Optimal Choice of Threshold in Two Level Processor Sharing

apport   de recherche ISSN 0249-6399 ISRN INRIA/RR--6215--FR+ENG Thème COM INSTITUT N A TION AL DE RECHERCHE EN INFORMA TIQUE ET EN A UTOMA TIQUE Optimal Choice of Threshold in T w o Lev el Processor Sharing K onstantin A v rachenko v — Pa trick Brown — Natalia Osipova N° 6215 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 Optimal Choie of Threshold in T w o Lev el Pro essor Sharing K onstan tin A vra henk o v ∗ , P atri k Bro wn † , Natalia Osip o v a ‡ § Thème COM  Systèmes omm unian ts Pro jet MAESTR O Rapp ort de re her he n ° 6215  June 2007  19 pages Abstrat: W e analyze the T w o Lev el Pro essor Sharing (TLPS) s heduling disipline with the h yp er-exp onen tial job size distribution and with the P oisson arriv al pro ess. TLPS is a on v enien t mo del to study the b enet of the le size based dieren tiation in TCP/IP net w orks. In the ase of the h yp er-exp onen tial job size distribution with t w o phases, w e nd a losed form analyti expression for the exp eted so journ time and an appro ximation for the optimal v alue of the threshold that minimizes the exp eted so journ time. In the ase of the h yp er-exp onen tial job size distribution with more than t w o phases, w e deriv e a tigh t upp er b ound for the exp eted so journ time onditioned on the job size. W e sho w that when the v ariane of the job size distribution inreases, the gain in system p erformane inreases and the sensitivit y to the  hoie of the threshold near its optimal v alue dereases. Key-w ords: T w o Lev el Pro essor sharing, Hyp er-Exp onen tial distribution, Laplae transform. ∗ INRIA Sophia An tip olis, F rane, e-mail: K.A vra henk o vsophia.inria.fr † F rane T eleom R&D, F rane, e-mail: P atri k.Bro wnorange-ftgroup.om ‡ 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. Le Choix du Seuil Optimal p our La File d'A tten te Munie d'une P olitique à T emps P artagé A v e Deux-Niv eaux Résumé : Nous étudions la le d'atten te m unie d'une p olitique a T emps P artagé a v e Deux- Niv eaux "T w o Lev el Pro essor Sharing" a v e distribution h yp er- exp onen tielle des temps de servie et a v e le pro essus d'arriv ée P oisson. TLPS est un mo dèle ommo de p our ordonner l'aès aux ressoures en fontion de la taille dans un réseau TCP/IP . Dans le as où la distribution du temps de servie est une distribution h yp er- exp onen tielle a v e deux phases, nous trouv ons une expression analytique p our le temps de rép onse mo y en. Aussi nous trouv ons une appro ximation de v aleur de seuil optimal qui réduit au minim um le temps de rép onse mo y en. Dans le as où la distribution du temps de servie a plus que deux phases, nous trouv ons une b orne sup érieure p our la fontion de temps de rép onse mo y en qui est onditionnée au temps de servie. Nous mon trons que quand la v ariane de la distribution des temps de servie augmen te, le gain dans l'exéution du système est onsidérable et il n'y a pas de sensibilité au  hoix du seuil sous optimal. Mots-lés : La le d'atten te m unie d'une p olitique à T emps P artagées a v e Deux-Niv eaux, réseau TCP/IP , distribution h yp er-exp onen tielle, Laplae transformen t. Optimal Choi e of Thr eshold in Two L evel Pr o  essor Sharing 3 1 In tro dution It has b een kno wn for a long time that a lev er s heduling of tasks an signian tly impro v e system p erformane. F or instane, Shortest Remaining Pro essing Time (SRPT) s heduling dis- ipline minimizes the exp eted so journ time [15 ℄. Ho w ev er, SRPT requires to k eep tra k of all jobs in the system and also requires the kno wledge of the remaining pro essing times. These requiremen ts are often not feasible in appliations. The examples of su h appliations are le size based dieren tiation in TCP/IP net w orks [3 , 9℄ and W eb serv er request dieren tiation [10 , 11 ℄. The T w o Lev el Pro essor Sharing (TLPS) s heduling disipline [12 ℄ helps to o v erome the ab o v e men tioned requiremen ts. It uses the dieren tiation of jobs aording to a threshold on the attained servie and giv es priorit y to the jobs with small sizes. A detail desription of the TLPS disipline is presen ted in the ensuing setion. Of ourse, TLPS pro vides a sub-optimal me hanism in omparison with SRPT. Nev ertheless, as w as sho wn in [1 ℄, when the job size distribution has a dereasing hazard rate, the p erformane of TLPS with appropriate  hoie of threshold is v ery lose to optimal. It turns out that the distribution of le sizes in the In ternet indeed has a dereasing hazard rate and often ould b e mo deled with a hea vy-tailed distributions. It is kno wn, that the hea vy-tailed distribution ould b e appro ximated with a h yp er-exp onen tial distribution with a signian t n um b er of phases [ 5, 8℄. Also in [ 7℄, it w as sho wn that the h yp er-exp onen tial distribution mo dels w ell the le size distribution in the In ternet. Therefore, in the presen t w ork w e analyze the TLPS system with h yp er-exp onen tial job size distribution. The pap er organization and main results are as follo ws. In Setion 2 w e pro vide the mo del form ulation, main denitions and equations. In Setion 3 w e study the TLPS disipline in the ase of the h yp er-exp onen tial job size distribution with t w o phases. It is kno wn that the In ternet onnetions b elong to t w o distint lasses with v ery dieren t sizes of transfer. The rst lass is omp osed of short HTTP onnetions and P2P signaling onnetions. The seond lass orre- sp onds to do wnloads (PDF les, MP3 les, MPEG les, et.). This fat pro vides motiv ation to onsider rst the h yp er-exp onen tial job size distribution with t w o phases. W e nd an analytial expression for the exp eted so journ time in the TLPS system. Then, w e presen t the appro ximation of the optimal threshold whi h minimizes the exp eted so journ time. W e sho w that the appro ximated v alue of the threshold tends to the optimal threshold when the seond momen t of the job size distribution funtion go es to innit y . W e sho w that the use of the TLPS s heduling disipline an lead to 45% gain in the exp eted so journ time in omparison with the standard Pro essor Sharing. W e also sho w that the system p erformane is not to o sensitiv e to the  hoie of the threshold around its optimal v alue. In Setion 4 w e analyze the TLPS disipline when the job size distribution is h yp er-exp onen tial with man y phases. W e pro vide an expression of the exp eted onditional so journ time as the solution of a system of linear equations. Also w e apply an iteration metho d to nd the expression of the exp eted onditional so journ time and using the resulting expression obtain an expliit and tigh t upp er b ound for the exp eted so journ time funtion. In the exp erimen tal results w e sho w that the relativ e error of the latter upp er b ound with resp et to the exp eted so journ time funtion is 6-7%. W e study the prop erties of the exp eted so journ time funtion when the parameters of the job size distribution funtion are seleted in a su h a w a y that with the inreasing n um b er of phases the v ariane inreases. W e sho w n umerially that with the inreasing n um b er of phases the relativ e error of the found upp er b ound dereases. W e also sho w that when the v ariane of the job size distribution inreases the gain in system p erformane inreases and the sensitivit y of the system to the seletion of the optimal threshold v alue dereases. W e put some te hnial pro ofs in the App endix. RR n ° 6215 4 K. A vr ahenkov, P. Br own, N. Osip ova 2 Mo del desription 2.1 Main denitions W e study the T w o Lev el Pro essor Sharing (TLPS) s heduling disipline with the h yp er- exp onen tial job size distribution. Let us desrib e the mo del in detail. Jobs arriv e to the system aording to a P oisson pro ess with rate λ . W e measure the job size in time units. Sp eially , as the job size w e dene the time whi h w ould b e sp en t b y the serv er to treat the job if there w ere no other jobs in the system. Let θ b e a giv en threshold. The jobs in the system that attained a servie less than θ are assigned to the high priorit y queue. If in addition there are jobs with attained servie greater than θ , su h a job is separated in to t w o parts. The rst part of size θ is assigned to the high priorit y queue and the seond part of size x − θ w aits in the lo w er priorit y queue. The lo w priorit y queue is serv ed when the high priorit y queue is empt y . Both queues are serv ed aording to the Pro essor Sharing (PS) disipline. Let us denote the job size distribution b y F ( x ) . By F ( x ) = 1 − F ( x ) w e denote the omple- men tary distribution funtion. The mean job size is giv en b y m = R ∞ 0 xdF ( x ) and the system load is ρ = λm . W e assume that the system is stable ( ρ < 1 ) and is in steady state. It is kno wn that man y imp ortan t probabilit y distributions asso iated with net w ork tra are hea vy-tailed. In partiular, the le size distribution in the In ternet is hea vy-tailed. A distribution funtion has a hea vy tail if e ǫx (1 − F ( x )) → ∞ as x → ∞ , ∀ ǫ > 0 . The hea vy- tailed distributions are not only imp ortan t and prev alen t, but also diult to analyze. Often it is helpful to ha v e the Laplae transform of the job size distribution. Ho w ev er, there is eviden tly no on v enien t analyti expression for the Laplae transforms of the P areto and W eibull distributions, the most ommon examples of hea vy-tailed distributions. In [5 , 8℄ it w as sho wn that it is p ossible to appro ximate hea vy-tailed distributions b y h yp er-exp onen tial distribution with a signian t n um b er of phases. A h yp er-exp onen tial distribution F N ( x ) is a on v ex om bination of N exp onen ts, 1 ≤ N ≤ ∞ , namely , F N ( x ) = 1 − N X i =1 p i e − µ i x , µ i > 0 , p i ≥ 0 , i = 1 , ..., N , and N X i =1 p i = 1 . (1) In partiular, w e an onstrut a sequene of h yp er-exp onen tial distributions su h that it on v erges to a hea vy-tailed distribution [5℄. F or instane, if w e selet p i = ν i γ 1 , µ i = η i γ 2 , i = 1 , ..., N , γ 1 > 1 , γ 1 − 1 2 < γ 2 < γ 1 − 1 , where ν = 1 / P i =1 ,..,N i − γ 1 , η = ν / m P i =1 ,...,N i γ 2 − γ 1 , then the rst momen t of the job size distribution is nite, but the seond momen t is innite when N → ∞ . Namely , the rst and the seond momen ts m and d for the h yp er-exp onen tial distribution are giv en b y: m = Z ∞ 0 x dF ( x ) = N X i =1 p i µ i , d = Z ∞ 0 x 2 dF ( x ) = 2 N X i =1 p i µ 2 i . (2) Let us denote F i θ = p i e − µ i θ , i = 1 , ..., N . (3) W e note that P i F i θ = F ( θ ) . The h yp er-exp onen tial distribution has a simple Laplae transform: L F ( x ) ( s ) = N X i =1 p i µ i s + µ i . INRIA Optimal Choi e of Thr eshold in Two L evel Pr o  essor Sharing 5 W e w ould lik e to note that the h yp er-exp onen tial distribution has a dereasing hazard rate. In [1 ℄ it w as sho wn, that when a job size distribution has a dereasing hazard rate, then with the seletion of the threshold the exp eted so journ time of the TLPS system ould b e redued in omparison to standard PS system. Th us, in our w ork w e use h yp er-exp onen tial distributions to represen t job size distribution funtions. In partiular, the appliation of the h yp er-exp onen tial job size distribution with t w o phases is motiv ated b y the fat that in the In ternet onnetions b elong to t w o distint lasses with v ery dieren t sizes of transfer. The rst lass is omp osed of short HTTP onnetions and P2P signaling onnetions. The seond lass orresp onds to do wnloads (PDF les, MP3 les, MPEG les, et.). So, in the rst part of our pap er w e lo ok at the ase of the h yp er-exp onen tial job size distribution with t w o phases and in the seond part of the pap er w e study the ase of more than t w o phases. 2.2 The exp eted so journ time in TLPS system Let us denote b y T T LP S ( x ) the exp eted onditional so journ time in the TLPS system for a job of size x . Of ourse, T T LP S ( x ) also dep ends on θ , but for exp eted onditional so journ time w e only emphasize the dep endene on the job size. On the other hand, w e denote b y T ( θ ) the o v erall exp eted so journ time in the TLPS system. Here w e emphasize the dep endene on θ as later w e shall optimize the o v erall exp eted so journ time with resp et to the threshold v alue. T o alulate the exp eted so journ time in the TLPS system w e need to alulate the time sp en t b y a job of size x in the rst high priorit y queue and in the seond lo w priorit y queue. F or the jobs with size x ≤ θ the system will b eha v e as the standard PS system where the servie time distribution is trunated at θ . Let us denote b y X n θ = Z θ 0 ny n − 1 F ( y ) dy (4) the n -th momen t of the distribution trunated at θ . In the follo wing setions w e will need X 1 θ = m − N X i =1 F i θ µ i , X 2 θ = 2 N X i =1 p i µ 2 i − 2 θ m − N X i =1 F i θ µ i ! − 2 N X i =1 F i θ µ 2 i . (5) The utilization fator for the trunated distribution is ρ θ = λ X 1 θ = ρ − λ N X i =1 F i θ µ i . (6) Then, the exp eted onditional resp onse time is giv en b y T T LP S ( x ) =      x 1 − ρ θ , x ∈ [0 , θ ] , W ( θ ) + θ + α ( x − θ ) 1 − ρ θ , x ∈ ( θ , ∞ ) . A ording to [12 ℄, here ( W ( θ ) + θ ) / (1 − ρ θ ) expresses the time needed to rea h the lo w priorit y queue. This time onsists of the time θ/ (1 − ρ θ ) sp en t in the high priorit y queue, where the o w is serv ed up to the threshold θ , plus the time W ( θ ) / (1 − ρ θ ) whi h is sp en t w aiting for the high priorit y queue to empt y . Here W ( θ ) = λX 2 θ / (2(1 − ρ θ )) . The remaining term α ( x − θ ) / (1 − ρ θ ) is the time sp en t in the lo w priorit y queue. T o nd α ( x ) w e an use the in terpretation of the lo w er priorit y queue as a PS system with bat h arriv als [4, 14 ℄. As w as sho wn in [ 12 ℄, α ′ ( x ) = dα/dx is the solution of the follo wing in tegral equation α ′ ( x ) = λn Z ∞ 0 α ′ ( y ) B ( x + y ) dy + λn Z x 0 α ′ ( y ) B ( x − y ) dy + bB ( x ) + 1 . (7) RR n ° 6215 6 K. A vr ahenkov, P. Br own, N. Osip ova Here n = F ( θ ) / (1 − ρ θ ) is the a v erage bat h size, B ( x ) = F ( θ + x ) /F ( θ ) is the omplemen tary trunated distribution and b = b ( θ ) = 2 λF ( θ )( W ( θ ) + θ ) / (1 − ρ θ ) is the a v erage n um b er of jobs that arriv e to the lo w priorit y queue in addition to the tagged job. The exp eted so journ time in the system is giv en b y the follo wing equations: T ( θ ) = Z ∞ 0 T T LP S ( x ) dF ( x ) , T ( θ ) = X 1 θ + W ( θ ) F ( θ ) 1 − ρ θ + 1 1 − ρ θ T B P S ( θ ) , (8) T B P S ( θ ) = Z ∞ θ α ( x − θ ) dF ( x ) = Z ∞ 0 α ′ ( x ) F ( x + θ ) dx. (9) INRIA Optimal Choi e of Thr eshold in Two L evel Pr o  essor Sharing 7 3 Hyp er-exp onen tial job size distribution with t w o phases 3.1 Notations In the rst part of our w ork w e onsider the h yp er-exp onen tial job size distribution with t w o phases. Namely , aording to (1) the um ulativ e distribution funtion F ( x ) for N = 2 is giv en b y F ( x ) = 1 − p 1 e − µ 1 x − p 2 e − µ 2 x , where p 1 + p 2 = 1 and p 1 , p 2 > 0 . The mean job size m , the seond momen t d , the parameters F i θ , X 1 θ , X 2 θ and ρ θ are dened as in Setion 2.1 and Setion 2.2 b y form ulas (2),(3),(5), (6) with N = 2 . W e note that the system has four free parameters. In partiular, if w e x µ 1 , ǫ = µ 2 /µ 1 , m , and ρ , the other parameters µ 2 , p 1 , p 2 and λ will b e funtions of the former parameters. 3.2 Expliit form for the exp eted so journ time T o nd T T LP S ( x ) w e need to solv e the in tegral equation (7). T o solv e (7) w e use the Laplae transform based metho d desrib ed in [6 ℄. Theorem 1. The exp e te d sojourn time in the TLPS system with the hyp er-exp onential job size distribution with two phases is given by T ( θ ) = X 1 θ + W ( θ ) F ( θ ) 1 − ρ θ + m − X 1 θ 1 − ρ + b ( θ )  µ 1 µ 2 ( m − X 1 θ ) 2 + δ ρ ( θ ) F 2 ( θ )  2(1 − ρ ) F ( θ )  µ 1 + µ 2 − γ ( θ ) F ( θ )  , (10) wher e δ ρ ( θ ) = 1 − γ ( θ ) ( m − X 1 θ ) = (1 − ρ ) / (1 − ρ θ ) and γ ( θ ) = λ/ (1 − ρ θ ) . Pr oof. W e an rewrite in tegral equation (7 ) in the follo wing w a y α ′ ( x ) = γ ( θ ) Z ∞ 0 α ′ ( y ) F ( x + y + θ ) dy + γ ( θ ) Z x 0 α ′ ( y ) F ( x − y + θ ) dy + b ( θ ) B ( x ) + 1 , α ′ ( x ) = γ ( θ ) X i =1 , 2 F i θ e − µ i x Z ∞ 0 α ′ ( y ) e − µ i y dy + γ ( θ ) Z x 0 α ′ ( y ) F ( x − y + θ ) dy + b ( θ ) B ( x ) + 1 . W e note that in the latter equation R ∞ 0 α ′ ( y ) e − µ i y dy , i = 1 , 2 are the Laplae transforms of α ′ ( y ) ev aluated at µ i , i = 1 , 2 . Denote L i = Z ∞ 0 α ′ ( y ) e − µ i y dy , i = 1 , 2 . Then, w e ha v e α ′ ( x ) = γ ( θ ) X i =1 , 2 F i θ L i e − µ i x + γ ( θ ) Z x 0 α ′ ( y ) F ( x − y + θ ) dy + b ( θ ) B ( x ) + 1 . No w taking the Laplae transform of the ab o v e equation and using the on v olution prop ert y , w e get L α ′ ( s ) = γ ( θ ) X i =1 , 2 F i θ L i s + µ i + γ ( θ ) X i =1 , 2 F i θ L α ′ ( s ) s + µ i + b ( θ ) F ( θ ) X i =1 , 2 F i θ s + µ i + 1 s , = ⇒ L α ′ ( s )   1 − γ ( θ ) X i =1 , 2 F i θ s + µ i   = γ ( θ ) X i =1 , 2 F i θ L i s + µ i + b ( θ ) F ( θ ) X i =1 , 2 F i θ s + µ i + 1 s . RR n ° 6215 8 K. A vr ahenkov, P. Br own, N. Osip ova Here L α ′ ( s ) = R ∞ 0 α ′ ( x ) e − sx dx is the Laplae transform of α ′ ( x ) . Let us note that L α ′ ( µ i ) = L i , i = 1 , 2 . Then, if w e substitute in to the ab o v e equation s = µ 1 and s = µ 2 , w e an get L 1 and L 2 as a solution of the linear system L 1 = 1  µ 1 + µ 2 − γ ( θ ) F ( θ )  δ ρ ( θ )  b ( θ ) 2 F ( θ )  µ 2 ( m − X 1 θ ) + δ ρ ( θ ) F ( θ )   + 1 µ 1 δ ρ ( θ ) , L 2 = 1  µ 1 + µ 2 − γ ( θ ) F ( θ )  δ ρ ( θ )  b ( θ ) 2 F ( θ )  µ 1 ( m − X 1 θ ) + δ ρ ( θ ) F ( θ )   + 1 µ 2 δ ρ ( θ ) . Next w e need to alulate T B P S ( θ ) . T B P S ( θ ) = Z ∞ 0 α ′ ( x ) F ( x + θ ) dx = Z ∞ 0 α ′ ( x ) X i =1 , 2 F i θ e − µ i x dx = X i =1 , 2 F i θ L i , T B P S ( θ ) = 1 − ρ θ 1 − ρ   m − X 1 θ + b ( θ )  µ 1 µ 2 ( m − X 1 θ ) 2 + δ ρ ( θ ) F 2 ( θ )  2 F ( θ )  µ 1 + µ 2 − γ ( θ ) F ( θ )    . Finally , b y (8) w e ha v e (10 ). 3.3 Optimal threshold appro ximation W e are in terested in the minimization of the exp eted so journ time T ( θ ) with resp et to θ . Of ourse, one an dieren tiate the exat analyti expression pro vided in Theorem 1 and set the result of the dieren tiation to zero. Ho w ev er, this will giv e a transenden tal equation for the optimal v alue of the threshold. In order to nd an appro ximate solution of T ′ ( θ ) = dT ( θ ) /dθ = 0 , w e shall appro ximate the deriv ativ e T ′ ( θ ) b y some funtion e T ′ ( θ ) and obtain a solution to e T ′ ( ˜ θ opt ) = 0 . Sine in the In ternet onnetions b elong to t w o distint lasses with v ery dieren t sizes of transfer (see Setion 2.1 ), then to nd the appro ximation of T ′ ( θ ) w e onsider a partiular ase when µ 2 << µ 1 . Let us in tro due a small parameter ǫ su h that µ 2 = ǫ µ 1 , ǫ → 0 , p 1 = 1 − ǫ ( mµ 1 − 1 ) 1 − ǫ , p 2 = ǫ ( mµ 1 − 1 ) 1 − ǫ . W e note that when ǫ → 0 the seond momen t of the job size distribution go es to innit y . W e then v erify that ˜ θ opt indeed on v erges to the minim um of T ( θ ) when ǫ → 0 . Lemma 2. The fol lowing ine quality holds: µ 1 ρ > λ . Pr oof. Sine p 1 > 0 and p 2 > 0 , w e ha v e the follo wing inequalit y mµ 1 > 1 . Then, m > 1 µ 1 . T aking in to aoun t that λm = ρ w e get ρ λ > 1 µ 1 . Consequen tly , w e ha v e that µ 1 ρ > λ . Prop osition 3. The derivative of T ( θ )  an b e appr oximate d by the fol lowing funtion: e T ′ ( θ ) = − e − µ 1 θ µ 1 c 1 + e − µ 2 θ µ 2 c 2 , wher e c 1 = λ ( mµ 1 − 1 ) µ 1 ( µ 1 − λ )(1 − ρ ) , c 2 = λ ( mµ 1 − 1 ) ( µ 1 − λ ) 2 . (11) Namely, T ′ ( θ ) − e T ′ ( θ ) = O ( µ 2 /µ 1 ) . INRIA Optimal Choi e of Thr eshold in Two L evel Pr o  essor Sharing 9 Pr oof. Using the analytial expression for b oth T ′ ( θ ) and e T ′ ( θ ) , w e get the T a ylor series for T ′ ( θ ) − e T ′ ( θ ) with resp et to ǫ , whi h sho ws that indeed T ′ ( θ ) − e T ′ ( θ ) = O ( ǫ ) . Th us w e ha v e found an appro ximation of the deriv ativ e of T ( θ ) . No w w e an nd an appro xi- mation of the optimal threshold b y solving the equation e T ′ ( θ ) = 0 . Theorem 4. L et θ opt denote the optimal value of the thr eshold. Namely, θ opt = arg min T ( θ ) . The value ˜ θ opt given by ˜ θ opt = 1 µ 1 − µ 2 ln  ( µ 1 − λ ) µ 2 (1 − ρ )  appr oximates θ opt so that T ′ ( ˜ θ opt ) = o ( µ 2 /µ 1 ) . Pr oof. Solving the equation e T ′ ( θ ) = 0 , w e get an analyti expression for the appro ximation of the optimal threshold: e θ opt = − 1 µ 1 (1 − ǫ ) ln  ǫ µ 1 (1 − ρ ) ( µ 1 − λ )  = 1 µ 1 − µ 2 ln  ( µ 1 − λ ) µ 2 (1 − ρ )  . Let us sho w that the ab o v e threshold appro ximation is greater than zero. W e ha v e to sho w that ( µ 1 − λ ) µ 2 (1 − ρ ) > 1 . Sine µ 1 > µ 2 and µ 1 ρ > λ (see Lemma 2 ), w e ha v e µ 1 > µ 2 = ⇒ µ 1 (1 − ρ ) > µ 2 (1 − ρ ) = ⇒ λ < µ 1 ρ < µ 1 − µ 2 (1 − ρ ) = ⇒ ( µ 1 − λ ) > µ 2 (1 − ρ ) . Expanding T ′ ( e θ opt ) as a p o w er series with resp et to ǫ giv es: T ′ ( e θ opt ) = ǫ 2 ( const 0 + c onst 1 ln ǫ + const 2 ln 2 ǫ ) , where const i , i = 1 , 2 are some onstan t v alues 1 with resp et to ǫ . Th us, T ′ ( e θ opt ) = o ( ǫ ) = o ( µ 2 /µ 1 ) , whi h ompletes the pro of. In the next prop osition w e  haraterize the limiting b eha vior of T ( θ opt ) and T ( e θ opt ) as ǫ → 0 . In partiular, w e sho w that T ( e θ opt ) tends to the exat minim um of T ( θ ) when ǫ → 0 . Prop osition 5. lim ǫ → 0 T ( θ opt ) = lim ǫ → 0 T ( e θ opt ) = m 1 − ρ − c 1 , wher e c 1 is given by (11 ). 1 The expressions for the onstan ts const i are um b ersome and an b e found using Maple ommand series. RR n ° 6215 10 K. A vr ahenkov, P. Br own, N. Osip ova Pr oof. W e nd the follo wing limit, when ǫ → 0 : lim ǫ → 0 T ( θ ) = m 1 − ρ − λ ( mµ 1 − 1 ) µ 1 ( µ 1 − λ )(1 − ρ ) + λ ( mµ 1 − 1 ) e − µ 1 θ µ 1 ( µ 1 − λ )(1 − ρ ) , lim ǫ → 0 T ( θ ) = m 1 − ρ − c 1 + c 1 e − µ 1 θ , where c 1 is giv en b y (11). Sine the funtion lim ǫ → 0 T ( θ ) is a dereasing funtion, the optimal threshold for it is θ opt = ∞ . Th us, lim ǫ → 0 T ( θ opt ) = lim θ →∞ lim ǫ → 0 T ( θ ) = m 1 − ρ − c 1 . On the other hand, w e obtain lim ǫ → 0 T ( e θ opt ) = m 1 − ρ − c 1 , whi h pro v es the prop osition. 3.4 Exp erimen tal results In Figure 1-2 w e sho w the plots for the follo wing parameters: ρ = 10 / 11 (default v alue), m = 2 0 / 11 , µ 1 = 1 , µ 2 = 1 / 10 , so λ = 1 / 2 and ǫ = µ 2 /µ 1 = 1 / 1 0 . Then, p 1 = 1 0 / 11 and p 2 = 1 / 1 1 . In Figure 1 w e plot T ( θ ) , T P S and T ( e θ opt ) . W e note, that the exp eted so journ time in the standard PS system T P S is equal to T (0 ) . W e observ e that T ( e θ opt ) orresp onds w ell to the optim um ev en though ǫ = 1 / 10 is not to o small. Let us no w study the gain that w e obtain using TLPS, b y setting θ = e θ opt , in omparison with the standard PS. T o this end, w e plot the ratio g ( ρ ) = T P S − T ( e θ opt ) T P S in Figure 2. The gain in the system p erformane with TLPS in omparison with PS strongly dep ends on ρ , the load of the system. One an see, that the gain of the TLPS system with resp et to the standard PS system go es up to 45% when the load of the system inreases. T o study the sensitivit y of the TLPS system with resp et to θ , w e nd the gain of the TLPS system with resp et to the standard PS system, w e plot in Figure 2 g 1 ( ρ ) = T P S − T ( 3 2 e θ opt ) T P S and g 2 ( ρ ) = T P S − T ( 1 2 e θ opt ) T P S . Th us, ev en with the 50% error of the e θ opt v alue, the system p erformane is lose to optimal. One an see that it is b eneial to use TLPS instead of PS in the ase of hea vy and mo derately hea vy loads. W e also observ e that the optimal TLPS system is not to o sensitiv e to the  hoie of the threshold near its optimal v alue, when the job size distribution is h yp er-exp onen tial with t w o phases. Nev ertheless, it is b etter to  ho ose larger rather than smaller v alues of the threshold. INRIA Optimal Choi e of Thr eshold in Two L evel Pr o  essor Sharing 11 0 5 10 15 20 25 30 35 40 0 2 4 6 8 10 12 14 16 18 20 θ T( θ ) T( θ opt ) T PS Figure 1: T ( θ ) - solid line, T P S ( θ ) - dash dot line, T ( e θ opt ) - dash line 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 ρ g( ρ ) g 1 ( ρ ) g 2 ( ρ ) Figure 2: g ( ρ ) - solid line, g 1 ( ρ ) - dash line, g 2 ( ρ ) - dash dot line 4 Hyp er-exp onen tial job size distribution with more than t w o phases 4.1 Notations In the seond part of the presen ted w ork w e analyze the TLPS disipline with the h yp er- exp onen tial job size distribution with more than t w o phases. As w as sho wn in [ 5, 7 , 8 ℄, the h yp er- exp onen tial distribution with a signian t n um b er of phases mo dels w ell the le size distribution in the In ternet. Th us, in this setion as the job size distribution w e tak e the h yp er-exp onen tial funtion with man y phases. Namely , aording to (1), F ( x ) = 1 − N X i =1 p i e − µ i x , N X i =1 p i = 1 , µ i > 0 , p i ≥ 0 , i = 1 , ..., N , 1 < N ≤ ∞ . In the follo wing w e shall write simply P i instead of P N i =1 . The mean job size m , the seond momen t d , the parameters F i θ , X 1 θ , X 2 θ and ρ θ are dened as in Setion 2.1 and Setion 2.2 b y form ulas (2 ),(3),(5 ), (6) for an y 1 ≤ N ≤ ∞ . The form ulas presen ted in Setion 2.2 an still b e used to alulate b ( θ ) , B ( x ) , W ( θ ) , γ ( θ ) , δ ρ ( θ ) , T T LP S ( x ) , T ( θ ) . W e shall also need the follo wing op erator notations: Φ 1 ( β ( x )) = γ ( θ ) Z ∞ 0 β ( y ) F ( x + y + θ ) dy + γ ( θ ) Z x 0 β ( y ) F ( x − y + θ ) dy , (12) Φ 2 ( β ( x )) = Z ∞ 0 β ( y ) F ( y + θ ) dy (13) for an y funtion β ( x ) . In partiular, for some giv en onstan t , w e ha v e Φ 1 ( c ) = c γ ( θ )( m − X 1 θ ) = c q , (14) Φ 2 ( c ) = c ( m − X 1 θ ) , (15) where q = γ ( θ )( m − X 1 θ ) = λ ( m − X 1 θ ) 1 − ρ θ = ρ − ρ θ 1 − ρ θ < 1 . (16) RR n ° 6215 12 K. A vr ahenkov, P. Br own, N. Osip ova The in tegral equation (7) an no w b e rewritten in the form α ′ ( x ) = Φ 1 ( α ′ ( y ))+ b ( θ ) F ( θ ) F ( x + θ ) + 1 . (17) and equation (9) for T B P S ( θ ) tak es the form T B P S ( θ ) = Φ 2 ( α ′ ( x )) . (18) 4.2 Linear system based solution Similarly to the rst part of the pro of of Theorem 1 , w e obtain the follo wing prop osition. Prop osition 6. T B P S ( θ ) = X i F i θ L i , with L i = L ∗ i + 1 δ ρ ( θ ) µ i , wher e the L ∗ i ar e the solution of the line ar system L ∗ p 1 − γ ( θ ) X i F i θ λ p + µ i ! = γ ( θ ) X i F i θ L ∗ i λ p + µ i + b ( θ ) F ( θ ) X i F i θ λ p + µ i , p = 1 , ..., N . (19) Unfortunately , the system ( 19 ) do es not seem to ha v e a tratable nite form analyti solution. Therefore, in the ensuing subsetions w e prop osed an alternativ e solution based on an op erator series and onstrut a tigh t upp er b ound. 4.3 Op erator series form for the exp eted so journ time Sine the op erator Φ 1 is a on tration [3, 4℄, w e an iterate equation ( 17 ) starting from some initial p oin t α ′ 0 . The initial p oin t ould b e simply a onstan t. As sho wn in [3 , 4℄ the iterations will on v erge to the unique solution of (17 ). Sp eially , w e mak e iterations in the follo wing w a y: α ′ n +1 ( x ) = Φ 1 ( α ′ n ( x ))+ b ( θ ) F ( θ ) F ( x + θ ) + 1 , n = 0 , 1 , 2 , ... (20) A t ev ery iteration step w e onstrut the follo wing appro ximation of T B P S ( θ ) aording to (18): T B P S n +1 ( θ ) = Φ 2 ( α ′ n +1 ( x )) . (21) Using (20) and (21 ), w e an onstrut the op erator series expression for the exp eted so journ time in the TLPS system. Theorem 7. The exp e te d sojourn time T ( θ ) in the TLPS system with the hyp er-exp onential job size distribution is given by T ( θ ) = X 1 θ + W ( θ ) F ( θ ) 1 − ρ θ + m − X 1 θ 1 − ρ + b ( θ ) F ( θ )(1 − ρ θ ) ∞ X i =0 Φ 2  Φ i 1 ( F ( x + θ ))  ! . (22) INRIA Optimal Choi e of Thr eshold in Two L evel Pr o  essor Sharing 13 Pr oof. F rom (20 ) w e ha v e α ′ n = q n α ′ 0 + n − 1 X i =1 q i + b ( θ ) F ( θ ) n − 1 X i =1 Φ i 1 ( F ( x + θ )) + b ( θ ) F ( θ ) F ( x + θ ) + 1 , and then from (21 ) and (14) it follo ws, that T B P S n ( θ ) = ( m − X 1 θ ) q n α ′ 0 + n − 1 X i =0 q i ! + b ( θ ) F ( θ ) Φ 2 n − 1 X i =0 Φ i 1 ( F ( x + θ )) !! . Using the fats (see (16 )): 1 . q < ρ < 1 = ⇒ q n → 0 as n → ∞ , 2 . ∞ X i =0 q i = 1 1 − q = 1 − ρ θ 1 − ρ , w e onlude that T B P S ( θ ) = lim n →∞ T B P S n ( θ ) = ( m − X 1 θ )  1 − ρ θ 1 − ρ  + b ( θ ) F ( θ ) ∞ X i =0 Φ 2  Φ i 1 ( F ( x + θ ))  ! . Finally , using (8) w e obtain ( 22 ). The resulting form ula (22 ) is diult to analyze and do es not ha v e a lear analyti expression. Using this result in the next subsetion w e nd an appro ximation,whi h is also an upp er b ound, of the exp eted so journ time funtion in a more expliit form. 4.4 Upp er b ound for the exp eted so journ time Let us start with auxiliary results. Lemma 8. F or any funtion β ( x ) ≥ 0 with β j = R ∞ 0 β ( x ) e − µ i x dx , if d ( β j µ j ) dµ j ≥ 0 , j = 1 , ..., N it fol lows, that Φ 2 (Φ 1 ( β ( x ))) ≤ q Φ 2 ( β ( x )) . Pr oof. See App endix. Lemma 9. F or the TLPS system with the hyp er-exp onential job size distribution the fol lowing statement holds: Φ 2 (Φ 1 ( α ′ ( x ))) ≤ q Φ 2 ( α ′ ( x )) . (23) Pr oof. W e dene α ′ j = R ∞ 0 α ′ ( x ) e − µ j x dx, j = 1 , ...N . As w as sho wn in [ 14 ℄, α ′ ( x ) has the follo wing struture: α ′ ( x ) = a 0 + X k a k e − b k x , a 0 ≥ 0 , a k ≥ 0 , b k > 0 , k = 1 , ..., N . Then, w e ha v e that α ′ ( x ) ≥ 0 and α ′ j = a 0 µ j + X k a k b k + µ j , j = 1 , ..., N , = ⇒ d ( α ′ j µ j ) dµ j = X k a k b k + µ j − X k a k µ j ( b k + µ j ) 2 = X k a k b k ( b k + µ j ) 2 ≥ 0 , j = 1 , ..., N , as a k ≥ 0 , b k > 0 , k = 1 , ..., N . So, then, aording to Lemma 8 w e ha v e (23 ). RR n ° 6215 14 K. A vr ahenkov, P. Br own, N. Osip ova Let us state the follo wing Theorem: Theorem 10. A n upp er b ound for the exp e te d sojourn time T ( θ ) in the TLPS system with the hyp er-exp onential job size distribution funtion with many phases is given by Υ( θ ) : T ( θ ) ≤ Υ ( θ ) = X 1 θ + W ( θ ) F ( θ ) 1 − ρ θ + m − X 1 θ 1 − ρ + b ( θ ) F ( θ )(1 − ρ ) X i,j F i θ F j θ µ i + µ j . (24) Pr oof. A ording to the reursion ( 20) w e ha v e for α ′ n ( x ) w e appro ximate α ′ ( x ) with the funtion e α ′ ( x ) , whi h satises the follo wing equation: e α ′ ( x ) = e α ′ ( x )Φ 1 (1) + b ( θ ) F ( θ ) F ( x + θ ) + 1 . Then, aording to (14 ) w e an nd the analytial expression for e α ′ ( x ) : e α ′ ( x ) = q e α ′ ( x ) + b ( θ ) F ( θ ) F ( x + θ ) + 1 , = ⇒ e α ′ ( x ) = 1 1 − q  b ( θ ) F ( θ ) F ( x + θ ) + 1  . W e tak e Υ B P S ( θ ) = Φ 2 ( e α ′ ( x )) as an appro ximation for T B P S ( θ ) = Φ 2 ( α ′ ( x )) . Then Υ B P S ( θ ) = Φ 2 ( e α ′ ( x )) = ( m − X 1 θ ) 1 − q + b ( θ ) F ( θ ) Φ 2 ( F ( x + θ )) = ( m − X 1 θ ) 1 − q + b ( θ ) F ( θ ) X i,j F i θ F j θ µ i + µ j . Let us pro v e, that T B P S ( θ ) ≤ Υ B P S ( θ ) , or equiv alen tly T B P S ( θ ) − Υ B P S ( θ ) = Φ 2 ( α ′ ( x )) − Φ 2 ( e α ′ ( x )) ≤ 0 . Let us lo ok at Φ 2 ( α ′ ( x )) − Φ 2 ( e α ′ ( x )) = = Φ 2 (Φ 1 ( α ′ ( x ))) + Φ 2  b ( θ ) F ( θ ) F ( x + θ ) + 1  −  q Φ 2 ( e α ′ ( x )) + Φ 2  b ( θ ) F ( θ ) F ( x + θ ) + 1  = Φ 2 (Φ 1 ( α ′ ( x ))) − q Φ 2 ( α ′ ( x )) + q (Φ 2 ( α ′ ( x )) − Φ 2 ( e α ′ ( x ))) = ⇒ Φ 2 ( α ′ ( x )) − Φ 2 ( e α ′ ( x )) = 1 1 − q (Φ 2 (Φ 1 ( α ′ ( x ))) − q Φ 2 ( α ′ ( x ))) . And from Lemma 9 and form ula (8) w e onlude that (24) is true. In this subsetion w e found the analytial expression of the upp er b ound of the exp eted so journ time in the ase when the job size distribution is a h yp er-exp onen tial funtion with man y phases. In the exp erimen tal results of the follo wing subsetion w e sho w that the obtained upp er b ound is also a lose appro ximation. The analyti expression of the upp er b ound whi h w e obtained is more lear and easier to analyze then the expression of the exp eted so journ time. It ould b e used in the future resear h on TLPS mo del. INRIA Optimal Choi e of Thr eshold in Two L evel Pr o  essor Sharing 15 4.5 Exp erimen tal results W e alulate T ( θ ) and Υ( θ ) for dieren t n um b ers of phases N of the job size distribution funtion. W e tak e N = 10 , 1 00 , 500 , 10 00 . T o alulate T ( θ ) w e nd the n umerial solution of the system of linear equations (19 ) using the Gauss metho d. Then using the result of Prop osition 6 w e nd T ( θ ) . F or Υ( θ ) w e use equation ( 24 ). As w as men tioned in Subsetion 2.1 , b y using the h yp er-exp onen tial distribution with man y phases, one an appro ximate a hea vy-tailed distribution. In our n umerial exp erimen ts, w e x ρ , m , and selet p i and µ i in a su h a w a y , that b y inreasing the n um b er of phases w e let the seond momen t d (see (2)) inrease as w ell. Here w e tak e ρ = 1 0 / 11 , λ = 0 . 5 , p i = ν i 2 . 5 , µ i = η i 1 . 2 , i = 1 , ..., N . In partiular, w e ha v e X i p i = 1 , = ⇒ ν = 1 P i i − 2 . 5 , X i p i µ i = m, = ⇒ η = ν m X i i − 1 . 3 . In Figure 3 one an see the plots of the exp eted so journ time and its upp er b ound as funtions of θ when N v aries from 10 up to 1000. In Figure 4 w e plot the relativ e error of the upp er b ound ∆( θ ) = Υ( θ ) − T ( θ ) T ( θ ) , when N v aries from 10 up to 1000. As one an see, the upp er b ound (24 ) is v ery tigh t. W e nd the maxim um gain of the exp eted so journ time of the TLPS system with resp et to the standard PS system. The gain is giv en b y g ( θ ) = T P S − T ( θ ) T P S . Here T P S is an exp eted so journ time in the standard PS system. Let us notie, that T P S = T (0) . The data and results are summarized in T able 1. N η d θ opt max θ g ( θ ) max θ ∆( θ ) 10 0.95 7.20 5 32.98 % 0.0640 100 1.26 32.28 12 45.75 % 0.0807 500 1.40 113.31 21 49.26 % 0.0766 1000 1.44 200.04 26 50.12 % 0.0743 T able 1: Inreasing the n um b er of phases With the inreasing n um b er of phases w e observ e that 1. the seond momen t d inreases; 2. the maxim um gain max θ g ( θ ) in exp eted so journ time in omparison with PS inreases; 3. the relativ e error of the upp er b ound ∆( θ ) with the exp eted so journ time dereases after the n um b er of phases b eomes suien tly large; 4. the sensitivit y of the system p erformane with resp et to the seletion of the sub-optimal threshold v alue dereases. Th us the TLPS system pro dues b etter and more robust p erformane as the v ariane of the job size distribution inreases. RR n ° 6215 16 K. A vr ahenkov, P. Br own, N. Osip ova 0 10 20 30 40 50 60 70 80 90 100 8 10 12 14 16 18 20 22 24 θ N=10 N=100 N=500 N=1000 T PS ϒ ( θ ) T( θ ) Figure 3: The exp eted so journ time T ( θ ) and its upp er b ound Υ( θ )) for N = 10 , 100 , 500 , 1000 0 10 20 30 40 50 60 70 80 90 100 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% θ N=10 N=100 N=500 N=1000 ∆ ( θ ) Figure 4: The relativ e error ∆( θ ) = ( T ( θ ) − Υ( θ )) /T ( θ ) for N = 10 , 100 , 500 , 1000 5 Conlusion W e analyze the TLPS s heduling me hanism with the h yp er-exp onen tial job size distribution funtion. In Setion 3 w e analyze the system when the job size distribution funtion has t w o phases and nd the analytial expressions of the exp eted onditional so journ time and the exp eted so journ time of the TLPS system. Connetions in the In ternet b elong to t w o distint lasses: short HTTP and P2P signaling onnetions and long do wnloads su h as: PDF, MP3, and so on. Th us, aording to this obser- v ation, w e onsider a sp eial seletion of the parameters of the job size distribution funtion with t w o phases and nd the appro ximation of the optimal threshold, when the v ariane of the job size distribution go es to innit y . W e sho w, that the appro ximated v alue of the threshold tends to the optimal threshold, when the seond momen t of the distribution funtion go es to innit y . W e found that the gain of the TLPS system ompared to the standard PS system ould rea h 45% when the load of the system inreases. Also the system is not to o sensitiv e to the seletion of the optimal v alue of the threshold. In Setion 4 w e ha v e studied the TLPS mo del when the job size distribution is a h yp er- exp onen tial funtion with man y phases. W e pro vide an expression of the exp eted onditional so journ time as a solution of the system of linear equations. Also w e apply the iteration metho d to nd the expression of the exp eted onditional so journ time in the form of op erator series and using the obtained expression w e pro vide an upp er b ound for the exp eted so journ time funtion. With the exp erimen tal results w e sho w that the upp er b ound is v ery tigh t and ould b e used as an appro ximation of the exp eted so journ time funtion. W e sho w n umerially , that the relativ e error b et w een the upp er b ound and exp eted so journ time funtion dereases when the v ariation of the job size distribution funtion inreases. The obtained upp er b ound ould b e used to iden tify an appro ximation of the optimal v alue of the optimal threshold for TLPS system when the job size distribution is hea vy-tailed. W e study the prop erties of the exp eted so journ time funtion, when the parameters of the job size distribution funtion are seleted in su h a w a y , that it appro ximates a hea vy-tailed distribution as the n um b er of phases of the job size distribution inreases. As the n um b er of phases inreases the gain of the TLPS system ompared with the standard PS system inreases and the sensitivit y of the system with resp et to the seletion of the optimal threshold dereases. INRIA Optimal Choi e of Thr eshold in Two L evel Pr o  essor Sharing 17 6 App endix: Pro of of Lemma 8 Let us tak e an y funtion β ( x ) > 0 and dene β j = R ∞ 0 β ( x ) e − µ j x dx, j = 1 , ..., N . Let us sho w for β ( x ) ≥ 0 that if d ( β j µ j ) dµ j ≥ 0 , j = 1 , ..., N , then it follo ws that Φ 2 (Φ 1 ( β ( x ))) ≤ q Φ 2 ( β ( x )) . As Z ∞ 0 Z x 0 β ( y ) F ( x − y + θ ) F ( x + θ ) dy dx = Z ∞ 0 Z ∞ 0 β ( y ) F ( x 1 + θ ) F ( x 1 + y + θ ) dx 1 dy and Φ 2 (Φ 1 ( β ( x ))) = γ ( θ ) Z ∞ 0 Z ∞ 0 β ( y ) F ( x + y + θ ) F ( x + θ ) dy dx + γ ( θ ) Z ∞ 0 Z x 0 β ( y ) F ( x − y + θ ) F ( x + θ ) dy dx, then Φ 2 (Φ 1 ( β ( x ))) = 2 γ ( θ ) Z ∞ 0 Z ∞ 0 β ( x ) F ( x + θ ) F ( x + y + θ ) dy dx = = 2 γ ( θ ) Z ∞ 0 β ( x ) X i,j F i θ F j θ µ i + µ j e − µ j x dx = 2 γ ( θ ) X i,j F i θ F j θ µ i + µ j β j . Also for Φ 2 ( β ( x )) , taking in to aoun t that q = γ ( θ ) P i F i θ µ i , w e obtain q Φ 2 ( β ( x )) = γ ( θ ) X i F i θ µ i X j F j θ Z ∞ 0 β ( x ) e − µ j x dx = γ ( θ ) X i,j F i θ F j θ µ i β j . Th us, a suien t ondition for the inequalit y Φ 2 (Φ 1 ( β ( x ))) ≤ q Φ 2 ( β ( x )) to b e satised is that for ev ery pair i, j : 2 µ i + µ j β j + 2 µ j + µ i β i ≤ 1 µ i β j + 1 µ j β i ⇐ ⇒ − ( β j µ j − β i µ i )( µ j − µ i ) ≤ 0 . The inequalit y is indeed satised when β j µ j is an inreasing funtion of µ j . W e onlude that Φ 2 (Φ 1 ( β ( x ))) ≤ q Φ 2 ( β ( x )) , whi h pro v es Lemma 8. RR n ° 6215 18 K. A vr ahenkov, P. Br own, N. Osip ova Referenes [1℄ S. Aalto and U. A y esta, Mean dela y analysis of m ultilev el pro essor sharing disiplines, In Pro eedings of IEEE INF OCOM 2006. [2℄ S. Aalto, U. A y esta, and E.Nyb erg, T w o-lev el pro essor sharing s heduling disiplines: mean dela y analysis, in Pro eedings of A CM SIGMETRICS/P erformane 2004. [3℄ K. A vra henk o v, U. A y esta, P . Bro wn, and E. Nyb erg, Dieren tiation b et w een short and long TCP o ws: Preditabilit y of the resp onse time, in Pro eedings of IEEE INF OCOM 2004. [4℄ K. A vra henk o v, U. A y esta, P . Bro wn, Bat h Arriv al Pro essor-Sharing with Appliation to Multi-Lev el Pro essor-Sharing S heduling, Queuing ystems 50, pp.459-480, 2005. [5℄ F. Baelli and D.R. MDonald, A sto  hasti mo del for the rate of non-p ersisten t TCP o ws, in Pro eedings of V alueT o ols 2006. [6℄ N. Bansal, Analysis of the M/G/1 pro essor-sharing queue with bulk arriv als, Op er ations R ese ar h L etters , v.31, no.5, 2003, pp.401-405. [7℄ R. El Ab douni Kha y ari, R. Sadre, and B.R. Ha v erk ort, Fitting w orld-wide w eb request traes with the EM-algorithm, Performan e Evaluation , v.52, no.2-3, 2003, pp.175-191. [8℄ A. F eldmann, W. Whitt, Fitting mixtures of exp onen tials to long-tail distributions to analyze net w ork p erformane mo dels, Performan e Evaluation , v.31, 1998, pp.245-258. [9℄ H. F eng and V. Misra, Mixed s heduling disiplines for net w ork o ws, A CM SIGMETRICS Performan e Evaluation R eview , v.31(2), 2003, pp.36-39. [10℄ L. Guo and L. Matta, Dieren tiated on trol of w eb tra: A n umerial analisys, In SPIE ITCOM, Boston, 2002. [11℄ M. Har hol-Balter, B. S hro eder, N. Bansal, and M. Agra w al, Size-based s heduling to im- pro v e w eb p erformane, A CM T r ansations on Computer Systems , v.21, no.2, 2003, pp.207- 233. [12℄ L. Kleinro  k, Queueing systems, vol. 2 , John Wiley and Sons, 1976. [13℄ Kleinro  k, L., R. R. Mun tz, and E. Ro demi h, The Pro essor-Sharing Queueing Mo del for Time-Shared Systems with Bulk Arriv als, Net w orks Journal,1,1-13 (1971). [14℄ N. Osip o v a, Bat h Pro essor Sharing with Hyp er-Exp onen tial servie time, INRIA Resear h Rep ort RR-6180, 2007. A v ailable at h ttp://hal.inria.fr/inria-00144389 [15℄ L.E. S hrage, A pro of of the optimalit y of the shortest remaining pro essing time disipline, Op er ations R ese ar h , v.16, 1968, pp. 687-690. INRIA Optimal Choi e of Thr eshold in Two L evel Pr o  essor Sharing 19 Con ten ts 1 In tro dution 3 2 Mo del desription 4 2.1 Main denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The exp eted so journ time in TLPS system . . . . . . . . . . . . . . . . . . . . . . 5 3 Hyp er-exp onen tial job size distribution with t w o phases 7 3.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Expliit form for the exp eted so journ time . . . . . . . . . . . . . . . . . . . . . . 7 3.3 Optimal threshold appro ximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.4 Exp erimen tal results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4 Hyp er-exp onen tial job size distribution with more than t w o phases 11 4.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 Linear system based solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.3 Op erator series form for the exp eted so journ time . . . . . . . . . . . . . . . . . . 12 4.4 Upp er b ound for the exp eted so journ time . . . . . . . . . . . . . . . . . . . . . . 13 4.5 Exp erimen tal results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5 Conlusion 16 6 App endix: Pro of of Lemma 8 17 RR n ° 6215 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 versit é - ZA C des V ignes 4, rue Jacques Monod - 91893 ORSA Y Cedex (Franc e) Unité de reche rche INRIA Lorraine : LORIA, T echnopôle de Nancy-Braboi s - Campus scienti fique 615, rue du Jardin Botani que - BP 101 - 54602 V illers-lè s-Nancy Cedex (France ) Unité de reche rche INRIA Rennes : IRISA, Campus uni versitai re de Beauli eu - 35042 Rennes Cede x (France) Unité de reche rche INRIA Rhône-Alpes : 655, ave nue de l’Europe - 38334 Montbonno t Saint-Ismier (France) Unité de recherch e INRIA Rocquencourt : Domaine de V oluceau - Rocquenc ourt - 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