Lower and Upper Bounds on FIFO Buffer Management in QoS Switches

Lower and Upper Bounds on FIFO Buffer Managemen in QoS Swiches Mahias Engler Mahias Wesermann Deparmen of Compuer Science RWTH Aachen 52056 Aachen, Germany {engler,marsu}@cs.rwh-aachen.de Absrac We consider he managemen of FIFO buffers for nework swiches providing differeniaed services. In each ime sep, an arbirary number of packes arrive and only one packe can be sen. The buffer can sore a limied number of packes and, due o he FIFO propery, he sequence of sen packes has o be a subsequence of he arriving packes. The differeniaed service model is absraced by aribuing each packe wih a value according o is service level. A buffer managemen sraegy can drop packes, and he goal is o maximize he sum of he values of sen packes. For only wo differen packe values, we inroduce he accoun sraegy and prove ha his sraegy achieves an opimal compeiive raio of 2 ( 5 + 4 2 3)/2 1.282 if he buffer size ends o infiniy and an opimal compeiive raio of ( 13 1)/2 1.303 for arbirary buffer sizes. For general packe values, he simple preempive greedy sraegy (PG) is sudied. We show ha PG achieves a compeiive raio of 3 1.732 which is he bes known upper bound on he compeiive raio of his problem. In addiion, we give a lower bound of 1 + 1/ 2 1.707 on he compeiive raio of PG which improves he previously known lower bound. As a consequence, he compeiive raio of PG canno be furher improved significanly. Suppored by he DFG gran WE 2842/1. A preliminary version of his paper appeared in Proceedings of he 14h Annual European Symposium on Algorihms (ESA), 2006. 1

1 Inroducion Qualiy of Service (QoS) guaranees for nework services allow providers o address he requiremens of cusomers by offering differen levels of service. In he nework seing, where raffic volumes may exceed nework capaciy, effecive managemen of buffers in swiches is a key o achieve QoS guaranees. We consider FIFO buffers, i.e., he buffer can sore a limied number of packes and, due o he FIFO propery, he sequence of sen packes has o be a subsequence of he arriving packes. By differeniaing service levels, packes of differen ypes may be reaed according o he level of service hey require. This model is absraced by aribuing each packe wih a value according o is service level. A buffer managemen sraegy can drop packes, and he goal is o maximize he sum of he values of sen packes. For only wo differen packe values, we inroduce he accoun sraegy and prove ha his sraegy achieves an opimal compeiive raio of 2 ( 5 + 4 2 3)/2 1.282 if he buffer size ends o infiniy and an opimal compeiive raio of ( 13 1)/2 1.303 for arbirary buffer sizes. For general packe values, he simple preempive greedy sraegy (PG) is sudied. We show ha PG achieves a compeiive raio of 3 1.732 which is he bes known upper bound on he compeiive raio of his problem. In addiion, we give a lower bound of 1 + 1/ 2 1.707 on he compeiive raio of PG which improves he previously known lower bound. As a consequence, he compeiive raio of PG canno be furher improved significanly. 1.1 The Model Time is sloed in ime seps. In each ime sep, an arbirary number of packes arrive, and, a he end of each ime sep, only one packe can be sen. Packes ha are no sen can be sored in a FIFO buffer wih a limied sorage capaciy for b packes. Iniially, he FIFO buffer is empy. Due o he FIFO propery, he sequence of sen packes has o be a subsequence of he arriving packes, i.e., if a packe p is sen before a packe p, p has arrived before p. The differeniaed service model is absraced by aribuing each packe p wih a value v(p) according o is service level. A buffer managemen sraegy can drop arriving packes, i.e., hese packes are never sored in he buffer, or can drop packes sored in he buffer, i.e., hese packes are deleed from he buffer and no sen. The goal of he buffer managemen sraegy is o maximize he sum of he values of sen packes. The noion of an online sraegy is inended o formalize he realisic scenario where he sraegy does no have knowledge abou he whole inpu sequence of arriving packes in advance. The online sraegy ges o know his sequence packe by packe and has o reac wihou knowledge abou he fuure. Online sraegies are ypically evaluaed in a compeiive analysis. In his kind of analysis he oal value produced by he online sraegy is compared wih he oal value produced by an opimal offline sraegy. For a given inpu sequence σ of arriving packes, le OPT(σ) denoe he oal value produced by an opimal offline sraegy. An online sraegy is denoed as c-compeiive if i produces oal value a leas OPT(σ)/c, for each inpu sequence σ of arriving packes. The value c is also called he compeiive raio of he online sraegy. 1.2 Previous Work Aiello e al. [1] inroduce he model of differeniaed services for FIFO buffers wihou preempion. Mansour, Pa-Shamir, and Lapid [11] add preempion and general packe values o his 2

model. Kesselman and Mansour [8] sudy he value of he los packes insead of he value of he sen packes. Kesselman e al. [7] show ha he greedy sraegy achieves a compeiive raio of 2. Kesselman, Mansour, and van See [9] inroduce he preempive greedy sraegy and prove ha his sraegy achieves a compeiive raio of 1.983. In addiion, hey give he previously bes known lower bound of (1 + 5)/2 1.618 on he compeiive raio of he preempive greedy sraegy. Bansal e al. [5] sudy a modificaion of he preempive greedy sraegy and show ha his sraegy achieves a compeiive raio of 7/4 which is he previously bes known upper bound on he compeiive raio of his problem. Noe ha heir modificaion does no improve he overall performance of he sraegy [6]. The bes known lower bound on he compeiive raio of his problem is 1.419 [9]. The following resuls refer o he case where only wo differen packe values are considered. Loker and Pa-Shamir [10] presen a sraegy ha achieves a compeiive raio of 1.30448. Kesselman e al. [7] show a lower bound of 1.282 on he compeiive raio. Andelman [2] presens a randomized sraegy ha achieves a compeiive raio of 5/4. Furher, he gives a lower bound of 1.197 on he compeiive raio of any randomized sraegy. Azar and Richer [4] exend he buffer managemen problem o muli-queues, i.e., several incoming queues have o be served by delivering packes ha arrive a hese queues hrough one oupu por, one packe per ime sep. They presen a generic echnique ha ransforms a sraegy for a single queue o a sraegy for several queues. They show ha he compeiive raio of he consruced sraegy is a mos wice he compeiive raio of he single queue sraegy. 1.3 Our Conribuions In Secion 2, only wo packe values are considered. We inroduce he accoun sraegy and prove ha his sraegy achieves an opimal compeiive raio of 2 ( 5 + 4 2 3)/2 1.282 if he buffer size ends o infiniy and an opimal compeiive raio of ( 13 1)/2 1.303 for arbirary buffer sizes. Noe ha his is he firs non-rivial opimal resul in his area. In Secion 3, general packe values are considered. We sudy he preempive greedy sraegy (PG) inroduced in [9]. This is a simple sraegy ha can be implemened efficienly. We show ha PG achieves a compeiive raio of 3 1.732 which is he bes known upper bound on he compeiive raio of his problem. In addiion, we give a lower bound of 1+1/ 2 1.707 on he compeiive raio of PG which improves he previously known lower bound of (1+ 5)/2 1.618. Hence, he gap beween upper and lower bound for PG narrows o approximaely 1/40. We conjecure ha he lower bound is igh. As a consequence, new approaches are needed, since he compeiive raio of PG canno be furher improved significanly. Based on our lower bound for PG and our opimal accoun sraegy for wo packe values, we propose an approach o ackle he problems of PG. 3

2 Two Packe Values In his secion, only wo packe values 1 and α > 1 are considered. A packe of value 1 is called 1-packe, and a packe of value α is called α-packe. Define 13 1 r := 1.303 and 2 r := 5 + 4 2 3 2 1.282. 2 The following heorem saes wo lower bounds on he compeiive raio of any deerminisic sraegy. The proof for he firs saemen of his heorem can be found, e.g., in [3], and he proof for he second saemen of his heorem can be found, e.g., in [7]. Theorem 1. Consider only wo packe values 1 and α > 1. 1. The compeiive raio of any deerminisic sraegy is a leas r, if he buffer size is 2. 2. The compeiive raio of any deerminisic sraegy is a leas r, if he buffer size ends o infiniy. The accoun sraegy (ACC) ries o preemp 1-packes from he buffer in order o avoid losing oo many α-packes in case of a buffer overflow. The number of preemped 1-packes has o be chosen carefully. Obviously, he oal number of preemped 1-packes should no exceed (x 1) imes he oal value of sen packes if we wan o achieve a compeiive raio of x. Hence, one basic idea of ACC is o preemp a mos (x 1) α 1-packes for each α-packe enering he buffer and a mos (x 1) 1-packes for each sen 1-packe. ACC ries o preemp as much 1-packes as possible wihou violaing his consrain. We define ACC(x) wih one parameer x 1 which is he compeiive raio we aim for and which is herefore used o deermine how aggressive he sraegy is wih respec o preempion. ACC(x) uses an accoun a which is iniially se o 0. Basically, each packe sen by ACC(x) increases he accoun by (x 1) imes is own value, and each preemped 1-packe decreases he accoun by 1. More precisely, for each ime sep, ACC(x) does he following. 1. For each arriving packe p, do he following. (a) If here is an unoccupied locaion in he buffer, sore p. Oherwise, if a 1-packe is sored in he buffer, drop he 1-packe which is closes o he fron of he buffer and sore p. (b) If p is an α-packe ha is sored in he buffer (observe ha sored α-packes are never dropped), increase he accoun a by (x 1) α. (c) If he buffer is compleely filled wih α-packes, rese he accoun a o 0. 2. Afer all packes have arrived, do he following. (a) As long as he firs packe is a 1-packe and a 1, drop his packe, which is called preemped, and decrease he accoun a by 1. (b) Send he firs packe. If his packe is a 1-packe, increase he accoun by (x 1). (c) If no packe is sored in he buffer, rese he accoun a o 0. 4

The following heorem shows ha ACC achieves opimal compeiive raios. Theorem 2. Consider only wo packe values 1 and α > 1. 1. ACC(r) achieves a compeiive raio of r for arbirary buffer sizes. 2. ACC(r ) achieves a compeiive raio of r if he buffer size ends o infiniy. Proof. We define a paricular opimal offline sraegy OPT (compare [10]). For each inpu sequence, he se of feasible work conserving schedules, i.e., he feasible schedules in which a packe is sen in each ime sep in which he buffer is no empy, is a maroid. Hence, a greedy sraegy can compue an opimal soluion. Firs, OPT considers all α-packes in increasing order of heir arrival, and hereafer, OPT considers all 1-packes in increasing order of heir arrival. We show ha he analysis can be resric o inpu sequences ha saisfy he following wo properies. 1. In each ime sep, excep for he b 1 las ones, ACC sends a packe, where b denoes he buffer size. 2. In each α-overflow ime sep, i.e., he buffer of ACC is compleely filled wih α-packes, exacly b α-packes and no 1-packes arrive. The following wo observaions show ha we can assume w.l.o.g. ha each inpu sequence saisfies he wo properies. Observaion 3. For each inpu sequence σ, i exiss an inpu sequence for which ACC has a leas he same compeiive raio and which saisfies he firs propery. Proof. Afer each ime sep in σ in which he buffer of ACC is empy, inser b 1 addiional ime seps in which no packes arrive. The se of packes sen by ACC does no change and he value of an opimal soluion can only increase. Hence, he compeiive raio of ACC for he alered inpu sequence is a leas as large as for he original sequence σ. Now, we pariion he inpu sequence ino subsequences. A new subsequence sars afer b 1 consecuive ime seps in which no new packes arrive. Obviously, we can assume ha here are never more han b 1 consecuive ime seps in which no new packes arrive. Fix a subsequence σ (i). The buffers of ACC and OPT are empy a he beginning of σ (i), since any packe sored in he buffers of size b is sen during one of he previous b ime seps and no new packes arrive in beween. Furhermore, he buffers of ACC and OPT are empy a he end of σ (i). However, he buffer of ACC is only empy for he las b 1 ime seps of σ (i), due o he consrucion of he subsequences. In all oher ime seps, a packe is sen. Finally noe ha he compeiive raio of ACC for one of he subsequences is a leas as large as for he original sequence σ. Observaion 4. For each inpu sequence σ, i exiss an inpu sequence for which ACC has a leas he same compeiive raio and which saisfies boh properies. Proof. In each α-overflow ime sep of σ, add b α-packes o he arriving packes. None of hese α-packes can be sored by ACC. The se of packes sen by ACC does no change and he value of an opimal soluion can only increase. Hence, he compeiive raio of ACC for he alered inpu sequence is a leas as large as for he original sequence σ. 5

For each α-overflow ime sep, we remove all arriving packes excep for b α-packes. The ses of packes sen by ACC and OPT do no change, since in each ime sep only he b mos valuable arriving packes are relevan. Now, fix an inpu sequence σ ha saisfies boh properies. We pariion σ ino ime inervals. A ime inerval ends wih an α-overflow ime sep, and he nex ime inerval begins wih he ime sep following his α-overflow. Le P i denoe he se of packes arriving in he i-h ime inerval, and le m denoe he oal number of differen ime inervals, i.e., each arriving packe in σ is in m i=1 P i. Le ACC 1 (P i ) (ACC α (P i )) denoe he subse of 1-packes (α-packes) in P i ha are sen by ACC, and le OPT 1 (P i ) (OPT α (P i )) denoe he subse of 1-packes (α-packes) in P i ha are sen by OPT. In order o show he heorem, we prove he claimed compeiive raio for each se of packes P i, i.e., we prove, for each P i, OPT 1 (P i ) + α OPT α (P i ) ACC 1 (P i ) + α ACC α (P i ) r (or r, respecively). (1) The following wo lemmaa give upper bounds on he number of packes sen by OPT. Lemma 5. ACC sends he same number of packes as OPT from each se P i wih i < m. Proof. We prove he lemma by inducion over i. Fix an i < m and assume ha ACC sends he same number of packes as OPT from each se P j wih j < i. As a consequence, ACC and OPT sar sending packes from P i in he same ime sep. Le denoe he las ime sep in which a packe from P i arrives, i.e., he α-overflow ime sep. In ime sep + b 1, ACC sends a packe from P i, since in ime sep he buffer of ACC is compleely filled wih α-packes and he las α-packe in he buffer is a packe from P i. OPT does no send more packes from P i han ACC. Each packe is sored in he buffer for a mos b 1 ime seps. As a consequence, afer ime sep + b 1, OPT can only send packes ha arrive afer ime sep, and hence, hese packes are no in P i. OPT does no send less packes from P i han ACC. Assume for conradicion ha OPT sends less packes from P i han ACC. As a consequence, in ime sep + b 1 a packe from P j wih j > i is sen by OPT. Hence, OPT does no send all α-packes from P i, since b α-packes arrive in ime sep. When one of hese α-packes no send by OPT was considered o be included in he schedule of OPT, i could have been added wihou making he schedule infeasible. This is a conradicion o our definiion of OPT. This concludes he proof of he lemma. Le D P m denoe he se of preemped 1-packes from P m, i.e., D := {p P m p is preemped by ACC}. Lemma 6. m i=1 ( OPT1 (P i ) + OPT α (P i ) ) m i=1 ( ACC1 (P i ) + ACC α (P i ) ) + D. 6

Proof. In he following, we add packes from D o he schedule of ACC, such ha he resuling schedule is maximal, i.e., he schedule becomes infeasible if anoher packe is added. As a consequence, he schedule of OPT conains he same number of packes as our modified schedule, since he se of feasible work conserving schedules is a maroid. Consider he las ime sep in which he buffer of ACC is compleely filled wih packes. Le D denoe he se of packes ha are eiher sored in he buffer of ACC a ime sep or arrive afer and ha are no conained in he schedule of ACC. Observe ha each packe in D is a preemped 1-packe from P m, since is he las ime sep in which he buffer of ACC is compleely filled wih packes. Hence, D D. Adding as much packes as possible from D o he schedule of ACC, such ha he resuling schedule is feasible, produces a maximal schedule. Obviously, adding an addiional packe ha is no in D o he schedule makes he schedule infeasible. Now, we are able o show Inequaliy (1) for P m. Combining Lemma 5 and Lemma 6 yields OPT 1 (P m ) + OPT α (P m ) ACC 1 (P m ) + ACC α (P m ) + D. Since ACC sends all α-packes from P m, OPT α (P m ) = ACC α (P m ). Hence, OPT 1 (P m ) + α OPT α (P m ) ACC 1 (P m ) + α ACC α (P m ) + D. (2) When he las packe of P m 1 arrives, he buffer of ACC is compleely filled wih α-packes and he accoun a is rese o 0. Hence, he preempion of laer arriving packes, i.e., packes in P m, is caused by packes from P m ha are sen by ACC. As a consequence, D (r 1) ( ACC 1 (P m ) + α ACC α (P m ) ). In combinaion wih Inequaliy (2), his gives OPT 1 (P m ) + α OPT α (P m ) ACC 1 (P m ) + α ACC α (P m ) +(r 1) ( ACC 1 (P m ) + α ACC α (P m ) ) = r ( ACC 1 (P m ) + α ACC α (P m ) ). To show Inequaliy (1) for each P i wih i < m, we need o know by how much he number of α-packes sen by OPT exceeds he number of α-packes sen by ACC. For a P i from which ACC sends only α-packes, Inequaliy (1) holds obviously. Consider a P i wih i < m from which ACC sends a leas one 1-packe and b + y α-packes (ACC sends a leas b α-packes from P i ). The only α-packes ha canno be sen by ACC are he ones arriving in he α-overflow ime sep. For each α-packe in he buffer of ACC a his ime sep ha is already sen by OPT, OPT can sore one addiional α-packe ha canno be sen by ACC. The following lemma gives an upper bound on he number of α-packes sen by OPT bu no by ACC. Lemma 7. Consider a se P i wih i < m from which ACC(x) sends a leas one 1-packe and b + y α-packes. A mos b 1 + x + y (x 1) (x 1) α + x α-packes in he buffer of ACC(x) are already sen by OPT righ before he α-overflow ime sep of P i. 7

Proof. Consider he laes ime sep before he α-overflow ime sep in which he number of α-packes in he buffer of ACC ha are already sen by OPT is increased from n 1 o n. Hence, ACC sends a 1-packe p and OPT sends an α-packe ha arrived afer p and is sored in he buffer of ACC. Each α-packe in he buffer of ACC ha is already sen by OPT arrived laer han p. Le q denoe he firs α-packe in he buffer of ACC ha is already sen by OPT, and le denoe he ime sep in which q arrives. Each α-packe in he buffer of ACC has increased he accoun a by (x 1) α. In addiion, he accoun a is increased by z (x 1), where z denoes he number of 1-packes sen by ACC from o 1. Observe ha he accoun a is no rese o 0 from o, since he 1- packe p is sored in he buffer from o. However, he value of he accoun a is less han 1 righ before p is sen by ACC, since oherwise p would have been preemped. Hence, a leas n (x 1) α + z (x 1) 1 1-packes are preemped from o. All he preemped 1-packes arrive before p. Since only one α-packe can be sen by OPT in each ime sep, a leas n 1 packes are sen from o. In fac, z + y n 1 packes are sen from o, where y denoes he number of α-packes sen by ACC from o. Noe ha y y. Afer he arrival of q in he ime sep, here are less or equal han b 1 oher packes in he buffer of ACC and all of hem arrived earlier han q. Unil ime sep, a leas n (x 1) α + z (x 1) 1 of hem are preemped, z + y n 1 of hem are sen, and p is sill in he buffer. Hence, n (x 1) α + z (x 1) 1 + z + y + 1 b 1. Alogeher, b 1 + x + y (x 1) b 1 + x + y (x 1) which concludes he proof of he lemma. Due o Lemma 5, b 1 + x + (n 1 z) (x 1) n (x 1) α + z (x 1) 1 + z + y + 1 +x + (n 1 z) (x 1) = n (x 1) α + n (x 1) + z + y + 1 n (x 1) α + n (x 1) + n = n ((x 1) α + x), OPT 1 (P i ) + α OPT α (P i ) ACC 1 (P i ) + α ACC α (P i ) Hence, i remains o show ha = α OPT α (P i ) ACC 1 (P i ) + α ACC α (P i ) OPT 1 (P i ) α OPT α (P i ) (α 1) ACC α (P i ) + OPT α. (P i ) (α x) OPT α (P i ) x (α 1) ACC α (P i ), for x := r and x := r wih b. Due o Lemma 7, his inequaliy is equivalen o ( ) b 1 + x + (x 1) y (α x) b + y + x (α 1) (b + y), (x 1) α + x 1 From o denoes he ime inerval from o excluding ime sep. 8

which is equivalen o b 1 + x + (x 1) y (α x) (x 1) α (b + y). (3) (x 1) α + x Suppose ha x := r and b. Observe ha Then, i follows ha ( ) 1 1/b + lim (α r r /b ) (r 1) α b (r 1) α + r ( ) 1 = (α r ) (r 1) α 0. (r 1) α + r lim (α r ) b ( b 1 + r (r 1) α + r ) (r 1) α b 0. Finally, Inequaliy (3) can be shown as follows b 1 + lim (α r r + (r 1) y ) (r 1) α (b + y) b (r 1) α + r ( ) b 1 + r + (r 1) y lim (α r ) (r 1) α (b + y) b (r 1) α + r ( ) b 1 + r lim (α r ) (r 1) α b 0. b (r 1) α + r Suppose ha x := r. Defining k := b + (r 1) y, we ge b 1 + r + (r 1) y (α r) (r 1) α (b + y) (r 1) α + r k 1 + r = (α r) (r 1) α (k + (2 r) y) (r 1) α + r k 1 + r (α r) (r 1) α k. (r 1) α + r We disinguish wo cases. Suppose ha k 5/2. Observe ha k 1 + r (α r) (r 1) α k (r 1) α + r ( k 1 + r (α r) (r 1) α + r ( 1 + r = (α r) (r 1) α + r ) (r 1) α k ) + (r 1)2 α 2 + (1 + r r 2 ) α r k. (r 1) α + r 9

The las erm decreases wih increasing k. Hence, Inequaliy (3) can be shown as follows ( ) k 1 + r (α r) (r 1) α k (r 1) α + r ( ) 3/2 + r 5(r 1) α (α r) 0. (r 1) α + r 2 Suppose ha 2 k < 5/2. Observe ha 0 k 1 + r (r 1) α + r < 3/2 + r (r 1) α + r 3/2 + r 2r 1 < 2. As a consequence, (k 1 + r)/((r 1) α + r) equals eiher 0 or 1. If i equals 0, Inequaliy (3) follows obviously. Oherwise, k (r 1) α + 1. If k = 2, his gives α 1/(r 1). Hence, Inequaliy (3) can be shown as follows (α r) k 1 + r (r 1) α + r (r 1) α k = (α r) 2(r 1) α ( 1 r 1 r ) 2 = 0. If k > 2, y 1, since oherwise k would be inegral. Hence, Inequaliy (3) can be shown as follows k 1 + r (α r) (r 1) α (k + (2 r) y) (r 1) α + r (α r) (r 1) α (k + (2 r)) This concludes he proof of he heorem. (α r) (r 1) α ((r 1) α + 1 + (2 r)) 0. 3 The Preempive Greedy Sraegy Kesselman, Mansour, and van See [9] inroduce he preempive greedy sraegy (PG) wih he parameer β > 1. When a packe p arrives, PG does he following. 1. Find he firs packe, i.e., he packe closes o he fron of he buffer, p, wih v(p ) v(p)/β. If such a packe p exiss, drop i (p is called preemped by p). 2. If here is an unoccupied locaion in he buffer, sore p in he buffer. 3. Oherwise, find a packe p wih he smalles value among he packes in he buffer. If v(p ) < v(p), drop p (p is called ejeced by p) and sore p in he buffer. Oherwise, drop p (p is called rejeced). Bansal e al. [5] sudy a modified version of PG. The only difference is ha sep 1 of PG is subsiued by he following. 1. Find he firs packe p, wih v(p ) v(p)/β and v(p ) is no larger han he value of he packe sored afer p in he buffer. If such a packe exiss, drop i. 10

Noe ha his modificaion does no improve he overall performance of he sraegy [6]. New approaches are needed, since, due o he following lower and upper bound, he compeiive raio of PG canno be furher improved significanly. A basic concep of PG is ha, for each arriving packe p, he firs packe whose value is a mos v(p)/β is preemped. A firs sigh, i seems more reasonable ha, insead, he packe wih he smalles value is preemped. Bu in fac, he preempion of he firs packe whose value is suiable small enough is a crucial propery o achieve a compeiive raio smaller han 2. However, his can urn ou o be a grea disadvanage as he firs inpu sequence in he following lower bound shows. This disadvanage diminishes wih increasing β. On he oher hand, oo few packes are preemped for larger β as he second inpu sequence in he following lower bound shows. An approach o ackle his problem migh be he following: If, for large β, he value of a single packe does no suffice o preemp anoher packe, he values of more han one packe are combined for preempion. Noe ha, in he case of only wo packe values, we achieve wih his idea an opimal sraegy. 3.1 Lower Bound The following heorem gives an lower bound on he compeiive raio of PG. Theorem 8. The compeiive raio of PG is a leas 1 + 1/ 2 1.707. Proof. Fix an even buffer size b. Depending on β, we disinguish he following wo cases. Suppose ha β 2 + 2. The inpu sequence consiss of n consecuive phases defined as follows. Phase 1 i < n consiss of b/2 ime seps. In ime sep 1 of he i-h phase, a firs b packes of value ε and finally b/2 packe of value β i arrive. In he remaining b/2 1 ime seps, new packes do no arrive. Phase n consiss of one ime sep. In his ime sep, b packes of value β n 1 arrive. For his inpu sequence, PG produces value lim ε 0 n 1 i=1 ( ) b 2 ε + b β n 1 = b β n 1, and he opimal value is n 1 ( ) b 2 βi + b β n 1 = b 3βn 2β n 1 β 2(β 1) i=1. Hence, he compeiive raio is Suppose ha β > 2 + 2. 3β n 2β n 1 β lim n 2(β n β n 1 ) = 1 + β 2(β 1) 1 + 1. 2 The inpu sequence consiss of n consecuive phases defined as follows. 11

Phase 1 consiss of b 1 ime seps. In ime sep 1, a firs b 1 packes of value 1 and finally one packe of value α < β arrive. In each of he remaining b 2 ime seps, one packe of value α arrives. Phase 1 < i < n consiss of b 1 ime seps. In each of hese ime seps, one packe of value α i arrives. Phase n consiss of one ime sep. In his ime sep, b packes of value α n 1 arrive. For his inpu sequence, PG produces value n 2 ((b 1) α i ) + b α n 1 = b αn 1 b αn 1 b 1 b α 1 i=0, and he opimal value is n 1 ((b 1) α i ) + b α n 1 = b i=1 ( ) 2 1 b α n α n 1 b 1 b α α 1. Hence, he compeiive raio is ( 2 1 b lim lim lim α β n b ) α n α n 1 b 1 b α n 1 b αn 1 b 1 b 2α n α n 1 α 2α 1 = lim lim α β n α n = lim = 1 + β 1 1 + 1. 1 α β α β 2 This concludes he proof of he heorem. 3.2 Upper Bound The following heorem gives an upper bound on he compeiive raio of PG. Theorem 9. PG achieves a compeiive raio of 3 1.732 for β = 2 + 3. Proof. Le OPT denoe an opimal offline sraegy. We assume ha OPT only sores packes in is buffer ha are sen by OPT. Furher, we assume ha, a he arrival of each packe, he buffer of PG is compleely filled wih packes. If here are unoccupied locaions in he buffer of PG, i is assumed ha dummy packes of value 0 are sored a hese locaions which are always a he end of he buffer. Hence, each arriving packe eiher preemps anoher packe, ejecs anoher packe, or is rejeced. Fix an inpu sequence of arriving packes. This inpu sequence can also be regarded as a sequence σ = σ 1 σ 2 of arrival and send evens, where each arrival of a new packe corresponds o an arrival even and each sending of a packe corresponds o a send even. The even sequence σ is pariioned ino ime seps, where he firs ime sep sars wih he firs even and a new ime sep sars righ afer each send even. Le S pg (S op ) denoe he se of packes sen by PG (OPT) by he end of even σ, i.e., all packes sen in he evens σ 1,..., σ (including σ ). Le B pg (B op ) denoe he se of packes sored in he buffer of PG (OPT) a he end of σ. For a packe p B pg, we call c (p) he charge of p a he end of σ. Furher, we call D he se of packes wih a deposi a he end α 12

of σ. Noe ha charges and deposis are wo independen conceps we use. Iniially, D 0 :=. The goal is o choose c (p) and D in such a way ha, for each even σ, he main inequaliy r v(p) + c (p) v(p) p S pg p B pg p S op D is rue, wih r := 3. As a consequence, his yields he heorem. Le pg ( op ) denoe he aleraions of he lef (righ) side of he main inequaliy a he even σ, i.e., pg := r v(p) + c (p) c 1 (p) and op := p S pg \S pg 1 p (S op D )\(S op 1 D 1) p B pg v(p). p B pg 1 Obviously, he main inequaliy is rue before he firs even, since packes have no been sen so far and he buffers and he se of packes wih a deposi are empy. Hence, i is sufficien o show, for each even σ, pg op, since his yields he main inequaliy. Firs, we give an inuiion for he basic ideas of he proof. Then, we presen he formal proof. The basic idea for he se D is simple. Packes sored exclusively in he buffer of OPT a he end of even σ, especially packes already sen by PG, could be a problem, if PG canno send a packe, i.e., he buffer of PG is empy, when hose packes are sen by OPT. The lef side of he main inequaliy is no increased a hese evens, and i is crucial for he proof ha he same is rue for he righ side of he main inequaliy. Hence, hese packes have o be conained in D. Inuiively, PG has already gained enough value o pay hese packes in advance, i.e., before hey are sen by OPT. The basic idea for c (p) is he following. In case of a send even σ in which OPT sends a much more valuable packe han PG ha is no in D 1, he righ side of he main inequaliy is increased by a large amoun and we have o compensae his by increasing he charge of packes sored in he buffer of PG. I is fairly unproblemaic o charge a packe up o (r 1) imes is own value because if such a packe is sen by PG and OPT in he same send even, he lef side of he main inequaliy is sill increased by he same amoun as he righ side of he main inequaliy. In any case, larger charges are only allowed for packes ha are exclusively in he buffer of PG. In case of a buffer overflow in he buffer of PG in which a charged packe is ejeced, his charge has o be ransferred o anoher packe in he buffer of PG. This is problemaic for an ejeced packe ha is charged by more han (r 1) imes is own value, since, afer his charge is ransferred o anoher packe in he buffer of PG, here migh be a packe charged by more han (r 1) imes is own value ha is no exclusively in he buffer of PG. Therefore we inroduce he concep of buddies. A packe sored exclusively in he buffer of PG migh be charged by 2(r 1) imes is own value only if here is anoher packe in he buffer of PG ha is no charged a all. We call he packe wih no charge buddy for he packe wih he high charge. Unforunaely, he precise definiion of charges is slighly more complicaed. Before we define he charges in deail, we need some preliminaries. For each wo packes p and p, we wrie p p if p arrives before p in he inpu sequence. Furher, for each packe p and he undefined symbol, p, p, and. Each p B pg can have assigned anoher 13

s (p) c (p) commen BC (r 2) v(p) buddy wih credi B 0 buddy U (r/β) v(p) + (2 r) v min (p) unproblemaic E (r 1) v(p) exclusively in B pg EB 2(r 1) v(p) exclusively in B pg, i.e., no in B op wih buddy Figure 1: Definiion of he charge c (p) of a packe p B pg a he end of even σ. The charges are lised in increasing order, e.g., a packe in sae E is a leas as much charged as a packe of same value in sae U. Noe ha he charge in case of sae BC is negaive. Furher, noe ha v min (p) v(p). If v min (p) = v(p), he charges in sae U and E are he same for packe p. p B pg as buddy if p p. However, each p B pg is assigned as buddy for a mos one oher packe. If p B pg has assigned anoher p B pg as buddy a he end of even σ, define b (p) := p, oherwise, define b (p) :=. Furher, for each p B pg, b (p) :=. Finally, for each p B pg, define v min (p) := min{v(p ) B pg p p}. Each p B pg is in one of he five saes BC, B, U, E, and EB. Le s (p) denoe he sae of p a he end of even σ, and define s ( ) :=. Le BC, B, U, E, and EB denoe he se of packes ha are in sae BC, B, U, E, and EB, respecively, a he end of even σ. The iniial sae of each packe is B, and dummy packes of value 0 are always in sae B. The charge c (p) of a packe p a he end of even σ is defined in Figure 1. Noe ha he charge of a packe, excep for packes in sae U, does no change as long as his packe says in he same sae. The charge of a packe in sae U can only increase, since v min (p) v min +1 (p). Le P denoe he se of packes ha are preemped by PG by he end of even σ. For each packe p, if p preemps anoher packe p, define d(p) := p, oherwise, define d(p) :=. A packe p ransiively preemps anoher packe p, if eiher d(p) = p, p preemps a packe ha ransiively preemps p, or p ejecs a packe ha ransiively preemps p. For each p P, if p is ransiively preemped by a packe p B pg, define ˆd (p ) := p, oherwise, define ˆd (p ) :=. For each p P, define ˆd (p ) :=. Figure 2 gives an overview of our noaion. In order o prove he heorem, we show he following five invarians by inducion over he even sequence σ. To shoren noaion, we define X := (P S pg ) (B op \ D ). op. I2: If p E EB, hen p B op. I3: If p EB, hen b (p) BC B. I4: If p X, hen ˆd (p) BC B. I5: If p B pg \ BC, hen b 1 (p) d(p). Observe ha he invarians have only o be verified in he following cases. I1: Always. I2: For each packe p (E EB ) \ (E 1 EB 1 ). 14

noaion S pg, S op B pg commen The se of packes sen by PG and OPT by he end of σ., B op The se of packes sored in he buffer of PG and OPT a he end of σ. P The se of packes preemped by PG by he end of σ. D The se of packes wih a deposi a he end of σ. X Shor noaion for (P S pg ) (B op \ D ). c (p) The charge of he packe p B pg a he end of σ, which is deermined by is sae. s (p) The sae of he packe p B pg. Each packe p B pg is in one of he five saes BC, B, U, E, or EB. v min (p) The value of he leas valuable packe sored in he buffer of PG in fron of p B pg. b (p) The buddy packe of he packe p. Equals if p has no buddy or p B pg. b 1 (p) The packe for which p is a buddy. Equals if p is no a buddy for anoher packe. d(p) The packe ha is preemped by p. ˆd (p) Equals if p does no preemp anoher packe. The packe p B pg ha ransiively preemped p. Equals if p was no preemped, i.e., p P, or here is no packe in he buffer of PG which ransiively preemped p. p p The packe p arrives before he packe p. Figure 2: Informal overview of our noaion. I3: For each packe p wih (p EB \ EB 1 ) (b 1 (p) (BC 1 B 1 ) \ (BC B )) (b 1 (p) b (p)). I4: For each packe p wih (p X \ X 1 ) ( ˆd 1 (p) (BC 1 B 1 ) \ (BC B )) ( ˆd 1 (p) ˆd (p)). I5: For each packe p wih (p (B pg \ BC ) \ (B pg 1 \ BC 1)) (b 1 1 (p) b 1 (p)). The following lemma is used o dramaically reduce he number of cases we have o consider. Whenever we encouner a siuaion during he inducion where B op P S pg B pg, we manipulae he buffer conens of OPT in such a way ha B op P S pg B pg. The five invarians coninue o hold afer his manipulaion. Thereafer, we can coninue he inducion. Lemma 10. Assume ha σ is he firs even wih B op conens of OPT can be manipulaed in such a way ha B op invarians coninue o hold. Proof. Assume ha σ is he firs even wih B op P S pg P S pg B pg P S pg. Then, he buffer B pg and he five B pg, i.e., he buffer of OPT conains a packe ha was ejeced or rejeced by PG. Since σ is he firs even wih B op 15

P S pg B pg, a packe p mus have been ejeced or rejeced by PG in σ. This also implies ha σ is an arrival even. In he following, we assume ha p is rejeced by PG bu sored in he buffer of OPT. The argumens for he case ha p is ejeced are analogous. Since OPT sores p in is buffer and he buffer of PG is compleely filled wih packes, here exiss a packe q B pg \ B op. The value v(q) of q has o be a leas as large as v(p). Oherwise, q would have been ejeced by PG and p would have been sored in he buffer of PG. Define v := v(p). Afer p arrived, we manipulae he buffer conens of OPT in he following way: The arrival ime of p is se o he arrival ime of q, i.e., he packes sored in he buffer of OPT are reordered such ha p is placed a he posiion of q if q would be conained in he buffer of OPT. This reordering does no change he se of packes sen by OPT and hence, does no change he oal value gained by OPT. In addiion, we manipulae he value of p. We increase he value of p o he value of q. Afer boh manipulaions, he aribues of he packe p B op \ B pg are idenical o he. As a consequence, p can be idenified wih q, i.e., we can assume ha packe q B pg \ B op p is acually he packe q and herefore sored in he buffer of PG. The Invarians I3, I4, and I5 are no effeced by our manipulaion, since changes are no made in he buffer of PG and q P S pg. If s (q) {E, EB}, Invarian I2 is no effeced eiher. Oherwise, se s (q) := U and, if q was in sae EB, se s (b 1 (q)) := U (due o I3 b 1 (q) exiss and is in sae BC or B in his case). Thus, Invarians I2 I5 coninue o hold. I remains o sudy he effec of our manipulaion on he main inequaliy. If s (q) {E, EB} he main inequaliy does no change. If q was in sae E and is sae changed o U, he lef side of he main inequaliy is decreased by a mos (r 1) v(q) ((r/β) v(q)+(2 r) v min (q)) = (2 r) (v(q) v min (q)) v(q) v, since r/β = 2r 3 and p is rejeced a σ. If q was in sae EB and is sae changed o U, he lef side of he main inequaliy is decreased by a mos 2(r 1) v(q) ((r/β) v(q)+(2 r) v min (q)) = v(q)+(r 2) v min (q) v(q)+(r 2) v. In his case, he sae of b 1 (q) changed from BC or B o U. This increases he lef side of he main inequaliy by a leas (r/β) v(b 1 (q)) + (2 r) v min (b 1 (q)) (r/β) v + (2 r) v (r 1) v. Hence, in oal he lef side of he main inequaliy is decreased by a mos v(q) + (r 2) v (r 1) v = v(q) v. Hence, he lef side of he main inequaliy is decreased by a mos v(q) v. As a consequence, we can only guaranee ha v(q) v + r v(p ) v(p ) p S pg l p S op l afer he las even σ l in he sequence of evens σ. This is no sufficien o show he heorem. Forunaely, by virually increasing he value of p we have also increased p S op v(p ) by l v(q) v, i.e., he real oal value of OPT is smaller by v(q) v. Finally, v(q) v + r PG(σ) = v(q) v + r v(p ) v(p ) = OPT(σ) + v(q) v. which concludes he proof of he lemma. p S pg l p S op l 16

case packes concerned verificaion a1 I2 I3 q, b 1 1 (q) s (q) =, b (b 1 1 (q)) = p B I4 q, {p ˆd 1 (p ) = q} ˆd (q) = p B, ˆd 1 (p ) = q ˆd (p ) = p B I5 p, q b 1 (p) = b 1 1 (q) q = d(p), q Bpg a2 I2 I3 q, b 1 1 (q) s (q) =, s (b 1 1 (q)) = s 1(b 1 I3 1 (q)) EB I4 q q I2 B op 1 q Bop X {p ˆd 1 (p ) = q} ˆd 1 (p ) = q I4 p X 1 {q} X I5 p, q b 1 (p) = d(p), q B pg a3 I2 I3 q, b 1 1 (q) s (q) =, s (b 1 1 (q)) = s 1(b 1 I3 1 (q)) EB I4 q, {p ˆd 1 (p ) = q} q D q X, ˆd 1 (p ) = q I4 p X 1 {q} X I5 p, q b 1 (p) = d(p), q B pg a4 I2 I3 q, b 1 1 (q) s (q) =, b (b 1 1 (q)) = p B I4 q, {p ˆd 1 (p ) = q} q P S pg X, ˆd 1 (p ) = q ˆd (p ) = p B I5 p, q b 1 (p) = b 1 1 (q) = d(p), q Bpg a5 I2 I3 q, b 1 1 (q) s (q) =, s (b 1 1 (q)) = s 1(b 1 I3 1 (q)) EB I4 q, {p ˆd 1 (p ) = q} q P S pg X, ˆd 1 (p ) = q I4 p X 1 X I5 p, q b 1 a6 I2 I3 q, b 1 1 (q), b 1(q) I4 q, {p ˆd 1 (p ) = q} q P S pg b 1 (q) I5 p, q b 1 b 1 (q) (p) = = d(p), q B pg s (q) =, s (b 1 1 (q)) = s 1(b 1 I3 1 (q)) EB, b 1 (q) U X, ˆd 1 (p ) = q I4 p X 1 X b 1 (q) P S pg X (p) = = d(p), q B pg (b 1 (q)) = d(b 1 (q)) b 1 Figure 3: Verificaion of he Invarians I2 I5 for he Cases a1 a6. Fix an arrival even σ in which a packe p arrives. We disinguish he following cases. If no menioned oherwise, everyhing remains unchanged a even σ. We only consider he Invarian I1. For he verificaion of he Invarians I2 I5, see Figure 3. p preemps anoher packe q a1: q B 1 BC 1 17

Changes: b (b 1 1 (q)) := p and s (p) := B a2: q E 1 EB 1 Changes: s (p) := U 0 = op (r/β) v(p) 2(r 1) v(q) (r/β) v(p) 2(r 1) v(p)/β = ((2 r)/β) v(p) > 0 = op a3: q U 1 Changes: s (p) := U and D := D 1 {q} p ejecs anoher packe q (r/β) v(p) ((r/β) v(q) + (2 r) v(q)) (r/β) (β v(q)) (r 1) v(q) = v(q) = op a4: q B 1 BC 1 Changes: s (p) := B and b (b 1 1 (q)) := p a5: q E 1 U 1 Changes: s (p) := U 0 = op (r/β) v(p) + (2 r) v 1(p) min (r 1) v(q) (r/β) v(q) + (2 r) v(q) (r 1) v(q) = ((2r 3) + (2 r) (r 1)) v(q) = 0 = op a6: q EB 1 Changes: s (p) := U and s (b 1 (q)) := U (Due o I3, b 1 (q) B 1 BC 1.) p is rejeced (r/β) v(p) + (2 r) v 1(p) min +(r/β) v(b 1 (q)) + (2 r) v 1(b min 1 (q)) 2(r 1) v(q) (r/β) v(q) + (2 r) v(q) +(r/β) v(q) + (2 r) v(q) 2(r 1) v(q) = (2((2r 3) + (2 r)) 2(r 1)) v(q) = 0 = op Changes: (Due o Lemma 10, p is also no sored in he buffer of OPT.) 18

b9, b13 BC b15 b11, b13, b14 B b10, b12, b15 b4 b6, b14 b8 U E EB Figure 4: Possible sae ransiions a a send even. The labels a he edges specify he cases in which he respecive sae ransiion could occur. Fix a send even σ in which PG sends packe p and OPT sends packe q. Noe ha due o Lemma 10, q P 1 S pg 1 Bpg 1. Since a new dummy packe of value 0 is sored in he buffer of PG afer a packe is sen, a packe u B B pg \B pg 1 exiss wih s (u B ) = B. We can assign u B as buddy o anoher packe a his even, since u B B pg 1. We disinguish he following cases. If no menioned oherwise, everyhing remains unchanged a even σ. We only consider he Invarian I1. For he verificaion of he Invarians I2 I5, see Figure 5, Figure 6, and Figure 7. In Figure 4, we depic he possible sae ransiions a σ. q P 1 S pg 1 b1: q D 1 and p B 1 BC 1 Changes: D := D 1 {p} {p ˆd 1 (p ) = p} r v(p) v(p) + v(p)/β i op b2: q D 1 and p B op \ (B 1 BC 1 ) Changes: D := D 1 {p} (Due o I2, p U 1.) r v(p) ((r/β) v(p) + (2 r) v 1(p)) min v(p) = op b3: q D 1 and p B op (B 1 BC 1 ) Changes: i=1 r v(p) 2(r 1) v(p) 0 = op b4: q D 1 and p B 1 BC 1 Changes: s ( ˆd 1 (q)) := U, D := D 1 {p} {p ˆd 1 (p ) = p} {q q ˆd 1 (q ) = ˆd 1 (q)}, b (b 1 1 ( ˆd 1 (q))) := u B (Due o I4, ˆd 1 (q) BC 1 B 1.) r v(p) + (r/β) v( ˆd 1 (q)) v(p) + v(p)/β i + v( ˆd 1 (q))/β i v(p) + i=1 v(p)/β i + i=1 i=1 q, ˆd 1 (q )= ˆd 1 (q) v(q ) op 19

case packes concerned verificaion b1 I2 I3 p, b 1 1 (p) s (p) =, b 1 1 (p) = I4 p, {p ˆd 1 (p ) = p} p D p X, ˆd 1 (p ) = p p D p X I5 p, b 1 (p), u B p B pg b2 I2 and I3 p, b 1 1 (p) b3 I4 p, {p ˆd 1 (p ) = p} p B op I5 p, b 1 (p), u B p B pg, b 1 (b 1 (p)) = d(b 1 (p)), d(u B ) = s (p) =, b 1 1 (p) = \ D X, ˆd 1 (p ) = p I4 p X 1 X, b 1 (b 1 (p)) = d(b 1 (p)), d(u B ) = b4 I2 I3 p, b 1 1 (p), b 1 1 ( ˆd 1 (q)) s (p) =, b 1 1 (p) =, b (b 1 1 ( ˆd 1 (q))) = u B B I4 p, {p ˆd 1 (p ) = p} p D p X, ˆd 1 (p ) = p p D p X q q B op q X {p ˆd 1 (p ) = ˆd 1 (q)} ˆd 1 (p ) = ˆd 1 (q) p D {q} p X I5 p, b 1 (p) p B pg, b 1 (b 1 (p)) = d(b 1 (p)) ˆd 1 (q), u B ( ˆd 1 (q)) = d( ˆd 1 (q)), d(u B ) = b5 I2 and I3 p, b 1 1 (p), b 1 b 1 1 ( ˆd 1 (q)) b6 I4 p, {p ˆd 1 (p ) = p} p B op q q B op s (p) =, b 1 1 (p) =, b (b 1 1 ( ˆd 1 (q))) = u B B \ D X, ˆd 1 (p ) = p p X 1 X q X {p ˆd 1 (p ) = ˆd 1 (q)} ˆd 1 (p ) = ˆd 1 (q) p D {q} p X I5 p, b 1 (p) p B pg, b 1 (b 1 (p)) = d(b 1 (p)) ˆd 1 (q), u B ( ˆd 1 (q)) = d( ˆd 1 (q)), d(u B ) = b 1 Figure 5: Verificaion of he Invarians I2 I5 for he Cases b1 b6. b5: q D 1 and p B op \ (B 1 BC 1 ) Changes: s ( ˆd 1 (q)) := U, D := D 1 {p} {q q ˆd 1 (q ) = ˆd 1 (q)}, b (b 1 1 ( ˆd 1 (q))) := u B (Due o I2, p U 1. Due o I4, ˆd 1 (q) BC 1 B 1.) r v(p) ((r/β) v(p) + (2 r) v 1(p)) min + (r/β) v( ˆd 1 (q)) v(p) + (r/β) v( ˆd 1 (q)) v(p) + i=1 v( ˆd 1 (q))/β i op b6: q D 1 and p B op (B 1 BC 1 ) Changes: s ( ˆd 1 (q)) := U, D := D 1 {q q ˆd 1 (q ) = ˆd 1 (q)}, b (b 1 1 ( ˆd 1 (q))) := u B 20

b7: q = p (Due o I4, ˆd 1 (q) BC 1 B 1.) r v(p) 2(r 1) v(p) + (r/β) v( ˆd 1 (q)) (r/β) v( ˆd 1 (q)) v( ˆd 1 (q))/β i op i=1 Changes: (Due o I2, p U 1 BC 1 B 1.) q B pg 1 \ {p} r v(p) ((r/β) v(p) + (2 r) v 1(p)) min v(p) = op b8: q U 1 Changes: b (q) := u B, s (q) := EB r v(p) c 1 (p) +2(r 1) v(q) ((r/β) v(q) + (2 r) v 1(q)) min r v(p) 2(r 1) v(p) b9: q BC 1 Changes: b (b 1 1 (q)) := u B, s (q) := E +2(r 1) v(q) ((r/β) v(q) + (2 r) v(p)) = v(q) = op r v(p) c 1 (p) + (r 1) v(q) (r 2) v(q) v(q) = op b10: q B 1 and v(p) < v(q)/β Changes: b (q) := u B, s (q) := EB (Due o I5, b 1 1 (q) d(q), i.e., b 1 1 b11: q B 1 and v(p) v(q)/β and p EB 1 Changes: b (b 1 1 (q)) := u B, s (q) := E (q) Bpg 1, since v(p) < v(q)/β.) r v(p) c 1 (p) + 2(r 1) v(q) > v(q) = op = r v(p) c 1 (p) + (r 1) v(q) v(p) + (r 1) v(q) (1/β + (r 1)) v(q) = v(q) = op b12: q B 1 and v(p) v(q)/β and p EB 1 and b 1 1 (q) = Changes: b (q) := u B, s (q) := EB r v(p) c 1 (p) + 2(r 1) v(q) v(q) = op 21

case packes concerned verificaion b7 I2 I3 p, b 1 1 (p) s (p) =, b 1 1 (p) = I4 p p B op p X, {p ˆd 1 (p ) = p} ˆd 1 (p ) = p p p p B op X I5 p, b 1 (p), u B p B pg b8 I2 q q B op I3 p, b 1 1 (p), q, b 1 (b 1 (p)) = d(b 1 (p)), d(u B ) = s (p) =, b 1 1 (p) =, b (q) = u B B I4 p p B op p X, {p ˆd 1 (p ) = p} ˆd 1 (p ) = p p p q p B op X I5 p, b 1 (p), u B p B pg, b 1 (b 1 (p)) = d(b 1 (p)), d(u B ) = b9 I2 q q B op I3 p, b 1 1 (p), b 1 1 (q) s (p) =, b 1 1 (p) =, b (b 1 1 (q)) = u B B I4 p p B op p X, {p ˆd 1 (p ) = p} ˆd 1 (p ) = p p p q p B op X {p ˆd 1 (p ) = q} ˆd 1 (p ) = q p q p B op X I5 p, b 1 (p) p B pg q, u B b 1 b10 I2 q q B op I3 p, b 1 1 (p), q, b 1 1 (q), b 1 (q) =, d(u B ) = I4 p p q p B op (b 1 (p)) = d(b 1 (p)) s (p) =, b 1 1 (p) =, b (q) = u B B, b 1 I5 1 (q) = X {p ˆd 1 (p ) = p} ˆd 1 (p ) = p p p q p B op {p ˆd 1 (p ) = q} ˆd 1 (p ) = q p q p B op I5 p, b 1 (p), u B p B pg b11 I2 q q B op I3 p, b 1 1 (p), b 1 1 (q), b 1 I4 p p q p B op X X (b 1 (p)) = d(b 1 (p)), d(u B ) = s (p) =, b 1 1 (p) =, b (b 1 1 (q)) = u B B X {p ˆd 1 (p ) = p} ˆd 1 (p ) = p p p q p B op {p ˆd 1 (p ) = q} ˆd 1 (p ) = q p q p B op I5 p, b 1 (p), u B p B pg, b 1 X X (b 1 (p)) = d(b 1 (p)), d(u B ) = Figure 6: Verificaion of he Invarians I2 I5 for he Cases b7 b11. b13: q B 1 and v(p) v(q)/β and p EB 1 and b 1 (p) b 1 1 (q) Changes: s (b 1 (p)) := E, b (b 1 1 (q)) := u B, s (q) := E (Due o I5, b 1 1 (q) d(q), i.e., v(b 1(p)) v(q)/β. Due o I3, b 1 (p) B 1 22

case packes concerned verificaion b12 I2 q q B op I3 p, b 1 1 (p), q, b 1 1 (q) s (p) =, b 1 1 (p) =, b (q) = u B B, b 1 I4 p p q p B op X {p ˆd 1 (p ) = p} ˆd 1 (p ) = p p p q p B op {p ˆd 1 (p ) = q} ˆd 1 (p ) = q p q p B op I5 p, b 1 (p), u B p B pg b13 I2 q, b 1 (p) q B op I3 p, b 1 1 (p), b 1 1 (q) I4 p p q p B op X 1 (q) = X, b 1 (b 1 (p)) = d(b 1 (p)), d(u B ) =, b 1 (p) b 1 1 (q) q b 1(p) B op s (p) =, b 1 1 (p) =, b (b 1 1 (q)) = u B B X {p ˆd 1 (p ) = p} ˆd 1 (p ) = p p p q p B op {p ˆd 1 (p ) = q} ˆd 1 (p ) = q p q p B op I5 p, b 1 (p), u B p B pg b14 I2 q q B op I3 p, b 1 1 (p), b 1 1 (q), b 1 I4 p p q p B op X X (b 1 (p)) = d(b 1 (p)), d(u B ) = s (p) =, b 1 1 (p) =, b (b 1 1 (q)) = u B B X {p ˆd 1 (p ) = p} ˆd 1 (p ) = p p p q p B op {p ˆd 1 (p ) = q} ˆd 1 (p ) = q p q p B op X {p ˆd 1 (p ) = b 1 (p)} ˆd 1 (p ) = b 1 (p) p D p X I5 p, b 1 (p), u B p B pg b15 I2 q q B op I3 p, b 1 q, b 1 1 (p) 1 (q) I4 p p q p B op X, b 1 (b 1 (p)) = d(b 1 (p)), d(u B ) = s (p) =, b 1 1 (p) = b (q) = u B B, b (b 1 1 (q)) = b 1(p) BC X {p ˆd 1 (p ) = p} ˆd 1 (p ) = p p p q p B op {p ˆd 1 (p ) = q} ˆd 1 (p ) = q p q p B op, b 1 (p) BC, d(u B ) = I5 p, b 1 (p), u B p B pg X Figure 7: Verificaion of he Invarians I2 I5 for he Cases b12 b15. X BC 1.) r v(p) 2(r 1) v(p) + (r 1) v(q) + (r 1) v(b 1 (p)) (2 r) v(q)/β + (r 1) v(q) + (r 1) v(q)/β = (1/β + (r 1)) v(q) = v(q) = op b14: q B 1 and v(p) v(q)/β and p EB 1 and b 1 1 (q) b 1(p) and v(b 1 (p)) > 2v(q) Changes: s (b 1 (p)) := U, D := D 1 {p ˆd 1 (p ) = b 1 (p)}, b (b 1 1 (q)) := u B, s (q) = E 23

(Due o I3, b 1 (p) B 1 BC 1.) r v(p) c 1 (p) + (r/β) v(b 1 (p)) c 1 (b 1 (p)) + (r 1) v(q) (2 r) v(p) + (r/β) v(b 1 (p)) + (r 1) v(q) = (2 r) v(p) + (3r 5)/2 v(b 1 (p)) +(r 1) v(q) + v(b 1 (p))/(β 1) (2 r) v(q)/β + (3r 5) v(q) + (r 1) v(q) +v(b 1 (p))/(β 1) = v(q) + v(b 1 (p))/β i op i=1 b15: q B 1 and v(p) v(q)/β and p EB 1 and b 1 1 (q) b 1(p) and v(b 1 (p)) 2v(q) Changes: s (b 1 (p)) := BC, b (b 1 1 (q)) := b 1(p), b (q) := u B, s (q) := EB (Due o I3, b 1 (p) B 1 BC 1.) = r v(p) c 1 (p) +(r 2) v(b 1 (p)) c 1 (b 1 (p)) + 2(r 1) v(q) This concludes he proof of he heorem. References (2 r) v(p) + (r 2) v(b 1 (p)) + 2(r 1) v(q) (2 r) v(q)/β + (r 2) 2v(q) + 2(r 1) v(q) = v(q) = op [1] W. Aiello, Y. Mansour, S. Rajagopolan, and A. Rosen. Compeiive queue policies for differeniaed services. Journal of Algorihms, 55(2):113 141, 2005. [2] N. Andelman. Randomized queue managemen for DiffServ. In Proceedings of he 17h ACM Symposium on Parallel Algorihms and Archiecures (SPAA), pages 1 10, 2005. [3] N. Andelman, Y. Mansour, and A. Zhu. Compeiive queueing policies for QoS swiches. In Proceedings of he 14h ACM-SIAM Symposium on Discree Algorihms (SODA), pages 761 770, 2003. [4] Y. Azar and Y. Richer. Managemen of muli-queue swiches in QoS neworks. Algorihmica, 43(1 2):81 96, 2005. [5] N. Bansal, L. Fleischer, T. Kimbrel, M. Mahdian, B. Schieber, and M. Sviridenko. Furher improvemens in compeiive guaranees for QoS buffering. In Proceedings of he 31s Inernaional Colloquium on Auomaa, Languages and Programming (ICALP), pages 196 207, 2004. [6] W. Jawor. Three dozen papers on online algorihms. SIGACT News, 36(1):71 85, 2005. [7] A. Kesselman, Z. Loker, Y. Mansour, B. Pa-Shamir, B. Schieber, and M. Sviridenko. Buffer overflow managemen in QoS swiches. SIAM Journal on Compuing, 33(3):563 583, 2004. 24

[8] A. Kesselman and Y. Mansour. Loss-bounded analysis for differeniaed services. Journal of Algorihms, 46(1):79 95, 2003. [9] A. Kesselman, Y. Mansour, and R. van See. Improved compeiive guaranees for QoS buffering. Algorihmica, 43(1 2):63 80, 2005. [10] Z. Loker and B. Pa-Shamir. Nearly opimal FIFO buffer managemen for wo packe classes. Compuer Neworks, 42(4):481 492, 2003. [11] Y. Mansour, B. Pa-Shamir, and O. Lapid. Opimal smoohing schedules for real-ime sreams. Disribued Compuing, 17(1):77 89, 2004. 25