A Constant Complexity Fair Scheduler with O(log N) Delay Guarantee

Transcription

1 A Constant Complexity Fair Scheduler with O(lo N) Delay Guarantee Den Pan and Yuanyuan Yan 2 Deptment of Computer Science State University of New York at Stony Brook, Stony Brook, NY 79 denpan@cs.sunysb.edu 2 Depart of Electrical and Computer Enineerin State University of New York at Stony Brook, Stony Brook, NY 79 yan@ece.sunysb.edu Abstract. Many Internet multimedia applications require the support of network services with fairness and delay uarantees. Currently, there are two types of fair schedulers in the literature. The time stamp based schedulers achieve ood fairness and delay uarantees but have hih O(lo N) time complexity, where N is the number of incomin flows. While the round robin based schedulers reach O() complexity, their delay uarantees are O(N). Aimin at constant time complexity as well as ood fairness and delay uarantees, we desin a new fair scheduler suitable for variable lenth packets in this paper. Fast Credit Based (FCB) fair schedulin, the alorithm we propose, provides O(lo N) fairness and delay uarantees, by trackin and minimizin the difference between the service a flow reserves and that it actually receives. It reduces the time complexity to O() by utilizin approximation and synchronization. To compare FCB with other fair schedulers on their end-to-end delay performance, simulations are conducted in NS2 for various packet lenths, and the results show that FCB achieves short end-to-end delay and handles variable lenth packets efficiently. Keywords: Fair schedulin, shared output link, bandwidth uarantee, delay uarantee. Introduction The boomin Internet multimedia applications require the next eneration of networks to provide services with bandwidth and delay uarantees, besides the traditional best effort services. A fair scheduler works at a shared output link or ateway to provide incomin flows with uaranteed bandwidth and delay performance. It ensures that the difference between the services that any flow reserves and that it actually receives is bounded within a specified rane, reardless of the lenth of the time interval. Fair schedulers in the literature can be classified into three types: )Time stamp based. Time stamp based fair schedulers, such as WFQ [] and W F 2 Q [2], compute time stamps for each incomin packet, and schedule packets in the order of the time stamps. They usually provide excellent fairness and delay uarantees, but have hih O(lo N) time complexity due to the operation to sort packets, where N is the number of flows of the ateway. 2) Round robin based. Round robin based fair schedulers, such as DRR [3] and SRR [4], serve the flows in a round robin manner, so that each flow has an equal opportunity of consumin bandwidth. They

2 achieve O() time complexity, but have poor delay bounds, usually proportional to N, as each flow has to wait for all other flows before transmittin the next packet. 3) Combination of both. Some recently proposed alorithms, such as BSFQ [5] and Stratified Round Robin [6], attempt to obtain the tiht delay bound of time stamp schedulers as well as the low complexity of round robin schedulers, by adoptin a basic round robin like schedulin policy plus time stamp based schedulin on a reduced number of units. They improve the time complexity by reducin the number of items that need to be sorted, but still have O(N) worst case delay because of the round robin nature. The followin factors must be considered to desin a ood fair scheduler. ) Bandwidth uarantee. The scheduler should ensure each flow its reserved bandwidth, so that well-behaved flows can be protected from malicious behavior. 2) Delay uarantee. Bandwidth uarantee should be obtained in an efficient way, and therefore well-behaved flows can have short and bounded packet delay. 3) Low complexity. The scheduler should have low time complexity to achieve hih speed schedulin. Especially, constant time complexity enables the scheduler to maintain performance when the number of flows increases. 4) Capability to schedule variable lenth packets. Some schedulers, such as SRR [4], consider only fixed lenth packets, and can not handle variable lenth packets efficiently. They have to sement the incomin variable lenth packets into fixed lenth cells before schedulin, and the receivers also need extra buffer space to reassemble the sementations back to the oriinal packets. In this paper we aim at desinin a fair scheduler for variable lenth packets with constant time complexity as well as ood fairness and delay uarantees. Fast Credit Based (FCB) fair schedulin, the alorithm we propose, adopts a credit based policy, and provides fairness and delay uarantees by trackin and minimizin the difference between the service a flow should receive and that it has actually received in the alorithm. FCB achieves O() complexity by utilizin approximation and synchronization, and we theoretically prove that FCB provides O(lo N) fairness and delay uarantees. To compare FCB with other fair schedulers on their end-to-end delay performance, simulations are conducted in NS2 with packet lenth followin different statistical distributions, and the results show that FCB matches WFQ with reduced complexity, and that FCB handles variable lenth packets efficiently. 2 Fast Credit Based Fair Schedulin In this section, we present the Fast Credit Based (FCB) fair schedulin alorithm. We first introduce some definitions, and then describe the alorithm. In the followin, a ateway (or shared output link) with N incomin flows {f,..., f N is considered. 2. Terminoloies The reservation r i of flow f i is its reserved bandwidth normalized with respect to the total bandwidth of the ateway. By the definition, we have that < r i, and to avoid overbookin, N r i. The credit (t) of flow f i is the rate that the flow should ideally receive bandwidth from the ateway accordin to its reservation at time t, i.e., ( r i (t), flow (t) = P (t) r fi is backloed at time t (t), otherwise where (t) is the set of backloed flows at time t. There may be the case that the sum of the reservations of all the backloed flows is less than the total bandwidth of the ateway. As in the ideal GPS [7] fairness system, the excessive bandwidth is reallocated to avoid wastin available transmission capacity. Therefore, if there is

3 at least one backloed flow at time t, after the reallocation of the excessive bandwidth, the sum of the credits of all the flows is equal to unit, i.e., N (t) =. The debit d i (t) of flow f i is the rate that the flow actually consumes bandwidth at time t. At any time, either the ateway is idle, or one of the flows is transmittin a packet throuh it. In the latter case, the transmittin flow exclusively consumes all the available bandwidth, and all other flows ( do not use any bandwidth, thus, if flow f i is transmittin a packet throuh the ateway at time t d i(t) =, otherwise To achieve fairness, balance is defined to record the up-to-date bandwidth usae of each flow. The balance b i (t) of flow f i is the accumulated difference of its reserved bandwidth and actually received bandwidth till time t, i.e., b i (t) = t (x)dx t d i(x)dx. From the definition, it is easy to see that the followin equation holds, b i (t ) = b i (t) + t t (x)dx t t d i (x)dx. We define a busy period to be the lonest interval durin which the ateway is never idle. It is sufficient to consider only one busy period, since the system state can be safely re-initialized at the beinnin of the next busy period. Assumin that t, t, t 2,... are the time points within a busy period that a new packet beins transmission throuh the ateway, we call them the schedulin points. Or, in other words, durin the interval between any two sequential schedulin points, a sinle packet is transmitted throuh the ateway. Use L to denote the maximum packet lenth. Generally, we have L t a+ t a, since t a+ t a = t a+ t a dx = lenth of the transmitted packet L. 2.2 Alorithm Description FCB achieves fairness and delay uarantees by restrictin the absolute value of the balance. Since the balance of a flow records the up-to-date difference of its reserved bandwidth and actually received bandwidth, maintainin a small difference helps to ensure the fairness property of the alorithm. Consequently, the ideal stratey is to always choose the backloed flow with the larest balance to transmit, so that it can have debit and reduce its balance. And for the flows that use more bandwidth than what they deserve and have neative balances, they should be penalized and not allowed to transmit in order to recover their balances. Unfortunately, always choosin the larest balance will incur hih O(lo N) time complexity. FCB utilizes approximation to simplify the operation. When FCB selects a flow to transmit, it does not need to have the larest balance, but instead, its balance could be less than the larest one, where is the ranularity of the approximation. The value of affects the performance of FCB, and usually a smaller value of leads to better fairness and delay uarantees. In order to achieve the approximation, 2L+ synchronization holes are used in FCB, as shown in Fiure. Each hole can hold only one flow whose balance is in a specific rane. Suppose the most recently scheduled flow is f k, which is scheduled at time t a, and define lastb to be its balance at t a, i.e., lastb = b k (t a ). Then, at time t a, i.e., the next schedulin point, a flow with its balance in the rane (lastb L + (u ), lastb L + u] can be placed into the u th hole. Fiure 2 describes FCB usin pseudo code, and the detailed explanation follows. Before the schedulin, the balance of each flow is initialized to zero (b i = ). Each time when the alorithm needs to select a flow to transmit, all the holes are first set to empty (hole[] = ). Then each flow tests if the hole (hole[ b i lastb+l (hole[ b i lastb+l ]) correspondin to its balance value is free ] == ), and if yes, fills the hole with itself by settin hole[ b i lastb+l ] = i. Theorem assures that at least one hole must be filled, and it also may happen that there are more than one filled holes. Next, the flow with the larest balance is selected from all the

4 2L+ hole[ ]: hole[u]: hole[2]: hole[]: i 2L+ lastb L+ 2L+ lastb L+( ) lastb L+u lastb L+(u ) lastb L+2 lastb L+ lastb L for each flow f i do { initialize balance b i = ; lastb = ; while true { for ( = ; 2L+ ); + + ) { hole[] = ; for each flow f i do { if (hole[ b i lastb+l ] == ) hole[ b i lastb+l ] = i; k = ; for ( = ; 2L+ ); + + ) { if (hole[] ) k = hole[]; flow f k sends a packet throuh the ateway, say, from t a to t a+; lastb = b k ; for each flow f i do { update balance R by R t b i+ = a+ t t a (x)dx a+ t a d i(x)dx; Fi.. FCB uses synchronization holes to achieve approximation. The flows whose balances are in the same rane compete for a sinle hole. Fi. 2. Pseudo code description of the FCB fair schedulin alorithm. filled holes and ranted to transmit a packet, and its balance value is assined to lastb for the next round of schedulin. After the packet transmission, the balance of each flow is updated accordinly. Althouh the balance update formula includes interal computation, which seems not efficient to implement, the credit and debit values are actually fixed durin each schedulin point interval ([t a, t a+ )), and therefore the interal can be simply computed as multiplication. Theorem If lastb is at most less than the larest balance of all the flows at time t a, then the larest balance at time t a is in the rane (lastb L, lastb + L + ]. Proof. Suppose that at time t a, flow f has the larest balance, b (t a ) b i (t a ) for any i N, and f k is selected to transmit. Then, lastb = b k (t a ) for time t a, and because b k (t a ) is at most less than b (t a ), we obtain lastb + b (t a ). Also suppose that at time t a, flow f l has the larest balance, b l (t a ) b i (t a ) for any i N. We prove that lastb L < b l (t a ) lastb + L +. On the one hand, when t [t a, t a ), d (t), and we have b l (t a) b (t a) = b (t a ) + t a c (x)dx t a d (x)dx > b (t a ) t a d (x)dx b (t a ) L And by b (t a ) b k (t a ), it follows that b l (t a ) > b k (t a ) L = lastb L. On the other hand, when t [t a, t a ), d l (t), and we can obtain b l (t a) = b l (t a ) + t a c l (x)dx t a d l (x)dx b l (t a ) + t a c l (x)dx b l (t a ) + L Usin the fact that b l (t a ) b (t a ) and b (t a ) lastb+, we obtain b l (t a ) lastb+l+. Next, we show that the precondition of the theorem, that the balance of the scheduled flow is at most less than the larest balance, is always uaranteed by FCB. Since the balance value of all the holes is in the rane (lastb L, lastb L + 2L+ ], and lastb L + 2L+ lastb + L +, from the above proof, we know that the balance b l (t a ) of f l must be in the

5 correspondin rane of one of the holes, say, hole[u]. If hole[u] is filled by f l, there is no doubt that f l is scheduled at time t a, because it has the larest balance. If hole[u] is filled by another flow, say, f m, then f m is scheduled, because all the holes with reater correspondin ranes must be empty. Since f m and f l belon to the same hole, the difference of their balances must be less than. The holes can be implemented by synchronization locks available in most operatin systems. At the beinnin, every lock is free. A flow tests if the lock is free before rabs it. If the lock is free, the flow sets the lock, and thereafter this lock is no loner available to other flows. Due to the approximation mechanism and the synchronization locks, FCB only needs to compare at most 2L+ flows in the filled holes, and 2L+ is a constant when is fixed. Thus, FCB achieves O() time complexity. 3 Performance Analysis In this section, we analyze the fairness and delay performance of FCB by considerin their relationship to the rane of the balance. As will be seen, FCB provides O(lo N) fairness and delay uarantees. 3. Bounds on Balance The balance represents the difference between the services that a flow requests and that it actually consumes, and its value rane is closely related to the fairness performance of FCB. First, we have the followin lemma. Lemma At any time, the sum of the credits of all the flows is equal to the sum of the debits of all the flows, i.e., N (t) = N d i(t), Proof. We prove the lemma by considerin two possible cases. Case : There is at least one backloed flow at time t. In this case, the full bandwidth utilization property applies, i.e., N (t) =. On the other hand, since FCB is work conservative and there is a backloed flow, the ateway should be busy, and one backloed flow, say, f k, is transmittin a packet throuh the ateway. Thus, N d i(t) = d k (t) =. Case 2: There is no backloed flow at time t. By the definition, the credit of any flow is, and N (t) =. Also, since there is no backloed flow, the ateway must be idle, i.e., N d i(t) =. In both cases, we have N (t) = N d i(t). Theorem 2 At any time, the sum of the balances of all the flows is, i.e., N b i(t) =. Proof. By the definition of the balance, N b i(t) = N Z t (x)dx Z t d i(x)dx = Z t N (x) And usin Lemma, we have N b i(t) =. It should be noted that, if a flow ends with non-zero balance, the theorem will be eopardized. For a purely technical reason, a flow should not be assined credit when its balance equals its queue lenth. The followin theorem shows that the balance of any flow is lower bounded by a constant. Theorem 3 The lower bound of the balance of any flow is ( L), i.e., b i (t) L. Proof. The balance of a flow only decreases when it has debit, or in other words, when the flow is transmittin a packet. Therefore, to compute the lower bound of the balance, we only need to consider the time that a flow finishes transmittin a packet. N d i(x) dx

6 Suppose that flow f k transmits a packet durin the interval [t a, t a ). Accordin to the schedulin policy, b k (t a ) is at most less than the larest balance at time t a, and from Theorem 2, the larest balance at any time should be reater than or equal to zero. Therefore, we have b k (t a ). And b k (t a ) = b k (t a ) + t a t a c k (x)dx t a t a d k (x)dx. It is obvious that ta t a c k (x)dx and t a t a d k (x)dx L, and we obtain b k (t a ) L. Next, we derive the upper bound of the balance. In order to simplify the problem, in the rest of this section, we assume that each flow is always backloed and its credit is kept as a constant (t) =. Because the excessive bandwidth of the idle flows is reallocated to other backloed flows which we are considerin, this assumption does not weaken the enerality of the results, but it makes the analysis much easier. Similar to the fact that the balance of a flow decreases as it is transmittin a packet, its balance increases when it is idle, because it has positive credit and zero balance. Thus, in order to compute the upper bound of the balance, we only need to consider the time when a flow beins sendin a packet. The basidea of the proof is to assume that the maximum value of the balance is reached by one flow at a specific time point, and consider the sum of the balances of the flows that have been recently scheduled before that specific time point. We trace back by includin one more flow into consideration each time, until finally there is only one flow outside of the set of flows we are considerin. Then, Theorem 3 can be applied to derive the bound of the maximum value we assumed. First, we explain the notations to be used in the proof. M: the maximum value of the balance. t m : the time that the maximum balance is reached. F (t a, t m ): the set of flows that bein to transmit packets at schedulin points t a,..., t m, where t a t m. Without loss of enerality, we assume that flow f reaches the maximum balance M at time t m. Then, f must bein to transmit at t m, otherwise b (t m+ ) = b (t m ) + t m+ c dx tm+ t m d (x)dx = M + t m+ t m c dx > M. Also assume that the sequence of the flows added into F (t a, t m ) when considerin the sum of the balances is f, f 2,..., f N. In other words, when we look back from the schedulin point t m, f 2 is the most recently scheduled flow. For example, if f 2 was most recently scheduled at t b (t b < t m ) before time t m, we have F (t m, t m ) = {f, F (t b+, t m ) = {f, and F (t b, t m ) = {f, f 2. If f 3 was most recently scheduled at t c (t c < t b ), then F (t c+, t m ) = {f, f 2 and F (t c, t m ) = {f, f 2, f 3. T (t a, t m ): the sum of the balances of all the flows in F (t a, t m ) at time t a, i.e., T (t a, t m ) = f i F (t b a,t m) i(t a ). We next introduce some supportin lemmas. Due to space limitation, the proofs of these lemmas are omitted. Lemma 2 Suppose F (t a, t m ) = {f,..., f k and F (t a, t m ) = {f,..., f k, f k+. Then, T (t a, t m) k + k T (t a, t m) L k Lemma 3 Suppose F (t a, t m ) = {f,..., f k and F (t a, t m ) = F (t a, t m ) = {f,..., f k. Then, T (t a, t m) T (t a, t m) Lemma 4 Suppose F (t m, t m ) = {f and F (t a, t m ) = {f,..., f k. Then, k T (t a, t m) km kl k =i k k i + The upper bound on the balance is iven in the followin theorem. t m

7 Theorem 4 The upper bound of the balance of any flow is ((L + ) ln N + C), i.e., b i (t) < (L + ) ln N + C, where C is a constant. Proof. Suppose at time t a, F (t a, t m ) = {f,..., f, and at time t a, F (t a, t m ) = {f,..., f N. By Lemma 4, T (t a, t m) (N )M (N )L (N ) =i On the one hand, since only f N transmits a packet durin the interval [t a, t a ), when t [t a, t a ), d i (t) = for any i N, and b i (t a ) = b i (t a ) t a t a dx b i (t a ) L. Thus, b i(t a ) T (t a, t m) L (N )M (N )L On the other hand, since flow f N Applyin Theorem 2, we have b i(t a ) = b N (t a ) Combinin the above two inequalities, we obtain or, (N )M (N )L M =i N + i + + L Since <, it follows that M < N + i + + L = i + =i (N ) i + is selected to transmit at time t a, b N (t a ). (N ) i + =i N + i + + L = N + ( + L) = In the above inequality, = is the (n )th harmonic number [8], and = = ln N + γ ɛ N where γ is the Euler s constant and there exists a constant K such that for a sufficiently lare N, ɛ N < K N. Since the value of ( ) is bounded, we can obtain M < ( +L) ln N +C where C is a constant. 3.2 Fairness Guarantee Two fairness measures are commonly used: Golestani measure [9] and Bennet-Zhan measure [2]. While the former compares the relative amount of service received by two different flows, the latter compares the absolute amount of service a flow would receive in the ideal model and the service it receives in the desined alorithm. In this paper we adopt the more accurate Bennet-Zhan measure for analyzin the fairness performance of FCB, and we have the followin theorem. Theorem 5 Durin any time interval, the difference between the bandwidth a flow reserves and that it actually receives in FCB is bounded by L < service difference < (L + ) ln N + C Proof. By the definitions of the credit and balance, credit (t) is the bandwidth flow f i should receive at time t accordin to its reservation, and debit d i (t) is the bandwidth that f i actually consumes at time t. Therefore, the balance is exactly the accumulated service difference, and the bounds of the balance are also the bounds of the service difference. =

8 3.3 Delay Guarantee Besides fairness, packet delay is another important performance measure for a practical fair schedulin alorithm. The delay of a packet is the interval from the time when a packet enters the queue of its flow to the time it is sent throuh the ateway. The followin theorem ives the packet delay of FCB. Theorem 6 The packet delay of flow f i in FCB is bounded by max, L (L + ) ln N C + q < packet delay < (L + ) ln N + C + + L + q where q is the queue lenth of f i after the packet arrives. Proof. Suppose that a packet of flow f i arrives at time t a and leaves the ateway at time t b, and that the queue lenth is q after this packet is put in the queue. Then, by the definition of the balance, we have b i(t b ) b i(t a) = Z tb t a dx Z tb t a d i(x)dx = (t b t a) q Because L (L + ) ln N C < b i (t b ) b i (t a ) < (L + ) ln N + C + + L, we obtain L (L + ) ln N C + q (L + ) ln N + C + + L + q < t b t a < Since the packet delay is always reater than, the lower bound should be adusted if it is smaller than. 4 Simulation Results We compare FCB with other fair schedulers on their end-to-end delay performance by simulation in NS2 []. Two implementations of FCB with = L/ and = L/3 respectively are included to see the effect of the ranularity value on the alorithm. Also, since FCB is desined to handle variable lenth packets, we conducted the simulations under the traffic with packet lenth followin different statistical distributions: constant packet lenth (Fiure 4(a)), packet lenth followin uniform distribution (Fiure 4(b)), and packet lenth followin normal distribution (Fiure 4(c)). The maximum packet lenth L is set to 3 bytes in all the situations. WFQ N N5 N2 N3 N4 M 6K bps, ms M2 9K bps, 2 ms Fi. 3. The network topoloy in the simulations. and DRR are used as the comparison counterparts. WFQ is the earliest and a typical time stamp based fair scheduler. Althouh WFQ has ood fairness and delay uarantees, it has a hih O(lo N) time complexity. By comparin with WFQ, we show that FCB matches the performance of WFQ with reduced complexity. DRR is a representative of the round robin based scheduler family. It has O() time complexity but O(N) worst case delay. By comparin with DRR, we demonstrate that, with the same complexity, FCB achieves better performance than DRR. The network topoloy for the simulations is illustrated in Fiure 3. We set up ten CBR flows between N and N 5 with reserved bandwidth from K bps to K bps in an increment

9 Probability (%).5.5 Constant packet lenth Packet lenth (bytes) Probability (%) Packet x 3 lenth followin uniform distribution Packet lenth (bytes) Probability (%) Packet lenth followin normal distribution Packet lenth (bytes) (a) (b) (c) Fi. 4. The packet lenth distributions. (a) Constant packet lenth. (b) Packet lenth followin uniform distribution. (c) Packet lenth followin normal distribution. Averae delay (ms) Constant packet lenth WFQ DRR FCB(=L/) FCB(=L/3) Averae delay (ms) Packet lenth followin uniform distribution WFQ DRR 8 FCB(=L/) FCB(=L/3) Averae delay (ms) Packet lenth followin normal distribution WFQ DRR 8 FCB(=L/) FCB(=L/3) Reserved bandwidth (K bps) Reserved bandwidth (K bps) Reserved bandwidth (K bps) (a) (b) (c) Fi. 5. Averae end-to-end delay of the normal flows in different alorithms. (a) Constant packet lenth. (b) Packet lenth followin uniform distribution. (c) Packet lenth followin normal distribution. step of. The ten flows behave normally and enerate packets at rates equal to their reserved bandwidths. Another five ill-behaved flows are desined to conest the network, each of which is assined K bps reserved bandwidth but enerates data at K bps. Two of the ill-behaved flows are from N to N 5, and three are from M to M 2. We are interested in the averae end-to-end packet delay of the ten normal flows under different schedulers. Fiure 5(a) ives the simulations results when the packet lenth is constant. As can be seen, the two FCB implementations and WFQ have similar performances, and their delay is shorter than that of DRR. Between the two FCB implementations, the one with larer ranularity value has relatively loner delay for flows with lare reserved bandwidths and shorter delay for flows with small reserved bandwidths, which can be explained by the fact that, with larer ranularity of approximation, flows with small balances have better chances to be scheduled. Under DRR, flows with different reserved bandwidths have rouhly the same delay, and the reason is that, due to the round robin policy, each flow has to wait a full cycle before sendin packets. Fiure 5(b) and Fiure 5(c) show the averae delay when packet lenth follows uniform distribution and normal distribution, respectively. Similar conclusions can be drawn that FCB matches the performance of WFQ with reduced complexity, and that DRR has relatively larer delay which is not sensitive to the reserved bandwidth of a flow. The results also show that FCB is robust in handlin variable lenth packets, in the sense that the delay does not increase dramatically comparin with that under constant packet lenth. 5 Conclusions In this paper, we have proposed the new Fast Credit Based (FCB) fair schedulin alorithm. FCB ensures the reserved bandwidth of a flow by trackin and minimizin the difference be-

10 tween the service the flow should receive and actually receives. By introducin approximation and synchronization, FCB successfully reduces the time complexity to O(). Also, we theoretically prove that FCB provides O(lo N) fairness and delay uarantees. Finally, simulations are conducted to compare the end-to-end delay performance of FCB with those of WFQ and DRR, and the results show that FCB matches the performance of WFQ with reduced complexity, and that FCB is able to efficiently schedule variable lenth packets. Acknowledement The research work was supported in part by the U.S. National Science Foundation under rant numbers CCR-7385 and CCR References. A. Demers, S. Keshav, S. Shenker, Analysis and simulation of a fair queuein alorithm, ACM SIGCOMM 89, vol. 9, no. 4, pp. 3-2, Austin, T, Sep H. Zhan, WF2Q: worst-case fair weihted fair queuein, IEEE INFOCOM 96, pp. 2-28, San Francisco, CA, Mar M. Shreedhar, G. Varhese, Efficient fair queuin usin deficit round robin, IEEE/ACM Trans. Networkin, vol. 4, no. 3, pp , Jun C. Guo, SRR: an O() time complexity packet scheduler for flows in multi-service packet networks, ACM SIGCOMM, pp , San Dieo, CA, Au S. Cheun and C. Pencea, BSFQ: bin sort fair queuin, IEEE INFOCOM 2, pp , New York, Jun S. Ramabhadran, J. Pasquale, Stratified round robin: a low complexity packet scheduler with bandwidth fairness and bounded delay, ACM SIGCOMM 3, pp , Karlsruhe, Germany, Au A. Parekh, R. Gallaer, A eneralized processor sharin approach to flow control in interated services networks: the sinle node case, IEEE/ACM Trans. Networkin, vol., no. 3, pp , Jun J. Conway, R. Guy, The Book of Numbers, New York: Spriner-Verla, pp. 43 and , S. Golestani, A self-clocked fair queuein scheme for broadband applications, IEEE INFO- COM 94, pp , Toronto, Canada, Jun