Lower and Upper Bounds on FIFO Buffer Management in QoS Switches
|
|
- Marjory Gibbs
- 6 years ago
- Views:
Transcription
1 Lower and Upper Bounds on FIFO Buffer Managemen in QoS Swiches Mahias Engler Mahias Wesermann Deparmen of Compuer Science RWTH Aachen Aachen, Germany 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 ( )/ if he buffer size ends o infiniy and an opimal compeiive raio of ( 13 1)/ 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 which is he bes known upper bound on he compeiive raio of his problem. In addiion, we give a lower bound of 1 + 1/ 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),
2 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 ( )/ if he buffer size ends o infiniy and an opimal compeiive raio of ( 13 1)/ 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 which is he bes known upper bound on he compeiive raio of his problem. In addiion, we give a lower bound of 1 + 1/ 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
3 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 In addiion, hey give he previously bes known lower bound of (1 + 5)/ 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 [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 Kesselman e al. [7] show a lower bound of 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 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 ( )/ if he buffer size ends o infiniy and an opimal compeiive raio of ( 13 1)/ 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 which is he bes known upper bound on he compeiive raio of his problem. In addiion, we give a lower bound of 1+1/ on he compeiive raio of PG which improves he previously known lower bound of (1+ 5)/ 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
4 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 := and 2 r := 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 > The compeiive raio of any deerminisic sraegy is a leas r, if he buffer size is 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 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
5 The following heorem shows ha ACC achieves opimal compeiive raios. Theorem 2. Consider only wo packe values 1 and α > 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
6 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
7 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
8 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
9 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
10 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 (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
11 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/ Proof. Fix an even buffer size b. Depending on β, we disinguish he following wo cases. Suppose ha β 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 > β n 2β n 1 β lim n 2(β n β n 1 ) = 1 + β 2(β 1) The inpu sequence consiss of n consecuive phases defined as follows. 11
12 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 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 for β = 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
13 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
14 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
15 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
16 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
17 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
18 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
19 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
20 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
21 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
22 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
23 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
24 (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): , [2] N. Andelman. Randomized queue managemen for DiffServ. In Proceedings of he 17h ACM Symposium on Parallel Algorihms and Archiecures (SPAA), pages 1 10, [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 , [4] Y. Azar and Y. Richer. Managemen of muli-queue swiches in QoS neworks. Algorihmica, 43(1 2):81 96, [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 , [6] W. Jawor. Three dozen papers on online algorihms. SIGACT News, 36(1):71 85, [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): ,
25 [8] A. Kesselman and Y. Mansour. Loss-bounded analysis for differeniaed services. Journal of Algorihms, 46(1):79 95, [9] A. Kesselman, Y. Mansour, and R. van See. Improved compeiive guaranees for QoS buffering. Algorihmica, 43(1 2):63 80, [10] Z. Loker and B. Pa-Shamir. Nearly opimal FIFO buffer managemen for wo packe classes. Compuer Neworks, 42(4): , [11] Y. Mansour, B. Pa-Shamir, and O. Lapid. Opimal smoohing schedules for real-ime sreams. Disribued Compuing, 17(1):77 89,
T L. t=1. Proof of Lemma 1. Using the marginal cost accounting in Equation(4) and standard arguments. t )+Π RB. t )+K 1(Q RB
Elecronic Companion EC.1. Proofs of Technical Lemmas and Theorems LEMMA 1. Le C(RB) be he oal cos incurred by he RB policy. Then we have, T L E[C(RB)] 3 E[Z RB ]. (EC.1) Proof of Lemma 1. Using he marginal
More informationApproximation Algorithms for Unique Games via Orthogonal Separators
Approximaion Algorihms for Unique Games via Orhogonal Separaors Lecure noes by Konsanin Makarychev. Lecure noes are based on he papers [CMM06a, CMM06b, LM4]. Unique Games In hese lecure noes, we define
More informationSOLUTIONS TO ECE 3084
SOLUTIONS TO ECE 384 PROBLEM 2.. For each sysem below, specify wheher or no i is: (i) memoryless; (ii) causal; (iii) inverible; (iv) linear; (v) ime invarian; Explain your reasoning. If he propery is no
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationExpert Advice for Amateurs
Exper Advice for Amaeurs Ernes K. Lai Online Appendix - Exisence of Equilibria The analysis in his secion is performed under more general payoff funcions. Wihou aking an explici form, he payoffs of he
More informationInventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions
Muli-Period Sochasic Models: Opimali of (s, S) Polic for -Convex Objecive Funcions Consider a seing similar o he N-sage newsvendor problem excep ha now here is a fixed re-ordering cos (> 0) for each (re-)order.
More informationLecture 4 Notes (Little s Theorem)
Lecure 4 Noes (Lile s Theorem) This lecure concerns one of he mos imporan (and simples) heorems in Queuing Theory, Lile s Theorem. More informaion can be found in he course book, Bersekas & Gallagher,
More informationDesigning Information Devices and Systems I Spring 2019 Lecture Notes Note 17
EES 16A Designing Informaion Devices and Sysems I Spring 019 Lecure Noes Noe 17 17.1 apaciive ouchscreen In he las noe, we saw ha a capacior consiss of wo pieces on conducive maerial separaed by a nonconducive
More informationChristos Papadimitriou & Luca Trevisan November 22, 2016
U.C. Bereley CS170: Algorihms Handou LN-11-22 Chrisos Papadimiriou & Luca Trevisan November 22, 2016 Sreaming algorihms In his lecure and he nex one we sudy memory-efficien algorihms ha process a sream
More information5. Stochastic processes (1)
Lec05.pp S-38.45 - Inroducion o Teleraffic Theory Spring 2005 Conens Basic conceps Poisson process 2 Sochasic processes () Consider some quaniy in a eleraffic (or any) sysem I ypically evolves in ime randomly
More informationSeminar 4: Hotelling 2
Seminar 4: Hoelling 2 November 3, 211 1 Exercise Par 1 Iso-elasic demand A non renewable resource of a known sock S can be exraced a zero cos. Demand for he resource is of he form: D(p ) = p ε ε > A a
More informationComments on Window-Constrained Scheduling
Commens on Window-Consrained Scheduling Richard Wes Member, IEEE and Yuing Zhang Absrac This shor repor clarifies he behavior of DWCS wih respec o Theorem 3 in our previously published paper [1], and describes
More information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More informationLongest Common Prefixes
Longes Common Prefixes The sandard ordering for srings is he lexicographical order. I is induced by an order over he alphabe. We will use he same symbols (,
More informationLinear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and
More informationRobotics I. April 11, The kinematics of a 3R spatial robot is specified by the Denavit-Hartenberg parameters in Tab. 1.
Roboics I April 11, 017 Exercise 1 he kinemaics of a 3R spaial robo is specified by he Denavi-Harenberg parameers in ab 1 i α i d i a i θ i 1 π/ L 1 0 1 0 0 L 3 0 0 L 3 3 able 1: able of DH parameers of
More informationSTATE-SPACE MODELLING. A mass balance across the tank gives:
B. Lennox and N.F. Thornhill, 9, Sae Space Modelling, IChemE Process Managemen and Conrol Subjec Group Newsleer STE-SPACE MODELLING Inroducion: Over he pas decade or so here has been an ever increasing
More information1 Review of Zero-Sum Games
COS 5: heoreical Machine Learning Lecurer: Rob Schapire Lecure #23 Scribe: Eugene Brevdo April 30, 2008 Review of Zero-Sum Games Las ime we inroduced a mahemaical model for wo player zero-sum games. Any
More informationSolutions for Assignment 2
Faculy of rs and Science Universiy of Torono CSC 358 - Inroducion o Compuer Neworks, Winer 218 Soluions for ssignmen 2 Quesion 1 (2 Poins): Go-ack n RQ In his quesion, we review how Go-ack n RQ can be
More informationFinish reading Chapter 2 of Spivak, rereading earlier sections as necessary. handout and fill in some missing details!
MAT 257, Handou 6: Ocober 7-2, 20. I. Assignmen. Finish reading Chaper 2 of Spiva, rereading earlier secions as necessary. handou and fill in some missing deails! II. Higher derivaives. Also, read his
More informationNotes for Lecture 17-18
U.C. Berkeley CS278: Compuaional Complexiy Handou N7-8 Professor Luca Trevisan April 3-8, 2008 Noes for Lecure 7-8 In hese wo lecures we prove he firs half of he PCP Theorem, he Amplificaion Lemma, up
More informationFinal Spring 2007
.615 Final Spring 7 Overview The purpose of he final exam is o calculae he MHD β limi in a high-bea oroidal okamak agains he dangerous n = 1 exernal ballooning-kink mode. Effecively, his corresponds o
More informationKEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering
More information11!Hí MATHEMATICS : ERDŐS AND ULAM PROC. N. A. S. of decomposiion, properly speaking) conradics he possibiliy of defining a counably addiive real-valu
ON EQUATIONS WITH SETS AS UNKNOWNS BY PAUL ERDŐS AND S. ULAM DEPARTMENT OF MATHEMATICS, UNIVERSITY OF COLORADO, BOULDER Communicaed May 27, 1968 We shall presen here a number of resuls in se heory concerning
More informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More informationThe Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales
Advances in Dynamical Sysems and Applicaions. ISSN 0973-5321 Volume 1 Number 1 (2006, pp. 103 112 c Research India Publicaions hp://www.ripublicaion.com/adsa.hm The Asympoic Behavior of Nonoscillaory Soluions
More informationLecture 20: Riccati Equations and Least Squares Feedback Control
34-5 LINEAR SYSTEMS Lecure : Riccai Equaions and Leas Squares Feedback Conrol 5.6.4 Sae Feedback via Riccai Equaions A recursive approach in generaing he marix-valued funcion W ( ) equaion for i for he
More informationBernoulli numbers. Francesco Chiatti, Matteo Pintonello. December 5, 2016
UNIVERSITÁ DEGLI STUDI DI PADOVA, DIPARTIMENTO DI MATEMATICA TULLIO LEVI-CIVITA Bernoulli numbers Francesco Chiai, Maeo Pinonello December 5, 206 During las lessons we have proved he Las Ferma Theorem
More informationMath 334 Fall 2011 Homework 11 Solutions
Dec. 2, 2 Mah 334 Fall 2 Homework Soluions Basic Problem. Transform he following iniial value problem ino an iniial value problem for a sysem: u + p()u + q() u g(), u() u, u () v. () Soluion. Le v u. Then
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More informationOptimal Server Assignment in Multi-Server
Opimal Server Assignmen in Muli-Server 1 Queueing Sysems wih Random Conneciviies Hassan Halabian, Suden Member, IEEE, Ioannis Lambadaris, Member, IEEE, arxiv:1112.1178v2 [mah.oc] 21 Jun 2013 Yannis Viniois,
More informationRandom Walk with Anti-Correlated Steps
Random Walk wih Ani-Correlaed Seps John Noga Dirk Wagner 2 Absrac We conjecure he expeced value of random walks wih ani-correlaed seps o be exacly. We suppor his conjecure wih 2 plausibiliy argumens and
More informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
More informationBasic definitions and relations
Basic definiions and relaions Lecurer: Dmiri A. Molchanov E-mail: molchan@cs.u.fi hp://www.cs.u.fi/kurssi/tlt-2716/ Kendall s noaion for queuing sysems: Arrival processes; Service ime disribuions; Examples.
More informationCHAPTER 12 DIRECT CURRENT CIRCUITS
CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As
More informationFamilies with no matchings of size s
Families wih no machings of size s Peer Franl Andrey Kupavsii Absrac Le 2, s 2 be posiive inegers. Le be an n-elemen se, n s. Subses of 2 are called families. If F ( ), hen i is called - uniform. Wha is
More information8. Basic RL and RC Circuits
8. Basic L and C Circuis This chaper deals wih he soluions of he responses of L and C circuis The analysis of C and L circuis leads o a linear differenial equaion This chaper covers he following opics
More informationE β t log (C t ) + M t M t 1. = Y t + B t 1 P t. B t 0 (3) v t = P tc t M t Question 1. Find the FOC s for an optimum in the agent s problem.
Noes, M. Krause.. Problem Se 9: Exercise on FTPL Same model as in paper and lecure, only ha one-period govenmen bonds are replaced by consols, which are bonds ha pay one dollar forever. I has curren marke
More informationLecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.
Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in
More informationEXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
More information4 Sequences of measurable functions
4 Sequences of measurable funcions 1. Le (Ω, A, µ) be a measure space (complee, afer a possible applicaion of he compleion heorem). In his chaper we invesigae relaions beween various (nonequivalen) convergences
More informationElectrical and current self-induction
Elecrical and curren self-inducion F. F. Mende hp://fmnauka.narod.ru/works.hml mende_fedor@mail.ru Absrac The aricle considers he self-inducance of reacive elemens. Elecrical self-inducion To he laws of
More informationInventory Control of Perishable Items in a Two-Echelon Supply Chain
Journal of Indusrial Engineering, Universiy of ehran, Special Issue,, PP. 69-77 69 Invenory Conrol of Perishable Iems in a wo-echelon Supply Chain Fariborz Jolai *, Elmira Gheisariha and Farnaz Nojavan
More information2. Nonlinear Conservation Law Equations
. Nonlinear Conservaion Law Equaions One of he clear lessons learned over recen years in sudying nonlinear parial differenial equaions is ha i is generally no wise o ry o aack a general class of nonlinear
More information3.1 More on model selection
3. More on Model selecion 3. Comparing models AIC, BIC, Adjused R squared. 3. Over Fiing problem. 3.3 Sample spliing. 3. More on model selecion crieria Ofen afer model fiing you are lef wih a handful of
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More information2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes
Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion
More informationLet us start with a two dimensional case. We consider a vector ( x,
Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our
More informationChallenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k
Challenge Problems DIS 03 and 0 March 6, 05 Choose one of he following problems, and work on i in your group. Your goal is o convince me ha your answer is correc. Even if your answer isn compleely correc,
More informationSolutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
More informationHamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation:
M ah 5 7 Fall 9 L ecure O c. 4, 9 ) Hamilon- J acobi Equaion: Weak S oluion We coninue he sudy of he Hamilon-Jacobi equaion: We have shown ha u + H D u) = R n, ) ; u = g R n { = }. ). In general we canno
More informationOnline Learning Approaches in Maximizing Weighted Throughput
Online Learning Approaches in Maximizing Weighed Throughpu Zhi Zhang, Fei Li, Songqing Chen Deparmen of Compuer Science George Mason Universiy Fairfax, Virginia 22030 Email: {zzhang8, lifei, sqchen}@cs.gmu.edu
More informationOnline Convex Optimization Example And Follow-The-Leader
CSE599s, Spring 2014, Online Learning Lecure 2-04/03/2014 Online Convex Opimizaion Example And Follow-The-Leader Lecurer: Brendan McMahan Scribe: Sephen Joe Jonany 1 Review of Online Convex Opimizaion
More informationVehicle Arrival Models : Headway
Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where
More informationSupplement for Stochastic Convex Optimization: Faster Local Growth Implies Faster Global Convergence
Supplemen for Sochasic Convex Opimizaion: Faser Local Growh Implies Faser Global Convergence Yi Xu Qihang Lin ianbao Yang Proof of heorem heorem Suppose Assumpion holds and F (w) obeys he LGC (6) Given
More informationEconomics 8105 Macroeconomic Theory Recitation 6
Economics 8105 Macroeconomic Theory Reciaion 6 Conor Ryan Ocober 11h, 2016 Ouline: Opimal Taxaion wih Governmen Invesmen 1 Governmen Expendiure in Producion In hese noes we will examine a model in which
More informationLogic in computer science
Logic in compuer science Logic plays an imporan role in compuer science Logic is ofen called he calculus of compuer science Logic plays a similar role in compuer science o ha played by calculus in he physical
More informationGuest Lectures for Dr. MacFarlane s EE3350 Part Deux
Gues Lecures for Dr. MacFarlane s EE3350 Par Deux Michael Plane Mon., 08-30-2010 Wrie name in corner. Poin ou his is a review, so I will go faser. Remind hem o go lisen o online lecure abou geing an A
More informationThis document was generated at 1:04 PM, 09/10/13 Copyright 2013 Richard T. Woodward. 4. End points and transversality conditions AGEC
his documen was generaed a 1:4 PM, 9/1/13 Copyrigh 213 Richard. Woodward 4. End poins and ransversaliy condiions AGEC 637-213 F z d Recall from Lecure 3 ha a ypical opimal conrol problem is o maimize (,,
More informationh[n] is the impulse response of the discrete-time system:
Definiion Examples Properies Memory Inveribiliy Causaliy Sabiliy Time Invariance Lineariy Sysems Fundamenals Overview Definiion of a Sysem x() h() y() x[n] h[n] Sysem: a process in which inpu signals are
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationTwo Popular Bayesian Estimators: Particle and Kalman Filters. McGill COMP 765 Sept 14 th, 2017
Two Popular Bayesian Esimaors: Paricle and Kalman Filers McGill COMP 765 Sep 14 h, 2017 1 1 1, dx x Bel x u x P x z P Recall: Bayes Filers,,,,,,, 1 1 1 1 u z u x P u z u x z P Bayes z = observaion u =
More informationHomework 4 (Stats 620, Winter 2017) Due Tuesday Feb 14, in class Questions are derived from problems in Stochastic Processes by S. Ross.
Homework 4 (Sas 62, Winer 217) Due Tuesday Feb 14, in class Quesions are derived from problems in Sochasic Processes by S. Ross. 1. Le A() and Y () denoe respecively he age and excess a. Find: (a) P{Y
More informationSome Ramsey results for the n-cube
Some Ramsey resuls for he n-cube Ron Graham Universiy of California, San Diego Jozsef Solymosi Universiy of Briish Columbia, Vancouver, Canada Absrac In his noe we esablish a Ramsey-ype resul for cerain
More informationEcon107 Applied Econometrics Topic 7: Multicollinearity (Studenmund, Chapter 8)
I. Definiions and Problems A. Perfec Mulicollineariy Econ7 Applied Economerics Topic 7: Mulicollineariy (Sudenmund, Chaper 8) Definiion: Perfec mulicollineariy exiss in a following K-variable regression
More informationLanguages That Are and Are Not Context-Free
Languages Tha re and re No Conex-Free Read K & S 3.5, 3.6, 3.7. Read Supplemenary Maerials: Conex-Free Languages and Pushdown uomaa: Closure Properies of Conex-Free Languages Read Supplemenary Maerials:
More informationOptimality Conditions for Unconstrained Problems
62 CHAPTER 6 Opimaliy Condiions for Unconsrained Problems 1 Unconsrained Opimizaion 11 Exisence Consider he problem of minimizing he funcion f : R n R where f is coninuous on all of R n : P min f(x) x
More informationMatrix Versions of Some Refinements of the Arithmetic-Geometric Mean Inequality
Marix Versions of Some Refinemens of he Arihmeic-Geomeric Mean Inequaliy Bao Qi Feng and Andrew Tonge Absrac. We esablish marix versions of refinemens due o Alzer ], Carwrigh and Field 4], and Mercer 5]
More informationINTRODUCTION TO MACHINE LEARNING 3RD EDITION
ETHEM ALPAYDIN The MIT Press, 2014 Lecure Slides for INTRODUCTION TO MACHINE LEARNING 3RD EDITION alpaydin@boun.edu.r hp://www.cmpe.boun.edu.r/~ehem/i2ml3e CHAPTER 2: SUPERVISED LEARNING Learning a Class
More informationMatlab and Python programming: how to get started
Malab and Pyhon programming: how o ge sared Equipping readers he skills o wrie programs o explore complex sysems and discover ineresing paerns from big daa is one of he main goals of his book. In his chaper,
More informationOn Boundedness of Q-Learning Iterates for Stochastic Shortest Path Problems
MATHEMATICS OF OPERATIONS RESEARCH Vol. 38, No. 2, May 2013, pp. 209 227 ISSN 0364-765X (prin) ISSN 1526-5471 (online) hp://dx.doi.org/10.1287/moor.1120.0562 2013 INFORMS On Boundedness of Q-Learning Ieraes
More informationOn Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature
On Measuring Pro-Poor Growh 1. On Various Ways of Measuring Pro-Poor Growh: A Shor eview of he Lieraure During he pas en years or so here have been various suggesions concerning he way one should check
More informationTechnical Report Doc ID: TR March-2013 (Last revision: 23-February-2016) On formulating quadratic functions in optimization models.
Technical Repor Doc ID: TR--203 06-March-203 (Las revision: 23-Februar-206) On formulaing quadraic funcions in opimizaion models. Auhor: Erling D. Andersen Convex quadraic consrains quie frequenl appear
More informationHamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t
M ah 5 2 7 Fall 2 0 0 9 L ecure 1 0 O c. 7, 2 0 0 9 Hamilon- J acobi Equaion: Explici Formulas In his lecure we ry o apply he mehod of characerisics o he Hamilon-Jacobi equaion: u + H D u, x = 0 in R n
More informationSMT 2014 Calculus Test Solutions February 15, 2014 = 3 5 = 15.
SMT Calculus Tes Soluions February 5,. Le f() = and le g() =. Compue f ()g (). Answer: 5 Soluion: We noe ha f () = and g () = 6. Then f ()g () =. Plugging in = we ge f ()g () = 6 = 3 5 = 5.. There is a
More informationCMU-Q Lecture 3: Search algorithms: Informed. Teacher: Gianni A. Di Caro
CMU-Q 5-38 Lecure 3: Search algorihms: Informed Teacher: Gianni A. Di Caro UNINFORMED VS. INFORMED SEARCH Sraegy How desirable is o be in a cerain inermediae sae for he sake of (effecively) reaching a
More informationInternational Journal of Scientific & Engineering Research, Volume 4, Issue 10, October ISSN
Inernaional Journal of Scienific & Engineering Research, Volume 4, Issue 10, Ocober-2013 900 FUZZY MEAN RESIDUAL LIFE ORDERING OF FUZZY RANDOM VARIABLES J. EARNEST LAZARUS PIRIYAKUMAR 1, A. YAMUNA 2 1.
More informationChapter 6. Systems of First Order Linear Differential Equations
Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More informationThe Arcsine Distribution
The Arcsine Disribuion Chris H. Rycrof Ocober 6, 006 A common heme of he class has been ha he saisics of single walker are ofen very differen from hose of an ensemble of walkers. On he firs homework, we
More informationModule 2 F c i k c s la l w a s o s f dif di fusi s o i n
Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms
More informationR t. C t P t. + u t. C t = αp t + βr t + v t. + β + w t
Exercise 7 C P = α + β R P + u C = αp + βr + v (a) (b) C R = α P R + β + w (c) Assumpions abou he disurbances u, v, w : Classical assumions on he disurbance of one of he equaions, eg. on (b): E(v v s P,
More informationRobust estimation based on the first- and third-moment restrictions of the power transformation model
h Inernaional Congress on Modelling and Simulaion, Adelaide, Ausralia, 6 December 3 www.mssanz.org.au/modsim3 Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion Nawaa,
More informationarxiv: v1 [cs.ds] 3 Mar 2016
Tigh Analysis of a Muliple-Swap Heurisic for Budgeed Red-Blue Median Zachary Friggsad Yifeng Zhang arxiv:1603.00973v1 [cs.ds] 3 Mar 2016 Deparmen of Compuing Science Universiy of Albera {zacharyf,yifeng2}@ualbera.ca
More informationApplication of a Stochastic-Fuzzy Approach to Modeling Optimal Discrete Time Dynamical Systems by Using Large Scale Data Processing
Applicaion of a Sochasic-Fuzzy Approach o Modeling Opimal Discree Time Dynamical Sysems by Using Large Scale Daa Processing AA WALASZE-BABISZEWSA Deparmen of Compuer Engineering Opole Universiy of Technology
More information= ( ) ) or a system of differential equations with continuous parametrization (T = R
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More informationINDEPENDENT SETS IN GRAPHS WITH GIVEN MINIMUM DEGREE
INDEPENDENT SETS IN GRAPHS WITH GIVEN MINIMUM DEGREE JAMES ALEXANDER, JONATHAN CUTLER, AND TIM MINK Absrac The enumeraion of independen ses in graphs wih various resricions has been a opic of much ineres
More informationStochastic models and their distributions
Sochasic models and heir disribuions Couning cusomers Suppose ha n cusomers arrive a a grocery a imes, say T 1,, T n, each of which akes any real number in he inerval (, ) equally likely The values T 1,,
More informationPhysics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution
Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his
More informationThe Contradiction within Equations of Motion with Constant Acceleration
The Conradicion wihin Equaions of Moion wih Consan Acceleraion Louai Hassan Elzein Basheir (Daed: July 7, 0 This paper is prepared o demonsrae he violaion of rules of mahemaics in he algebraic derivaion
More informationReading from Young & Freedman: For this topic, read sections 25.4 & 25.5, the introduction to chapter 26 and sections 26.1 to 26.2 & 26.4.
PHY1 Elecriciy Topic 7 (Lecures 1 & 11) Elecric Circuis n his opic, we will cover: 1) Elecromoive Force (EMF) ) Series and parallel resisor combinaions 3) Kirchhoff s rules for circuis 4) Time dependence
More informationInequality measures for intersecting Lorenz curves: an alternative weak ordering
h Inernaional Scienific Conference Financial managemen of Firms and Financial Insiuions Osrava VŠB-TU of Osrava, Faculy of Economics, Deparmen of Finance 7 h 8 h Sepember 25 Absrac Inequaliy measures for
More informationSZG Macro 2011 Lecture 3: Dynamic Programming. SZG macro 2011 lecture 3 1
SZG Macro 2011 Lecure 3: Dynamic Programming SZG macro 2011 lecure 3 1 Background Our previous discussion of opimal consumpion over ime and of opimal capial accumulaion sugges sudying he general decision
More information6. Stochastic calculus with jump processes
A) Trading sraegies (1/3) Marke wih d asses S = (S 1,, S d ) A rading sraegy can be modelled wih a vecor φ describing he quaniies invesed in each asse a each insan : φ = (φ 1,, φ d ) The value a of a porfolio
More informationCash Flow Valuation Mode Lin Discrete Time
IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728,p-ISSN: 2319-765X, 6, Issue 6 (May. - Jun. 2013), PP 35-41 Cash Flow Valuaion Mode Lin Discree Time Olayiwola. M. A. and Oni, N. O. Deparmen of Mahemaics
More information( ) a system of differential equations with continuous parametrization ( T = R + These look like, respectively:
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More information6.2 Transforms of Derivatives and Integrals.
SEC. 6.2 Transforms of Derivaives and Inegrals. ODEs 2 3 33 39 23. Change of scale. If l( f ()) F(s) and c is any 33 45 APPLICATION OF s-shifting posiive consan, show ha l( f (c)) F(s>c)>c (Hin: In Probs.
More informationRemoving Useless Productions of a Context Free Grammar through Petri Net
Journal of Compuer Science 3 (7): 494-498, 2007 ISSN 1549-3636 2007 Science Publicaions Removing Useless Producions of a Conex Free Grammar hrough Peri Ne Mansoor Al-A'ali and Ali A Khan Deparmen of Compuer
More informationChapter 7 Response of First-order RL and RC Circuits
Chaper 7 Response of Firs-order RL and RC Circuis 7.- The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial
More information