts. W e sho w that the exp eted so journ time is smaller for the p oliy with the w eigh v etor loser to the optimal p oliy v etor, pro vided b y cµ -rule. So, w e pro v e the monotoniit y of the exp eted so journ time for the DPS p oliy aording to the w eigh t v etor seletion. The result is pro v ed with some restritions on system parameters. The found restritions on the system parameters are su h that the result is true for systems su h as the mean v alues of the job lass size distributions are v ery dieren t from ea h other. W e found, that to pro v e the main result it is suien t to giv e the same w eigh ts to the lasses with similar means. The found restrition is a suien t and not a neessary ondition on a system parameters. When the load of the system dereases, the ondition b eomes less strit. A  kno wledgmen t I w ould lik e to thank K. A vra henk o v, P . Bro wn and U. A y esta for fruitful disussions and sug- gestions. 6 App endix In the follo wing pro of in the notations w e do not use the dep endeny of the parameters on g to simplify the notations. W e onsider that v etor g ∈ G , or g 1 ≥ g 2 . . . ≥ g M . T o simplify the notations let us use P k instead of P M k =1 . Lemma 8. V etor y = 1 ′ ( E − B ) − 1 M satises y 1 ≥ y 2 ≥ . . . ≥ y M , RR n ° 6475 10 N. Osip ova if the follo wing is true: µ i +1 µ i ≤ 1 − ρ, for ev ery i = 1 , . . . , M . Pr o of. Using the results of the follo wing Lemmas w e pro v e the statemen t of Lemma 8 and giv e the pro of for Remark 9 . Let us giv e the follo wing notations ˜ µ = µ T ( E − D ) − 1 , (14) ˜ A = M − 1 AM ( E − D ) − 1 . (15) Let us notie the follo wing ( E − D ) − 1 j = 1 1 − P k g k µ j g j + µ k g k = 1 1 − ρ + P k µ j g j µ k ( µ j g j + µ k g k ) > 0 , j = 1 , . . . , M , ˜ A ij = µ j g j µ i ( µ i g i + µ j g j ) 1 − P k g k µ j g j + µ k g k = µ j g j µ i ( µ i g i + µ j g j ) 1 − ρ + P k µ j g j µ k ( µ j g j + µ k g k ) > 0 , i, j = 1 , . . . , M Let us giv e the follo wing notation f ( x ) = X k x µ k ( x + µ k g k ) . Then ( E − D ) − 1 j = 1 1 − ρ + f ( µ j g j ) , j = 1 , . . . , M , ˜ A ij = µ j g j µ i ( µ i g i + µ j g j )(1 − ρ + f ( µ j g j )) , i, j = 1 , . . . , M . Let us rst pro v e additional Lemma. Lemma 11. Matrix ˜ A = M − 1 AM ( E − D ) − 1 is a p ositive  ontr ation. Pr o of. Matrix ˜ A is a p ositiv e op erator as elemen ts of matries M and A are p ositiv e and elemen ts of matrix ( E − D ) − 1 are p ositiv e. Let Ω = { X | x i ≥ 0 , i = 1 , . . . , M } . If X ∈ Ω , then ˜ AX ∈ Ω . Then to pro v e that matrix ˜ A is a on tration it is enough to sho w that ∃ q , 0 < q < 1 , || ˜ AX || ≤ q || X || , ∀ X ∈ Ω . (16) As X ∈ Ω , then w e an tak e || X || = 1 ′ X = P i x i . Then 1 ′ ˜ AX = X j x j X i ˜ A ij = X j x j P i µ j g j µ i ( µ j g j + µ i g i ) (1 − ρ + g ( µ j g j )) = = X j x j f ( µ j g j ) 1 − ρ + f ( µ j g j ) = X j x j  1 − 1 − ρ 1 − ρ + f ( µ j g j )  = = X j x j − (1 − ρ ) X j x j 1 − ρ + f ( µ j g j ) . INRIA Comp arison of the Disriminatory Pr o  essor Sharing Poliies 11 Let us nd the v alue of q , whi h satises ondition (16). 1 ′ ˜ AX ≤ 1 ′ X , X j x j − (1 − ρ ) X j x j 1 − ρ + f ( µ j g j ) ≤ q X j x j 1 − (1 − ρ ) P j x j 1 − ρ + f ( µ j g j ) P j x j ≤ q . As f ( µ j g j ) > 0 then 0 < 1 − (1 − ρ ) P j x j 1 − ρ + f ( µ j g j ) P j x j < 1 . Let us dene δ in the follo wing w a y: δ = 1 1 − ρ + max j f ( µ j g j ) < P j x j 1 − ρ + f ( µ j g j ) P j x j . Then 1 − (1 − ρ ) P j x j 1 − ρ + f ( µ j g j ) P j x j < 1 − (1 − ρ ) δ. Let us notie that max j f ( µ j g j ) alw a ys exists as the v alues of µ j g j , j = 1 , . . . , M are nite. Then w e an selet q = 1 − (1 − ρ ) δ, 0 < q < 1 . Whi h ompletes the pro of. Lemma 12. If y (0) 1 = [0 , . . . , 0 ] , (17) y ( n ) = ˜ µ + y ( n − 1) ˜ A, n = 1 , 2 , . . . , (18) then y ( n ) → y , when n → ∞ . Pr o of. Let us presen t y in the follo wing w a y . As B = E − A − D , then y = 1( E − B ) − 1 M , y M − 1 ( E − D − A ) = 1 , y M − 1 ( E − D ) = − y M − 1 A + 1 , y ( E − D ) − 1 M = − y M − 1 A ( E − D ) − 1 M + 1 ( E − D ) − 1 M . As matrixes D and M are diagonal, the M D = D M and then y = µ T ( E − D ) − 1 + y M − 1 AM ( E − D ) − 1 , where µ = [ µ 1 , . . . , µ M ] . A ording to notations (14 ) and (15) w e ha v e the follo wing y = ˜ µ + y ˜ A. Let us denote y ( n ) = [ y ( n ) 1 , . . . , y ( n ) 1 ] , n = 0 , 1 , 2 , . . . and let dene y (0) 1 and y ( n ) b y (17 ) and ( 18 ). A ording to Lemma 11 reetion ˜ A is a p ositiv e reetion and is a on tration. Also ˜ µ i are p ositiv e. Then y ( n ) → y , when n → ∞ and w e pro v e the statemen t of Lemma 12 . RR n ° 6475 12 N. Osip ova Lemma 13. L et y ( n ) is dene d by (18 ) and y (0) is given by (17), then y ( n ) 1 ≥ y ( n ) 2 ≥ . . . ≥ y ( n ) M , n = 1 , 2 , . . . (19) if µ i +1 µ i ≤ 1 − ρ for every i = 1 , . . . , M . Pr o of. W e pro v e the statemen t (19 ) b y indution. F or y (0) the statemen t (19 ) is true. Let us assume that (19 ) is true for the ( n − 1 ) step, y ( n − 1) 1 ≥ y ( n − 1) 2 ≥ . . . ≥ y ( n − 1) M . T o pro v e the indution statemen t w e ha v e to sho w that y ( n ) 1 ≥ y ( n ) 2 ≥ . . . ≥ y ( n ) M , whi h is equal to that y ( n ) j ≥ y ( n ) p , if j ≤ p . As y ( n ) j = ˜ µ j + M X i =1 y ( n − 1) i ˜ A ij , then y ( n ) j − y ( n ) p = ˜ µ j + M X i =1 y ( n − 1) i ˜ A ij − ˜ µ p + M X i =1 y ( n − 1) i ˜ A ip ! = = ˜ µ j − ˜ µ p + M X i =1 y ( n − 1) i ( ˜ A ij − ˜ A ip ) . T o sho w that y ( n ) j − y ( n ) p w e need to sho w that ˜ µ j − ˜ µ p ≥ 0 and P M i =1 y ( n − 1) i ( ˜ A ij − ˜ A ip ) ≥ 0 , when j ≤ p . Let us sho w that to pro v e that P M i =1 y ( n − 1) i ( ˜ A ij − ˜ A ip ) ≥ 0 , j ≤ p it is enough to pro v e that P r i =1 ( ˜ A ij − ˜ A ip ) ≥ 0 , j ≤ p , r = 1 , . . . , M . If w e regroup this sum w e an get the follo wing M X i =1 y ( n − 1) i ( ˜ A ij − ˜ A ip ) = M X i =1 ( y ( n − 1) i − y ( n − 1) i +1 + y ( n − 1) i +1 − . . . − y ( n − 1) M + y ( n − 1) M )( ˜ A ij − ˜ A ip ) = = M − 1 X i =1 ( y ( n − 1) i − y ( n − 1) i +1 ) h ( ˜ A 1 j − ˜ A 1 p ) + ( ˜ A 2 j − ˜ A 2 p ) + . . . + ( ˜ A ij − ˜ A ip ) i + + y ( n − 1) M (( ˜ A 1 j − ˜ A 1 p ) + . . . + ( ˜ A ( M − 1) j − ˜ A ( M − 1) p ) + ( ˜ A M j − ˜ A M p )) = = M − 1 X i =1 ( y ( n − 1) i − y ( n − 1) i +1 ) r X k =1 ( ˜ A kj − ˜ A kp ) + y ( n − 1) M M X k =1 ( ˜ A kj − ˜ A kp ) . As y ( n − 1) i ≥ y ( n − 1) i +1 , i = 1 , . . . , M , aording to the indution step, then to sho w that P M i =1 y ( n − 1) i ( ˜ A ij − ˜ A ip ) ≥ 0 , j ≤ p it is enough to sho w that P r i =1 ( ˜ A ij − ˜ A ip ) ≥ 0 , j ≤ p , r = 1 , . . . , M . W e sho w this in Lemma 15 . In Lemma 14 w e sho w ˜ µ j ≥ ˜ µ p , j ≤ p , when µ i +1 µ i ≤ 1 − ρ for ev ery i = 1 , . . . , M . Then w e pro v e the indution statemen t and so pro v e the statemen t of Lemma 13 . Lemma 14. ˜ µ 1 ≥ ˜ µ 2 . . . ≥ ˜ µ M , (20) if µ i +1 µ i ≤ 1 − ρ for every i = 1 , . . . , M . INRIA Comp arison of the Disriminatory Pr o  essor Sharing Poliies 13 Pr o of. Let us ompare ˜ µ j and ˜ µ p , j ≤ p . If µ j = µ p and g j = g p , then ˜ µ j = ˜ µ p and (20 ) is satised. Let us denote f 2 ( x ) = X k g k x + µ k g k , whi h has the follo wing prop erties 0 < f 2 ( x ) < ρ. (21) Then ˜ µ i = µ i 1 − P j g j µ i g i + µ j g j = µ i 1 − f 2 ( µ i g i ) . Let us nd ˜ µ j − ˜ µ p = µ j 1 − f 2 ( µ j g j ) − µ p 1 − f 2 ( µ p g p ) = µ j − µ p − ( µ j f 2 ( µ p g p ) − µ p f 2 ( µ j g j )) (1 − f 2 ( µ j g j ))(1 − f 2 ( µ p g p )) . As (21 ) then µ j f 2 ( µ p g p ) − µ p f 2 ( µ j g j ) < µ j ρ. Then ˜ µ j − ˜ µ p > ( µ j − µ p ) (1 − f 2 ( µ j g j )(1 − f 2 ( µ p g p )))  1 − ρ  µ j µ j − µ p  = = ( µ j − µ p ) (1 − f 2 ( µ j g j )(1 − f 2 ( µ p g p ))) 1 − ρ 1 1 − µ p µ j !! ≥ 0 , when µ p µ j ≤ 1 − ρ. So, if µ p µ j ≤ 1 − ρ and g j ≥ g p , then ˜ µ j ≥ ˜ µ p . Let us sho w that if µ j > µ p and g j = g p , then ˜ µ j ≥ ˜ µ p . In this ase ˜ µ j − ˜ µ p = µ j 1 − f 2 ( µ j g j ) − µ p 1 − f 2 ( µ p g p ) = = µ j − µ p − ( µ j f 2 ( µ p g j ) − µ p f 2 ( µ j g j )) (1 − f 2 ( µ j g j ))(1 − f 2 ( µ p g j )) = ∆ 1 (1 − f 2 ( µ j g j ))(1 − f 2 ( µ p g j )) . Let us nd when ∆ 1 > 0 . ∆ 1 = µ j − µ p − µ j M X k =1 g k µ p g j + µ k g k − µ p M X k =1 g k µ j g j + µ k g k ! = = µ j − µ p − M X k =1 g k ( g j ( µ 2 j − µ 2 p ) + µ k g k ( µ j − µ p )) ( µ p g j + µ k g k )( µ p g j + µ k g k ) ! = ( µ j − µ p ) 1 − M X k =1 g k ( g j ( µ j + µ p ) + µ k g k ) ( µ p g j + µ k g k )( µ p g j + µ k g k ) ! . RR n ° 6475 14 N. Osip ova As 0 < µ j µ p g 2 j , k = 1 , . . . , M , g k µ k ( g j ( µ j + µ p ) + µ k g k ) < ( µ j µ p g 2 j + g j ( µ j + µ p ) µ k g k + µ 2 k g 2 k ) , k = 1 , . . . , M g k µ k ( g j ( µ j + µ p ) + µ k g k ) < ( µ j g j + µ k g k )( µ p g j + µ k g k ) , k = 1 , . . . , M g k ( g j ( µ j + µ p ) + µ k g k ) ( µ j g j + µ k g k )( µ p g j + µ k g k ) < 1 µ k , k = 1 , . . . , M . Then ∆ 1 > ( µ j − µ p ) 1 − M X k =1 1 µ k ! = 1 − ρ > 0 . Then w e pro v ed the follo wing: If µ j = µ p , g j = g p , then ˜ µ j = ˜ µ p , If µ j > µ p , g j = g p , then ˜ µ j > ˜ µ p , If µ p µ j ≤ 1 − ρ, g j ≥ g p , then ˜ µ j ≥ ˜ µ p . Setting p = j + 1 and remem b ering that µ 1 ≥ . . . ≥ µ M , w e get that ˜ µ 1 ≥ ˜ µ 2 . . . ≥ ˜ µ M is true when µ i +1 µ i ≤ 1 − ρ for ev ery i = 1 , . . . , M . That pro v es the statemen t of Lemma 14 . Returning ba k to the main Theorem 1 , Lemma 14 giv es ondition (5) as a restrition on a system parameters. Let us notie that for the job lasses su h for whi h the means are su h as µ i +1 µ i < 1 − ρ , if the w eigh ts giv en for these lasses are equal, then still ˜ µ i ≥ ˜ µ i +1 . This ondition giv es us as a result Remark 9 and Remark 2 . Lemma 15. r X i =1 ˜ A i 1 ≥ r X i =1 ˜ A i 2 ≥ . . . ≥ r X i =1 ˜ A iM , r = 1 , . . . , M . Pr o of. Let us remem b er ˜ A = M − 1 AM ( E − D ) − 1 . Then as ρ = P M k =1 1 µ k , then r X i =1 ˜ A ij = P r i =1 µ j g j µ i ( µ j g j + µ i g i ) 1 − P M k =1 g k µ j g j + µ k g k = P r i =1 µ j g j µ i ( µ j g j + µ i g i ) 1 − ρ + P M k =1 µ j g j µ k ( µ j g j + µ k g k ) Let us dene f 3 ( x ) = P r i =1 x µ i ( x + µ i g i ) 1 − ρ + P M k =1 x µ k ( x + µ k g k ) = h 1 ( x ) 1 − ρ + h 1 ( x ) + h 2 ( x ) , where h 1 ( x ) = r X i =1 x µ i ( x + µ i g i ) > 0 , h 2 ( x ) = M X j = r +1 x µ j ( x + µ j g j ) > 0 . Let us sho w that f 3 ( x ) is inreasing on x . F or that it enough to sho w that d f 3 ( x ) dx ≥ 0 . Let us onsider d f 3 ( x ) dx = h ′ 1 ( x )(1 − ρ ) + h ′ 1 ( x ) h 2 ( x ) − h 1 ( x ) h ′ 2 ( x ) (1 − ρ + h 1 ( x ) + h 2 ( x )) 2 INRIA Comp arison of the Disriminatory Pr o  essor Sharing Poliies 15 Sine h ′ 1 ( x ) > 0 and 1 − ρ > 0 : d f 3 ( x ) dx ≥ 0 if h ′ 1 ( x ) h 2 ( x ) − h 1 ( x ) h ′ 2 ( x ) ≥ 0 . Let us onsider h ′ 1 ( x ) h 2 ( x ) − h 1 ( x ) h ′ 2 ( x ) = r X i =1 g i ( x + µ i g i ) 2 M X k = r +1 x µ k ( x + µ k g k ) − r X i =1 x µ i ( x + µ i g i ) M X k = r +1 g k ( x + µ k g k ) 2 = = r X i =1 M X k = r +1  g i x ( x + µ i g i ) 2 ( x + µ k g k ) µ k − g k x µ i ( x + µ i g i )( x + µ k g k ) 2  = = r X i =1 M X k = r +1 x ( x + µ i g i )( x + µ k g k )  g i µ k ( x + µ i g i ) − g k µ i ( x + µ k g k )  = = r X i =1 M X k = r +1 x ( x + µ i g i )( x + µ k g k )  µ i g i ( x + g k µ k ) − µ k g k ( x + µ i g i ) µ i µ k ( x + µ k g k )( x + µ i g i )  = = r X i =1 M X k = r +1 x 2 ( µ i g i − µ k g k ) ( x + µ i g i ) 2 ( x + µ k g k ) 2 µ k µ i ≥ 0 , Then d f 3 ( x ) dx ≥ 0 and f 3 ( x ) is an inreasing funtion of x . As µ j g j ≥ µ p g p , j ≤ p , then w e pro v e the statemen t of Lemma 15 . Referenes [1℄ E. Altman, K. A vra henk o v, U. A y esta, A surv ey on disriminatory pro essor sharing, Queue- ing Systems, V olume 53, Num b ers 1-2, June 2006 , pp. 53-63(11). [2℄ E. Altman, T. Jimenez, and D. K ofman, DPS queues with stationary ergo di servie times and the p erformane of TCP in o v erload, in Pro eedings of IEEE Info om, Hong-K ong, Mar h 2004. [3℄ K. A vra henk o v, U. A y esta, P . Bro wn, R. Nunez-Queija, Disriminatory Pro essor Sharing Revisited, INF OCOM 2005. 24th Ann ual Join t Conferene of the IEEE Computer and Com- m uniations So ieties. Pro eedings IEEE, V ol. 2 (2005), pp. 784-795 v ol. 2. [4℄ T. Bu and D. T o wsley , Fixed p oin t appro ximation for TCP b eha viour in an A QM net w ork, in Pro eedings of A CM SIGMETRICS/P erformane, pp. 216-225, 2001. [5℄ S. K. Cheung, J. L. v an den Berg, R.J. Bou herie, R. Litjens, and F. Roijers, An analytial pa k et/o w-lev el mo delling approa h for wireless LANs with qualit y-of-servie supp ort, in Pro eedings of ITC-19, 2005. [6℄ G. F a y olle, I. Mitrani, R. Iasnogoro dski, Sharing a Pro essor Among Man y Job Classes, Journal of the A CM (JA CM),V olume 27, Issue 3 (July 1980), pp. 519 - 532, 1980, ISSN:0004- 5411. [7℄ L. Guo and I. Matta, S heduling o ws with unkno wn sizes: appro ximate analysis, in Pro eed- ings of the 2002 A CM SIGMETRICS in ternational onferene on Measuremen t and mo deling of omputer systems, 2002, pp. 276-277. [8℄ G. v an Kessel, R. Nunez-Queija, S. Borst, Dieren tiated bandwidth sharing with disparate o w sizes, INF OCOM 2005. 24th Ann ual Join t Conferene of the IEEE Computer and Com- m uniations So ieties. Pro eedings IEEE, V ol. 4 (2005), pp. 2425-2435 v ol. 4. RR n ° 6475 16 N. Osip ova [9℄ G. v an Kessel, R. Nunez-Queija, S. Borst, Asymptoti regimes and appro ximations for dis- riminatory pro essor sharing, SIGMETRICS P erform. Ev al. Rev., v ol. 32, pp. 44-46, 2004. [10℄ B. Kim and J. Kim, Comparison of DPS and PS systems aording to DPS w eigh ts, Com- m uniations Letters, IEEE, v ol. 10, issue 7, pp. 558-560, ISSN:1089-7798, July 2006. [11℄ L. Kleinro  k, Time-shared systems: A theoretial treatmen t, J.A CM, v ol.14, no. 2, pp. 242-261, 1967. [12℄ Y. Ha y el and B. T un, Priing for heterogeneous servies at a disriminatory pro essor sharing queue, in Pro eedings of Net w orking, 2005. [13℄ K. M. Rege, B. Sengupta, A deomp osition theorem and related results for the disriminatory pro essor sharing queue, Queueing Systems, Springer Netherlands ISSN 0257-0130, 1572-9443, Issue V olume 18, Num b ers 3-4 / Septem b er, 1994, pp. 333-351, 2005. [14℄ K. M. Rege, B. Sengupta, Queue-Length Distribution for the Disriminatory Pro essor- Sharing Queue, Op erations Resear h, V ol. 44, No. 4 (Jul. - Aug., 1996), pp. 653-657. [15℄ Rhonda Righ ter, "S heduling", in M. Shak ed and J. Shan thikumar (eds), "Sto  hasti Orders", 1994. INRIA Comp arison of the Disriminatory Pr o  essor Sharing Poliies 17 Con ten ts 1 In tro dution 3 2 Previous results and problem form ulation 4 3 Exp eted so journ time monotoniit y 4 4 Numerial results 8 5 Conlusion 9 6 App endix 9 RR n ° 6475 Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France) Unité de reche rche INRIA Futurs : Parc Club Orsay Uni versité - ZAC des V ignes 4, rue Jacques Monod - 91893 ORSA Y Cedex (Franc e) Unité de reche rche INRIA Lorraine : LORIA, T echnopôle de Nancy -Brabois - Campus scientifique 615, rue du Jardin Botani que - BP 101 - 54602 V illers-lè s-Nancy Cedex (France) Unité de reche rche INRIA Rennes : IRISA, Campus univ ersitai re de Beaulie u - 35042 Rennes Cedex (Franc e) Unité de reche rche INRIA Rhône-Alpes : 655, aven ue de l’Europe - 38334 Montbonnot Saint-Ismie r (France) Unité de recherch e INRIA Rocquen court : Domaine de V oluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France) Éditeur INRIA - Domaine de V olucea u - Rocquenc ourt, BP 105 - 78153 Le Chesnay Cedex (France) http://www.inria.fr ISSN 0249 -6399 apport   technique