ALLOCATING BANDWIDTH FOR BURSTY CONNECTIONS

Transcription

1 SIAM J. COMPUT. Vol. 30, No. 1, pp c 2000 Society for Indutrial and Applied Mathematic ALLOCATING BANDWIDTH FOR BURSTY CONNECTIONS JON KLEINBERG, YUVAL RABANI, AND ÉVA TARDOS Abtract. In thi paper, we undertake the firt tudy of tatitical multiplexing from the perpective of approximation algorithm. The baic iue underlying tatitical multiplexing i the following: in high-peed network, individual connection (i.e., communication eion) are very burty, with tranmiion rate that vary greatly over time. A uch, the problem of packing multiple connection together on a link become more ubtle than in the cae when each connection i aumed to have a fixed demand. We conider one of the mot commonly tudied model in thi domain: that of two communicating node connected by a et of parallel edge, where the rate of each connection between them i a random variable. We conider three related problem: (1) tochatic load balancing, (2) tochatic bin-packing, and (3) tochatic knapack. In the firt problem the number of link i given and we want to minimize the expected value of the maximum load. In the other two problem the link capacity and an allowed overflow probability p are given, and the objective i to aign connection to link, o that the probability that the load of a link exceed the link capacity i at mot p. In binpacking we need to aign each connection to a link uing a few link a poible. In the knapack problem each connection ha a value, and we have only one link. The problem i to accept a many connection a poible. For the tochatic load balancing problem we give an O(1)-approximation algorithm for arbitrary random variable. For the other two problem we have algorithm retricted to on-off ource (the mot common pecial cae tudied in the tatitical multiplexing literature), with a omewhat weaker range of performance guarantee. A tandard approach that ha emerged for dealing with probabilitic reource requirement i the notion of effective bandwidth thi i a mean of aociating a fixed demand with a burty connection that repreent it ditribution a cloely a poible. Our approximation algorithm make ue of the tandard definition of effective bandwidth and alo a new one that we introduce; the performance guarantee are baed on new reult howing that a combination of thee meaure can be ued to provide bound on the optimal olution. Key word. combinatorial optimization, approximation algorithm, tatitical multiplexing, effective bandwidth AMS ubject claification. 05C85, 68R10, 68Q20 PII. S Introduction. Motivation and previou work. The iue of admiion control and routing in high-peed network have inpired recent analytical work on network routing and bandwidth allocation problem in everal communitie (e.g., [10, 1, 5]). One Received by the editor October 24, 1997; accepted for publication (in revied form) June 21, 1999; publihed electronically April 25, Reearch for thi paper wa upported in part by NSF through grant DMS Portion of thi work were performed while viiting the IBM Almaden Reearch Center. Cornell Univerity, Department of Computer Science, Ithaca, NY (kleinberg@c.cornell. edu). Thi author i upported in part by an Alfred P. Sloan Reearch Fellowhip, an ONR Young Invetigator Award, and NSF Faculty Early Career Development Award CCR Technion Intitute, Computer Science Department, Haifa 32000, Irael (rabani@c.technion. ac.il). Mot of thi work wa done while viiting Cornell Univerity. Support at Cornell wa provided by ONR through grant N Thi work wa alo upported by grant from the fund for the promotion of ponored reearch and the fund for the promotion of reearch at the Technion, and by a David and Ruth Mokowitz Academic Lecturhip award. Cornell Univerity, Department of Computer Science, Ithaca, NY (eva@c.cornell.edu). The reearch of thi author wa upported in part by an NSF PYI award DMI , by NSF through grant DMS , and by ONR through grant N

2 192 JON KLEINBERG, YUVAL RABANI, AND ÉVA TARDOS line of work ha been directed toward the development of approximation algorithm and competitive on-line algorithm for admiion control and virtual circuit routing problem (ee the urvey by Plotkin [16]). The network model in thi line of work repreent the link of the network a edge of fixed capacity and connection a pair of vertice with a fixed bandwidth demand between them. The algorithm and their analyi are motivated by thi network flow perpective. In fact, however, traffic in high-peed network baed on aynchronou tranfer mode (ATM) and related technologie tend to be extremely burty. The tranmiion rate of a ingle connection can vary greatly over time; there can be infrequent period of very high peak rate, while the average rate i much lower. One can try to avoid thi iue by aigning each connection a demand equal to it maximum poible rate. The ue of uch a conervative approximation will enure that edge capacitie are never violated. But much of the trength of ATM come from the advantage of tatitical multiplexing the packing of uncorrelated, burty connection on the ame link. In particular, uppoe one i willing to tolerate a low rate of packet lo due to occaional violation of the link capacity. A the peak tate of different connection coincide only very rarely, one can pack many more connection than i poible via the above wort-cae approach and till maintain a very low rate of packet lo due to overflow. Queueing theorit recently have devoted a great deal of tudy to the analyi of tatitical multiplexing (ee the book edited by Kelly, Zachary, and Zeidin [13]). Typically, thi work model a ingle connection either a a dicrete random variable X, with Pr[X = ] indicating the fraction of the time that the connection tranmit at rate, or a a finite-tate Markov chain with a fixed tranmiion rate for each tate. (A much-dicued cae i when X i an on-off ource. In our context, uch a connection i equivalent to a weighted Bernoulli trial.) Thi line of work ha concentrated primarily on the cae of point-to-point tranmiion acro a et of parallel link; thi allow one to tudy the packing and load balancing iue that arie without the added complication of path election in a large network. One of the main concept that ha emerged from thi work ha been the development of a notion of effective bandwidth for burty connection. Thi i baed on the following natural idea. Suppoe one i willing to tolerate a rate p of overflow on each link. One firt aign a number β p (X) to each connection (i.e., random variable) X, indicating the effective amount of bandwidth required by thi connection. One then ue a tandard packing or load balancing algorithm to aign connection to link, uing the ingle number β p (X) a the demand of the connection X. Thi notion of effective bandwidth i indeed what underlie the modeling of routing problem a network flow quetion. Conenu ha more or le been reached (ee Kelly [12]) on a pecific formula for β p, firt tudied by Hui [10]: a caled logarithm of the moment-generating function of X. One of it attraction i that packing according to β p (X) alway provide a relatively conervative etimate in the following ene: If the um of the effective bandwidth of a et of independent connection doe not exceed the link capacity, then the probability that the um of their tranmiion rate exceed twice the capacity at any intant i at mot p. Problem tudied in thi paper. In thi paper, we undertake the firt tudy of the iue inherent in tatitical multiplexing from the perpective of approximation algorithm. We are motivated primarily by the following fact: the queueing theoretical work dicued above doe not attempt to prove that it method, baed on

3 BANDWIDTH FOR BURSTY CONNECTIONS 193 effective bandwidth, provide olution that are near-optimal on all (or even on typical) intance. Indeed, reearcher have recognized that claim about the power of the effective bandwidth approach depend critically on a number of fundamental aumption about the nature of the underlying traffic (e.g., de Veciana and Walrand [18]). Thu an analyi of tatitical multiplexing problem in the framework of approximation algorithm can provide tool for undertanding the performance guarantee that can be attained in thi domain. We mentioned above that the model tudied in thi area concentrate primarily on the cae of two communicating node connected by a et of parallel edge. Thu, the problem of aigning burty connection to edge i equivalent to that of aigning (burty) item to bin. A a reult, we have a direct connection between the tandard quetion addreed in tatitical multiplexing and tochatic verion of ome of the claical reource allocation problem in combinatorial optimization. We deign and analyze approximation algorithm for the following fundamental problem: Stochatic load balancing. An item i a dicrete random variable. We are given item X 1,..., X n. We want to aign each item to one of the bin 1,..., m o a to minimize the expected maximum weight in any bin. That i, we want to minimize E max i X j B i X j where B i i the et of item aigned to bin i. Stochatic bin-packing. We are given item a above, and we define the overflow probability of a ubet of thee item to be the probability that their um exceed 1. We are alo given a number p 0. We want to determine the minimum number of bin (of capacity 1) that we need in order to pack all the item, o that the overflow probability of the item in each bin i at mot p. Stochatic knapack. We are given p 0 and a et of item X 1,..., X n, with item X i having a value v i. We want to find a ubet of the item of maximum value, ubject to the contraint that it overflow probability i at mot p. Thu, the above problem provide u with a very concrete etting in which to try aeing the power of variou approache to the tatitical multiplexing of burty connection. Thee problem are alo the natural tochatic analogue of ome of the central problem in the area of approximation algorithm; and hence we feel that their approximability i of baic interet. Of coure, each of thee problem i NP-hard, ince the verion in which each item X i i determinitic (i.e., take a ingle value with probability 1) correpond to the minimum makepan, bin-packing, and knapack problem, repectively. However, the tochatic verion introduce coniderable additional complication. For example, we how that even given a et of item, determining it overflow probability i #P - complete (ee ection 2). Moreover, we alo how that imple approache uch a (i) applying Hui definition of effective bandwidth [10] to the item, and then (ii) running a tandard algorithm for the cae of determinitic weight (e.g., Graham lowet-fit makepan algorithm or firt-fit for bin packing) can lead to reult that are very far from optimal. Indeed, we how in ection 2 that in a certain precie ene there i no direct ue of effective bandwidth that can provide approximation reult a trong a thoe we obtain.,

4 194 JON KLEINBERG, YUVAL RABANI, AND ÉVA TARDOS 1.1. Our reult. Thi paper provide the firt approximation algorithm for thee load balancing and packing problem with tochatic item. Our algorithm make ue of effective bandwidth, and their analyi i baed on new reult howing, roughly, that it i poible to define a notion of effective bandwidth that can be ued to obtain bound on the value of the optimum. However, the relationhip between the effective bandwidth and the optimum are quite ubtle. In particular, while Hui definition i a ueful ingredient in our algorithm for the cae of load balancing, we how in the cae of bin-packing and knapack that it i neceary to ue a definition of effective bandwidth that i different from the tandard one. Our new effective bandwidth function β ha a number of additional propertie that make it analyi particularly tractable. In particular, it wa through β that we were able to etablih our baic relation between the function β and the value of the optimum for the cae of load balancing. Load balancing. Perhap our tronget reult i for the load balancing problem: we provide a contant-factor approximation algorithm for the optimum load for arbitrary random variable. With a omewhat larger contant, we can modify our algorithm to work in an on-line etting, in which item arrive in equence and mut be aigned to bin immediately. Let u give ome indication of the technique underlying thi algorithm. Firt, we mentioned above that the tandard effective bandwidth β p come with an upper bound guarantee: if the um of the effective bandwidth of a et of item i bounded by 1, then the probability that the total load of thee item exceed 2 i at mot p. (Thi fact i due originally to Hui [10] and ha been extended and generalized by Kelly [11], Elwalid and Mitra [4], and other.) Our proof of the contant approximation ratio ue a new lower bound guarantee for effective bandwidth. Suppoe we have a et of random variable X 1,..., X n, o that each X i i a weighted Bernoulli trial taking on the value 0 and 2 i for an integer 0 i log log p 1. We how that there i an abolute contant C 7 o that if the um of the effective bandwidth of the X i i at leat C, then the probability that their um exceed 1 i at leat p. A number of iue mut be reolved in order to ue thee bound in the deign and analyi of our algorithm. Firt, the upper bound guarantee hold only under ome retricting aumption on the item ize, which are not necearily valid for our input. Therefore, we have to handle exceptional item eparately. Second, our lower bound concern overflow probabilitie, wherea our objective function i the expected maximum load in any bin. Finally, we have to ue thi lower bound in the etting of arbitrary random variable, depite the fact that the concrete reult itelf applie only to a retricted type of random variable. Bin-packing and knapack. In the cae of the bin-packing and knapack problem we conider primarily on-off ource. In our context, uch a connection i equivalent to a weighted Bernoulli trial. Our emphai on on-off ource i in keeping with the focu of much of the literature (ee, e.g., the book [13]). With omewhat weaker performance guarantee, we can alo handle the more general cae of high-low ource: connection whoe rate are alway one of two poitive value. For the bin-packing problem with on-off item we give an algorithm that find a olution with at mot O( log p 1 log log p 1 )B + O(log p 1 ) bin, where B i the minimum poible number of bin. For the knapack problem we provide an O(log p 1 )- approximation algorithm. We alo provide contant-factor approximation algorithm

5 BANDWIDTH FOR BURSTY CONNECTIONS 195 for both problem, in which cae one i allowed to relax either the ize of the bin or the overflow probability by an arbitrary contant ε > 0. Our algorithm for bin-packing can be modified to work in an on-line etting, in which item arrive in equence and mut be aigned to bin immediately. Our algorithm are baed on a notion of effective bandwidth, but not the tandard one in the literature. In particular, the guarantee provided by the tandard definition i not trong enough for the bin-packing and knapack problem: it ay that if the um of the effective bandwidth of a et of item i bounded by 1, then the probability that the total load of thee item exceed 2 i at mot p. While uch a guarantee i trong enough for the load balancing problem a load of 2 i within a contant factor of a load of 1 it i inadequate for the bin-packing and knapack problem, which fix hard limit on the ize of each bin. Stronger guarantee without exceeding the link capacity were provided by Hui [10], Kelly [11], and Elwalid and Mitra [4] uing large overflow buffer. We provide uch tronger guarantee without reorting to overflow buffer. In particular, for item of large peak rate (the mot difficult cae for the tandard definition β), we make ue of our new effective bandwidth β to provide the deired performance guarantee Connection with tochatic cheduling. Although we have o far expreed thing in the context of burty traffic in a network, our reult on load balancing alo reolve a natural problem in the area of tochatic cheduling. There i a large literature on cheduling with tochatic requirement; the recent book on cheduling theory by Pinedo [15] give an overview of the important reult known in thi area. In a tochatic cheduling problem, the job proceing time are repreented by random variable; typical aumption are that thee proceing time are independent and identically ditributed, and that the ditribution i Poion or exponential. For ome of thee cae, algorithm have been developed that guarantee an aymptotically optimal chedule with high probability (e.g., Wei [19, 20]). We can naturally view our load balancing problem a a cheduling problem on m identical machine (the bin), with a et of n tochatic job (the item). Since the problem contain the NP-hard determinitic verion a a pecial cae, we cannot expect to find an optimal olution. What our load balancing reult provide i a contant approximation for the minimum makepan problem on m identical machine, when the proceing time of each job can have an arbitrary ditribution. One ditinction that arie in thee cheduling problem i the following: mut all the job be loaded onto their aigned machine immediately, or can we perform an aignment adaptively, learning the proceing time of earlier job a they finih? Our model, ince it i motivated by a circuit-routing application, take the firt approach. Thi i alo the approach taken by, e.g., Lehtonen [14], who conider the pecial cae of exponentially ditributed proceing time; that work left the cae of general ditribution which we handle here a an open problem. 2. Preliminary reult and example. For much of the paper, we will be dicuing random variable that are Bernoulli trial. We ay that a random variable X i a Bernoulli trial of type (q, ) if X take the value with probability q and the value 0 with probability 1 q. The load balancing, bin-packing, and knapack problem are all NP-complete even when all item are determinitic (i.e., they aume a ingle value with probability 1). A mentioned above, the introduction of tochatic item lead to new ource of intractability.

6 196 JON KLEINBERG, YUVAL RABANI, AND ÉVA TARDOS Theorem 2.1. Given Bernoulli trial X 1,..., X n, where X i i of type (q i, i ), it i #P -complete to compute Pr[ i X i > 1]. Proof. Memberhip in #P i eay to verify. We prove #P -hardne by a reduction from the problem of counting the number of feaible olution to a knapack problem. That i, given number y 1,..., y n and a bound B, we want to know how many ubet of {y 1,..., y n } add up to at mot B. We make two modification to thi problem which do not affect it tractability: (i) We aume that B = 1. (ii) We conider the complementary problem of counting the number of ubet of {y 1,..., y n } that um to more than B. Thu, given y 1,..., y n, we create Bernoulli trial X 1,..., X n uch that X i i of type ( 1 2, y i). Let p = Pr[ i X i > 1]. The theorem follow from the fact that the number of ubet of {y 1,..., y n } that um to more than 1 i equal to p 2 n. The ue of effective bandwidth i a major component in the deign of our approximation algorithm. We now give ome example to how that no direct ue of effective bandwidth will uffice in order to obtain the approximation guarantee preented in later ection. Thee example alo provide intuition for ome of the iue that arie in dealing with tochatic item. Firt we conider the load balancing problem. A natural approximation method one might conider here i Graham lowet-fit algorithm applied to the expected value of the item. However, thi fail to achieve a contant-factor approximation. Thi i a conequence of the following much more general fact. Let γ be any function from random variable to the nonnegative real number. If X 1,..., X n are random variable, and φ i an aignment of them to m bin, we ay that φ i γ-optimal if it minimize the maximum um of the γ-value of the item in any one bin. Theorem 2.2. For every function γ a above, there exit X 1,..., X n and a γ-optimal aignment φ of X 1,..., X n to m bin uch that the load of φ i Ω(log m/ log log m) time the optimum load. Proof. For an arbitrary function γ, we conider jut two kind of ditribution: a Bernoulli trial of type (m 1 2, 1) and a Bernoulli trial of type (1, 1). (Thi latter ditribution i imply a determinitic item of weight 1.) By recaling, aume that γ take the value 1 on Bernoulli trial of type (1, 1) and the value am 1 2 on Bernoulli trial of type (m 1 2, 1). We conider two cae. Cae 1. a ε log m log log m for ome ufficiently mall contant ε. In thi cae, we conider the following γ-optimal aignment: one item of type (1, 1) in each of the m m bin, and m/a item of type (m 1 2, 1) in each of the remaining m bin. ε log m With high probability, at leat log log m of the latter type of item will be on in the ame bin, and hence the load of thi aignment i Ω(log m/ log log m). By placing at mot one item of each type in every bin, one can obtain a load of 2 for thi problem. Cae 2. a >. In thi cae, conider the following γ-optimal aignment ε log m log log m φ: C m log m item of type (m 1 2, 1) in each of m 1 bin, for a ufficiently large contant C, and ac log m item of type (1, 1) in the mth bin. Thu, the load of φ i at leat ac log m. However, with high probability, the maximum load in the firt m 1 bin will be Θ(log m), and hence the aignment that evenly balance the item of both type ha load O((1 + a am m ) log m). Thi i better by a factor of Ω( a+m ). We now dicu a imilar phenomenon in the cae of bin-packing. Let u ay that a packing of item into bin i incompreible if merging any two of it bin reult in an infeaible packing. For the problem of packing determinitic item, a baic fact i that any incompreible packing i within a factor of 2 of optimal. In contrat, we can

7 BANDWIDTH FOR BURSTY CONNECTIONS 197 how the exitence of a et of tochatic item that can be packed in only two bin, but for which there i an incompreible packing uing Ω(p 1 2 ) bin. Theorem 2.3. Conider a bin-packing problem with overflow probability p. There exit et of weighted Bernoulli trial S 1 and S 2 with the following propertie. (i) S 1 = S 2 = Ω(p 1 2 ). (ii) All the item of S 1 can be packed in a ingle bin. (iii) All the item of S 2 can be packed in a ingle bin. (iv) One cannot pack one item from S 1 and two from S 2 together in one bin. Thu there i a packing of S 1 S 2 in two bin, but the packing that ue Ω(p 1 2 ) and place one item from each et in each bin i incompreible. Proof. Let p be the given overflow probability, q a real number lightly greater than p, and ε a mall contant. One can verify that the above propertie hold for the following two et of weighted Bernoulli trial: S 1 conit of εp 1 2 item of type (q, 1 p); S 2 conit of εp 1 2 item of type (1, p). Corollary 2.4. No algorithm which imply look at a ingle effective bandwidth number for each item can provide an approximation ratio better than Ω(p 1 2 ). Proof. Note the behavior of any effective bandwidth function γ in the example of the above theorem. If X S 1 and Y S 2, then we have jut argued that there exit a et of item whoe effective bandwidth add up to γ(x) + 2γ(Y ) and which cannot be packed into one bin. But the entire et of item can be packed into two bin; and it total effective bandwidth i εp 1 2 [γ(x) + γ(y )]. Thi example alo how that the firt-fit heuritic applied to a given item ordering can ue a number of bin that i Ω(p 1 2 ) time optimal. The effective bandwidth we ue. A dicued in the introduction, we will ue both the tandard definition of effective bandwidth β p and a new modified effective bandwidth β p that turn out to be neceary in the cae of bin-packing and i alo ued in proving our lower bound on optimality for the load balancing problem. For a random variable X, one define [10, 12] (2.1) β p (X) = log E[p X ] log p 1. For a Bernoulli trial X of type (q, ), we define it modified effective bandwidth by (2.2) β p(x) = min{, qp }. For a et of random variable R, we will ue the notation β p (R) = X R β p(x) and β p(r) = X R β p(x). We firt give an inequality relating our modified effective bandwidth to the tandard one. The proof follow from elementary calculu. Propoition 2.5. For a Bernoulli trial X, β p (X) β p(x). Proof. Firt, we etablih the following claim. (A) For a 1, define f(x) = a x 1 and g(x) = xa x ln a. Then f(x) g(x) for all x [0, 1]. We prove (A) by noting that and f (x) g (x) for all x [0, 1]. f(x) lim x 0 g(x) = 1,

8 198 JON KLEINBERG, YUVAL RABANI, AND ÉVA TARDOS Now if X i of type (q, ), then we have β p (X) = log (qp + (1 q)) log p 1 = log (1 + q(p 1)) log p 1. To prove the propoition, it i ufficient to how that β p (X) and β p (X) qp. The firt of thee tatement follow by taking logarithm bae p 1 of the inequality qp + (1 q) p. To how the econd, note that by Taylor inequality and by fact (A) β p (X) q(p 1) log p 1, q(p 1) log p 1 qp. 3. Stochatic load balancing. Let X 1, X 2,..., X n be mutually independent random variable taking nonnegative real value. We hall refer to them a item. Let φ : {1,..., n} {1,..., m} be a function aigning each item X i to one of m bin. We define the load of the aignment φ, denoted L(φ), to be the expected maximum load on any bin; that i, L(φ) = E[max i j φ 1 (i) X j]. We are intereted in deigning approximation algorithm for the problem of minimizing L(φ) over all poible aignment φ. Note that the maximum of the expectation would be eay to approximate by imply load balancing the expectation The algorithm for on-off item. In thi ubection we preent an O(1)- approximation algorithm for the cae of weighted Bernoulli trial; we then extend thi to handle arbitrary ditribution in the following ubection. For a Bernoulli trial of type (q, ), we can further aume that i a power of 2 by reducing all item ize to the nearet power of 2 we loe only a factor of 2 in the approximation ratio. Our load balancing algorithm i on-line. It proceed through iteration; in each iteration it maintain a current etimate of the optimum load, which will alway be correct to within a contant factor. An iteration can end in one of two way: the input can come to an end, or the iteration can fail. In the latter cae, the etimate of the optimum i doubled, and a new iteration begin. For eae of notation, the algorithm recale all modified ize that it ee o that the etimate in the current iteration i alway equal to 1. An item X i of type (q i, i ) i aid to be exceptional if i > 1, and normal otherwie. Throughout the algorithm, we define p = m 1 (recall that m i the number of bin) and C to be an abolute contant. (C = 18 i ufficient.) One iteration proceed a follow; uppoe that item X i ha jut been preented. (1) For each bin j, let B j denote the et of all nonexceptional item from thi iteration that have been aigned to j. (2) If X i i normal, then we aign it to the bin j with the mallet value of β p (B j ). If thi would caue β p (B j ) to exceed C, then the iteration fail. (3) Suppoe X i i exceptional. If the total expected ize of all exceptional item een in thi iteration (including X i ) exceed 1, then the iteration fail. Otherwie, X i i aigned to an arbitrary bin. To prove that thi algorithm provide a contant-factor approximation, we how that (i) if an iteration doe not fail, then the load of the reulting aignment i within a contant factor of the etimate for that iteration; and (ii) if iteration fail, then

9 BANDWIDTH FOR BURSTY CONNECTIONS 199 the load of any aignment mut be at leat a contant time the etimate for that iteration. We tart with (ii). Lower bounding the optimal olution. Firt we prove a lower bound on the optimal olution to the load balancing problem. Thi lower bound i the main new technical contribution of thi part, and will be ued alo in analyzing the bin-packing and knapack algorithm in the next two ection. In thi ubection we tate and prove the lower bound for the pecial cae of weighted Bernoulli trial. (In ection 3.2 we how how the general cae follow from the pecial cae.) Aume that X 1, X 2,..., X n are independent Bernoulli trial uch that X i i of type (q i, i ). We will ometime ay that item X i i on to refer to the event that X i = i. We ue the following baic claim repeatedly. Claim 3.1. Let E 1,..., E k be independent event, with Pr[E i ] = q i. Let E be the event that at leat one of thee event occur. Let q 1 be a number uch that i q i q. Then Pr[E ] 1 2 q. Proof. Let q = 1 k i q i. Pr[E ] = 1 (1 q i ) 1 (1 q) ( 1 q qi) i i 1 e i qi 1 e q q 1 2 q2 1 2 q. Our key technical lower bound i in the following lemma. Here p [0, 1] i a target probability (in thi ection we ue p = m 1 ). Lemma 3.2. Let X 1,..., X n be Bernoulli trial of type (q 1, 1 ),..., (q n, n ), repectively, uch that log 1 p 1 i 1 for each i, and each i i an invere power of 2. If i β p(x i ) 7, then Pr[ i X i 1] p. Proof. Our goal i to modify the given et of Bernoulli trial o a to obtain a new problem in which (i) the probability of the um exceeding 1 i no greater than originally and (ii) the probability of the um exceeding 1 i at leat p. If there i any X i for which β p(x i ) = i, we lower q i until q i = p i. Thi preerve the aumption that i β p(x i ) 7. Let be an invere power of two, and conider the et W () of item X i for which i =. We partition W () into et W () 1,..., W r () uch that for all j = 1, 2,..., r 1, 2p i X i W q () i 3p and i X j i W q () i < 2p. Thi can be done becaue q i p r for all X i W (). We define a et V () of Bernoulli trial Y () 1,..., Y () r 1, each of type (p, ). Intuitively, each Y () j approximate well the behavior of X i W X () i. j In particular, we how that the former i tochatically dominated by the latter. We will prove the following: (A) Pr[ (B) β p( V () ) 1; j Y () j 1] Pr[ i X i 1]; j Y () j 1] p. (C) Pr[ The claim clearly follow from (A) and (C). To prove (A), we how that Pr[ X i W X () i ] p = Pr[Y () j ]. The j expreion on the left-hand ide i imply the probability that any of the item in i on; by Claim 3.1, the fact that i X i W () j thi probability i at leat p, and (A) follow. W () j q i 2p, and the fact that p 1 2,

10 200 JON KLEINBERG, YUVAL RABANI, AND ÉVA TARDOS To prove (B), notice that β p(w r () ) 2p p = 2, and for 1 j < r, β p(w () j ) 3p p = 3. On the other hand, β p(y () j ) = p p =. Thu β p(v () ) 1 3 (β p(w () ) 2). Hence β p( V () ) = β p(v () ) β p(w () ) 2 3 = 1 β 3 p(w () ) 2 1, 3 where the lat inequality follow from the fact that β p(w () ) 7, and 2 becaue only take on the value of invere power of 2. To prove (C), recall that for all j,, β p(y () j ) = p p =. Now, let V denote a ubet of V () coniting of item whoe ize um to 1. That uch a et exit follow from (B) and the fact that all ize are invere power of 2. Let {Y 1,..., Y l } denote the item in V, and let 1,..., l denote their ize, repectively. Note that the probability that Y i i on i equal to p i. j Y () j The probability of the event 1 i at leat a large a the probability that all item in V are on. But thi latter probability i equal to l i=1 p i = p. The lower bound for exceptional item follow by an argument uing Claim 3.1. Lemma 3.3. Let X 1,..., X n be uch that L 1 n and i q i i L. Then for all φ, we have L(φ) 1 2 L. Proof. Without lo of generality, we may aume i q i i = L. Let q i = j i q j. Let E i denote the event that at leat one item among {X j } j i i on, and let q i = Pr[E i]. Note that becaue i q i i = L and i L for all i, we have i q i 1 and hence q i 1 for all i. Thu, by Claim 3.1, q i 1 2 q i. Write 0 = 0 and q n+1 = 0. Oberve that i q i i = i q i ( i i 1 ), becaue each i i counted with a multiplier of q i on the right-hand ide. Since Pr[X i i on and not E i+1 ] = q i q i+1, we have E[max{X 1,..., X n }] i i (q i q i+1) = i q i ( i i 1 ). Thu for any aignment φ we have L(φ) E[max{X 1,..., X n }] q i ( i i 1 ) i 1 q 2 i( i i 1 ) = 1 q i i = L. i Our main lower bound for the load balancing problem i the following lemma. Lemma 3.4. Suppoe that for all i, i i an invere nonnegative integral power of 2 (o i 1). Further uppoe that i β m 1(X i) 17m. Then, for all φ, L(φ) = Ω(1). Proof. Let φ be an arbitrary aignment of the item to bin. Let B 1,..., B m denote the et of item aigned to bin 1,..., m, repectively. Apply the following contruction: a long a ome et B i contain a ubets with β m 1(S) 8, we i

11 BANDWIDTH FOR BURSTY CONNECTIONS 201 put aide a minimal ubet S with thi property. Note that β m 1(S) 9 a the bandwidth of a ingle item of ize at mot 1 never exceed 1. When we can no longer find uch a ubet, then the et of remaining item R ha β m 1(R) 8m. Thu, thi contruction produce at leat m ubet, uch that each i aigned to a ingle bin by φ. We denote the firt m of thee ubet by W 1,..., W m. Call a Bernoulli trial X of type (q, ) mall if < 1/ log p 1. Uing the fact that mall item have p 2, we can ee that the effective bandwidth β p(x) of a mall item i at mot twice it expectation E[X] = q. Call a et W i dene if the et of mall item S i W i ha β m 1(S i) 1. If there exit a dene et W i, then the expected ize of W i i at leat 1 2. Since L(φ) i at leat a large a the expected ize of W i, L(φ) 1 2 and the lemma follow. Thu, we conider the cae in which no W i i dene. Let W i W i denote the et of item in W i which are not mall. Since W i i not dene, β m 1(W i ) 7. By Lemma 3.2, the probability that ize of W i exceed 1 i at leat m 1. Hence the probability that ome W i exceed 1 i at leat 1 (1 m 1 ) m 1 e 1. Since L(φ) E[max{W 1,..., W m}], the lemma follow. Recall that the algorithm maintain a current etimate. The iteration fail if the total effective bandwidth of the mall and normal item in a bin would exceed a contant C (we ue C = 18) or if the total expected ize of all exceptional item een in thi iteration exceed 1. Theorem 3.5. Let W denote the et of item preented to the algorithm in an iteration that fail. For any aignment φ of W to a et of m bin we have L(φ) = Ω(1), where 1 i the etimate for the iteration. Proof. Let φ be an arbitrary aignment of item in W to bin. An iteration can fail in one of two way: either becaue the expected total ize of exceptional item exceed 1, or becaue the aignment of the new item to any bin j would caue β p (B j ) to exceed C. In the firt cae, Lemma 3.3 implie that L(φ) 1 2. Concerning the econd cae, conider the moment at which the iteration fail. We have j β p(b j ) m(c 1) (becaue the new item ize, and therefore it effective bandwidth, cannot exceed 1). Recalling that C 18, Lemma 3.4 aert that L(φ) = Ω(1). Upper bounding the olution obtained. The following propoition i eentially due to Hui [10], who tated it with a = 2 and b = 1. We give a hort proof for the ake of completene. Propoition 3.6 (ee [10]). Let X 1,..., X n be independent random variable, and X = i X i. Let a > b. If i β p(x i ) b, then Pr[X a] p a b. Proof. Firt, if i β p(x i ) b, then i log E[p Xi ] log p b and hence i E[p Xi ] p b. Thu we have Pr[X a] = Pr[p X p a ] p a E[p X ] = p a i E[p Xi ] p a b, where the firt inequality follow from Markov inequality, the equation from the independence of the X i, and the lat inequality from inequality above. Lemma 3.7. Conider the aignment produced by any iteration of the algorithm. The load of thi aignment i O(1). (Recall that ize are caled o that 1 i the etimate for that iteration.) Proof. The expected ize of the um of exceptional item placed in thi iteration i at mot 1, o they only add at mot 1 to the expected maximum load. Let S j = X i B j X i. Let x 0. A β p (B j ) C, Pr[S j > x + C] m x by Propoition 3.6. Let S = max{s 1,..., S m }. We havepr[s y] j Pr[S j y].

12 202 JON KLEINBERG, YUVAL RABANI, AND ÉVA TARDOS Hence E[S ] = 0 = C C Pr[S x]dx C Pr[S x + C]dx m m x dx = C m 1 m ln m = C + O(1), C+1 Pr[S x]dx from which the lemma follow. Since the etimate increae geometrically, a conequence of Lemma 3.7 i the following theorem. Theorem 3.8. Let φ A be the aignment produced by the algorithm. Then L(φ A ) = O(1), where item ize are caled o that 1 i the etimate for the final iteration. Combining Theorem 3.8 and 3.5, we get our main reult. Theorem 3.9. The algorithm provide a contant-factor approximation to the minimum load Extenion to arbitrary ditribution. We may aume that the only value taken on by our random variable are power of 2. If not, other value are rounded down to a power of 2. A in the previou ection, thi increae our approximation guarantee by a factor of 2 at mot. Call a random variable that only take value that are power of 2 geometric. By the following claim we can reduce the problem for geometric item to the problem for Bernoulli trial item, which we have already olved. Lemma Let X be a geometric random variable. Then there exit a et of independent Bernoulli trial Y 1,..., Y k, with Y = i Y i, uch that Pr[X = ] = Pr[ Y < 2]. Proof. Suppoe that X take the value i with probability q i for i = 1,..., k. Suppoe that 1 > 2 > > k. We define Y i to be of type (q i, i), where q i = q i (1 q 1 q i 1 ). Notice that the event X = i, i = 1,..., k, are mutually excluive, and therefore q i i imply Pr[X = i X i ]. The et {q i } i the olution to and hence Pr[max j q 1 = q 1, q 2 = (1 q 1)q 2, q 3 = (1 q 1)(1 q 2)q 3, ( k 1 q k = ) (1 q i) q k, i=1 i 1 Y j = i ] = q i (1 q j) = q i = Pr[X = i ]. j=1

13 BANDWIDTH FOR BURSTY CONNECTIONS 203 A i > j>i j, the claim follow. The algorithm i eentially the ame a before. It ue the tandard definition of effective bandwidth (Equation (2.1)), which applie to any ditribution. The only change arie from the fact that we mut define what we mean by exceptional in thi cae. Each item X i i now divided into an exceptional part X i 1 {Xi>1} and a nonexceptional part X i 1 {Xi 1}. When the expected total value of all exceptional part exceed 1, the iteration fail; before thi, exceptional part are (necearily) jut packed together with their nonexceptional part. Theorem The algorithm provide a contant-factor approximation to the minimum load. Proof. Recall that in the cae of Bernoulli trial exceptional item could be packed in any bin. The upper bound argument follow a before, uing Propoition 3.6 for the nonexceptional part of the item. The lower bound argument require the approximation of each item by a um of Bernoulli trial uing Lemma We replace each item X i of a geometric random variable by the correponding independent Bernoulli trial and apply the lower bound of the previou ubection to the reulting et of Bernoulli trial. 4. Bin packing with tochatic on-off item. In thi ection we conider the bin packing problem with independent weighted Bernoulli trial, which we will refer to a item. In addition we are given an allowed probability of overflow p. The problem i to pack the item into a few bin of ize 1 each a poible, o that in each bin the probability that the total ize of the item in the bin exceed 1 i at mot p. We aume throughout that p 1 8 ; thi i conitent with routing application, where p i much maller than thi [4]. We develop approximation algorithm parameterized by a number ε, 0 < ε < 1 2. Our reult how that a olution whoe value i within a factor of O(ε 1 ) to optimal can be obtained if we relax either the bin ize or the overflow probability. That i, we compare the performance of our algorithm to the optimum for a lightly maller bin ize or overflow probability. Uing thee reult we then give an approximation algorithm without relaxing either the bin ize or the overflow probability. Our algorithm will be on-line, a before. The baic outline of the method i a follow. A in the load balancing algorithm, we will claify item according to their ize. For the cae with relaxed ize and/or probabilitie, an item will be mall if i 1/ log 2 p 1, large if i 1 2ε for the parameter ε, and normal otherwie. We pack uing the expectation for mall item, uing the effective bandwidth β p (X) for normal item, and we develop technique for packing large item baed on our verion of the effective bandwidth β p(x). It can in fact be hown that the tandard definition of effective bandwidth i not adequate for obtaining a trong enough approximation ratio. For a large item of type (q i, i ), we effectively dicretize it ize, and work with it effective ize i ; thi i the reciprocal of the minimum number of copie of weight i that will overflow a bin of ize 1: 1 i = min{j : j i > 1}. Notice that i < i for all i. An algorithm with relaxed bin ize and probability. We tart by decribing a impler verion of the algorithm in which we relax both the bin ize and the overflow probability. Each bin will contain item only of the ame type (mall, normal, or large). Each item i aigned a weight, according to which it i packed. Bin of each type can be packed according to any on-line bin-packing heuritic, applied to the weight; to be concrete, we will aume that the firt-fit heuritic i being ued.

14 204 JON KLEINBERG, YUVAL RABANI, AND ÉVA TARDOS Small item are given a weight equal to their expectation. A bin with mall item will be packed o that it total weight doe not exceed 1 6. Each normal item X i aigned a weight of β p (X). A bin of normal item will be packed o that it total weight doe not exceed ε. The et of large item can have at mot 2ε 1 different effective ize. They are claified into group by the following two criteria. (i) Each bin will only contain item of the ame effective ize. (ii) We ay that a large item X i of type (q i, i ) and effective ize ha large probability if q i p and normal probability otherwie. No bin will contain item of both large and normal probabilitie. We pack large probability item in bin o that fewer than 1 are in any bin. We pack normal probability item o that the um of the probabilitie of item in a bin doe not exceed p / e where e i the bae of the natural logarithm. We now argue that the algorithm yield a feaible packing in bin of ize 1 + ε. Firt we conider large item. If a bin contain item of effective ize = 1 k, then it will overflow if and only if at leat k item are on. Thi implie that bin with large probability item do not overflow even if all item are on. Large item with normal probability are handled by the following lemma, which involve an analyi of our modified effective bandwidth. Lemma 4.1. Let X 1,..., X n be independent Bernoulli trial of type {(q i, i )}, and aume that the effective ize i = and q i p for all i. Let X = i X i, and aume that i q i p / e. Then Pr[X 1] p. Proof. We get overflow in a bin if and only if at leat k item are on, where k = 1. Let I denote the et of all item. For a et of item S I of ize k, the probability that all item in S are on i i S q i. Thu the probability of overflow i at mot (4.1) q i. S I, S =k i S We claim that thi formula i maximized for a given um of probabilitie i q i if all probabilitie q i are all the ame. To ee thi, uppoe that we have two item X i, X j with different probabilitie, and conider modified item with probabilitie q i = q j = 1 2 (q i + q j ). We now oberve that the um of probabilitie ha remained the ame, but the probability of overflow i larger: the term of (4.1) that contain 0 or 1 of the value q i, q j contribute in total the ame a before, and term containing both are each increaed. Aume now that all item have the ame probability q. The um of the probabilitie of item i at mot p / e; hence, the number of thee item i at mot p / qe. Now the probability that k item are on i bounded by ( ( ) p 1 /q e p )q 1 1 q = p; 1/ q the inequality follow from the etimate ( ) n k ( en k )k. The feaibility for mall item follow eaily from Chernoff bound. Lemma 4.2. If X 1,..., X k be independent Bernoulli trial of type (q 1, 1 ),..., 1 (q k, k ), uch that i log 2 p, and 1 i E[X i] 1 6, then Pr[ i X i 1] < p. Proof. We ue Chernoff bound to bound the probability that the um exceed 1. With µ = 1 6 log p 1, we have Pr[ i X i > 1] < (e 5/6 /6) 6µ < 2 6µ = p. For the normal item, we apply Propoition 3.6 with a = 1 + ε and b = ε.

15 BANDWIDTH FOR BURSTY CONNECTIONS 205 We tate thi pecial cae here for eay reference. Lemma 4.3. Let X 1,..., X n be independent random variable, and X = i X i. Let ε > 0. If i β p(x i ) ε, then Pr[X 1 + ε] p. Theorem 4.4. The on-line algorithm find a packing of item in bin with the property that for each bin, the probability that the total ize of the item in that bin exceed 1 + ε i at mot p. Note that large and mall item are alo feaible with bin ize 1; it i only the normal item that require the relaxed bin ize. To prove the approximation ratio, we need to lower-bound the optimum. For mall item, Chernoff bound are ufficient; for normal item and large item of a given effective ize we make ue of a more careful analogue of Lemma 3.2. Lemma 4.7 and 4.8 will how that on large item of a given effective ize the number of bin ued by our algorithm i at mot a contant factor away from the minimum poible. Since there are only 2ε 1 different large effective ize, thi implie a bound of O(ε 1 ) on large item. Lemma 4.6 how that normal item with large total effective ize (more than 5(1 + 2ε)) have overflow probability more than p 1+3ε. Thi will imply that the number of bin ued for normal item i at mot an O(ε 1 ) factor away from optimal. Finally, mall item are again handled directly with Chernoff bound. Lemma 4.5. Let p < 1 2 and X 1,..., X k be independent Bernoulli trial of type (q 1, 1 ),..., (q k, k ), uch that i log 2 p 1. If i E[X i] 4, then Pr[ i X i > 1] > p. Proof. We ue Chernoff bound to bound the probability that the um exceed 1. With µ = 4 log p 1, we have [ ] Pr X i 1 e 1 2( 3 4) 2µ < p, i and hence Pr[ i X i > 1] > 1 p p. Next we conider normal item. In the load-balancing algorithm we proved a lower bound for effective bandwidth in Lemma 3.2; here we require a tronger verion of thi lemma. For later ue we tate the lemma with a parameter δ. Here we will ue it with δ = 1. Lemma 4.6. Let X 1,..., X k be independent Bernoulli trial of type (q 1, 1 ),..., 1 (q k, k ), uch that i log 2 p, and 1 i β p(x i ) (3δ + 2)(1 + 2ε); then [ ] Pr X i > δ > p δ(1+2ε)+ε. i Proof. Recall that β p (X) β p(x) for all Bernoulli trial; hence we have that i β p(x i ) (3δ + 2)(1 + 2ε). Further, we will round up the ize of each Bernoulli trial X i to an integer power of 1 + ε. Let X i denote the reulting rounded item, and let (q i, i ) denote it type. Rounding up cannot decreae the effective bandwidth, o we have that i β p(x i ) (3δ + 2)(1 + 2ε). Next we prove an analogue of Lemma 3.2 for the rounded item. We claim that with probability more than p δ(1+2ε)+ε, the total ize of the rounded item exceed δ(1 + ε). Notice that thi implie that the total ize of the original item exceed δ with probability more than p δ(1+2ε)+ε. The proof i analogou to the proof of Lemma 3.2.

16 206 JON KLEINBERG, YUVAL RABANI, AND ÉVA TARDOS We may aume without lo of generality that q i p i that (4.2) q i i p i (3δ + 2)(1 + 2ε). i for all i. Now we have We define the et of item W () for each ize ; partition W () into et W () 1,..., W r () q i < 3p for j = 1,..., r 1; and define the et V () of uch that 2p X i W () j Bernoulli trial Y () 1,..., Y () r, each of type 1 (p, ), a in the proof of Lemma 3.2. Next we want to argue that (i) the probability of the um j, Y () j exceeding exceeding δ(1 + ε), and (ii) the δ(1 + ε) i no greater than the probability of i X i probability of the um j, Y () j exceeding δ(1 + ε) i at leat p δ(1+2ε)+ε. To argue part (i) we how a before, uing Claim 3.1, that Pr[ X i () X W i j ] p = Pr[Y () j ]. The fact that p 1/2 follow from the aumption that i 1/ log 2 p 1 for all i. To how part (ii) we claim, uing the notation from the proof of Lemma 3.2, that β p( V () ) > δ(1 + ε). To prove thi, we note that a before we have β p(v () ) > β p (W () ) 2 3. Hence β p( V () ) = β p(v () ) > ( β p(w () ) 2 = 1 β 3 3 p(w () ) 2 ( ) = 1 β 3 p(w () ) 2(1 + ε) δ(1 + 2ε), ince β p(w () ) (3δ + 2)(1 + 2ε) by (4.2), and (1 + ε) ince only take on value that are integer power of (1 + ε) and at mot ε. We complete the proof of the lemma by howing Pr[ j Y () j > δ(1 + ε)] p δ(1+2ε)+ε. Note that for all j,, β p(y () j ) = p p =. Now, let V denote a ubet of V () coniting of item whoe ize um to a number in (δ(1 + 2ε), δ(1 + 2ε) + ε); uch a et can be choen a we have hown above that the um of all ize in V () i at leat δ(1 + 2ε), and all ize are at mot ε. Let {Y 1,..., Y l } denote the item in V, with 1,..., l equal to p i. The probability of the event denoting their ize. Note that the probability that Y i Y () j δ(1 + 2ε) > δ(1 + ε) j ) i on i i at leat a large a the probability that all item in V are on. probability i equal to But thi latter l l p i = p i=1 i > p δ(1+2ε)+ε. i=1 Next we conider a group of large item of effective ize. The packing created by the algorithm i clearly optimal for item of large probability.

17 BANDWIDTH FOR BURSTY CONNECTIONS 207 Lemma 4.7. If 1 large probability item of effective ize are in the ame bin, then the probability of overflow i more than p. Proof. Let X 1,..., X denote 1 large probability item of effective ize. Note that if all 1 item are on, then the total ize exceed the bin ize 1. The probability of item i i q i > p for all i. The probability that all item are on i therefore at leat i q i > (p ) 1 = p. Finally, conider large item of a given effective ize and normal probability. Lemma 4.8. Let X 1,..., X k be independent Bernoulli trial of effective ize 1 and probability q 1,..., q k, uch that log 2 p ; q 1 i p for all i, and i q i 3p /. Then Pr[ i X i > 1] > p. Proof. We need to argue that the probability that at leat 1 of the item are on exceed p. We partition the et of item into et W 1,..., W r+1 uch that 2p < X i W j q i 3p for j = 1,..., r, and r 1. Thi i poible a i q i 3p / and q i p for each i. By the aumption that p 1 2, Claim 3.1 implie that in any et W j for j = 1,..., r the probability that at leat one of the item i on the et i more than p. Now the probability that at leat one item i on in each of the firt 1 group i more than (p ) 1 = p. Thi implie the lemma. Now we are ready to prove the general bound. 1 Theorem 4.9. For a parameter ε log 2 p, the above on-line algorithm find 1 a packing of item in bin of ize 1 + ε uch that the number of bin ued i at mot O(ε 1 ) time the minimum poible number of bin in any packing with bin ize 1 and overflow probability at mot p 1+3ε. Proof. We how that the number of bin ued by our algorithm for mall, normal, and large item i within O(ε 1 ) of optimal. Firt, uppoe we ue B bin for mall item. Each bin i packed up to an expected value of at leat log p ince packing an extra mall item in the bin would exceed 1 the expected value of 1 6. It follow that the total expected value of all mall item i at leat B(log p 1 6) 6 log p. Hence, if fewer than B(log p 1 6) 1 24 log p bin are ued, ome bin will 1 overflow with probability exceeding p, by Lemma 4.5. Next, uppoe we ue B bin for normal item. Each bin i packed up to a β p - value of at leat 1 2ε ince adding a new normal item to a bin would exceed the total β p value of ε, and each normal item ha β p value at mot 1 2ε. Therefore, the total β p -value of normal item i at leat 1 εb 2εB. Hence, if fewer than 10(1+2ε) bin of ize 1 are ued for normal item, then Lemma 4.6 implie that ome bin will overflow with probability exceeding p. Finally, we conider large item of a given effective ize. We how that we are within a contant factor of optimal on thi et of item, where the contant doe not depend on ε; thu, ince there are only 2ε 1 different effective ize, our packing of large item will be within O(ε 1 ) of optimal. Firt, Lemma 4.7 implie that for each effective ize, the number of bin ued for large item of large probability i optimal. Now uppoe that we ue B bin for large item of normal probability and a given effective ize. Then the total probability of thi et of item i at leat Bp /2e. Therefore, if fewer than B/6e bin were ued for thi et of item, the item in at leat one bin would have total probability more than 3p /, and by Lemma 4.8 the probability of overflow would exceed p.