Input Queued Switches: Cell Switching vs. Packet Switching

Transcription

1 Input Queued Switches: Cell Switching vs. Pacet Switching Yashar Ganjali, Abtin Keshavarzian, Devavrat Shah Departents of EE and CS Stanford University Stanford, CA Abstract Input Queued(IQ) switches have been very well studied in the recent past. The ain proble in the IQ switches concerns scheduling. The ain focus of the research has been the fixed length pacet-nown as cells-case. The scheduling decision becoes relatively easier for cells copared to the variable length pacet case as scheduling needs to be done at a regular interval of fixed cell tie. In real traffic dividing the variable pacets into cells at the input side of the switch and then reassebling these cells into pacets on the output side achieve it. The disadvantages of this cell-based approach are the following: (a) bandwidth is lost as division of a pacet ay generate incoplete cells, and (b) additional overhead of segentation and reassebling cells into pacets. This otivates the pacet scheduling: scheduling is done in units of arriving pacet sizes and in non-preeptive fashion. In [7] the proble of pacet scheduling was first considered. They show that under any adissible Bernoulli i.i.d. arrival traffic a siple odification of Maxiu Weight Matching(MWM) algorith is stable, siilar to cell-based MWM [1-4]. In this paper, we study the stability properties of pacet based scheduling algorith for general adissible arrival traffic pattern. We first show that the result of [7] extends to general re-generative traffic odel instead of just adissible traffic, that is, pacet based MWM is stable. Next we show that there exists an adissible traffic pattern under which any wor-conserving (that is axial type) scheduling algorith will be unstable. This suggests that the pacet based MWM will be unstable too. To overcoe this difficulty we propose a new class of waiting algoriths. We show that waiting -MWM algorith is stable for any adissible traffic using fluid liit technique [6]. Keywords Pacet switching, scheduling, variable length pacets I. INTRODUCTION The two iportant design criteria for switching architectures are: (a) throughput of the syste, and (b) average delay. Aong different switching architectures Input Queued(IQ) switch architecture has been very attractive due to its low eory bandwidth requireents copared to other nown architectures. The crossbar constraints of an IQ switch requires it to schedule pacets to be transferred between inputs and outputs. The throughput and delay in IQ switch are heavily dependent on this scheduling decision. In past there has been a lot of research done to design good scheduling algoriths for IQ switches [1-3],[8]. In these studies there is an iplicit assuption that the switch wors with fixed-size cells. In other words, they all assue that whenever a pacet arrives to the syste, it is divided into equal-sized cells, and after the switching is done, the cells are re-assebled in the for of the original pacet before leaving the syste. Contrary to this coon assuption, we consider systes in which the switch directly wors on pacets without breaing the into cells. We call such a switching syste a pacet-based syste copared to the cell-based systes, which only deal with the fixed-size cells. Using fixed-size cells in the switch aes the ipleentation of the scheduling algorith of the switch uch easier copared to the variable-length pacets, but the following are the two ain disadvantages with fixedsized cell approach: (i) Pacets arriving at input side need to be segented into cells, requiring a special input segentation odule; and at the output side these cells need to be re-assebled. This induces significant ipleentation overhead. (ii) Pacets ay generate incoplete cells because a cell should not contain data belonging to two different pacets. This can result in significant bandwidth loss. For exaple, if cell size is 64 bytes and pacet size is 40 bytes then the aount of bandwidth lost is 24/64 37%!. This otivates the study of pacet scheduling algoriths. The pacet-based algoriths have been studied before in [7]. It is iportant to first understand the throughput region for the case of pacet-scheduling algoriths. Naturally there is soe siilarity between pacet-based and cell-based scheduling. For cell-based scheduling it is nown that Maxiu Weight Matching(MWM) algorith is stable [1-6] for any adissible traffic. In [7] it is shown that canonical odification of the cell-based MWM for pacet-based, which we denote as PB-MWM, achieves 100%

2 throughput for any adissible Bernoulli i.i.d. traffic with pacet lengths being bounded (rather probabilistically bounded and independen. In this paper we first study the PB-MWM algorith. We study the throughput properties of the PB-MWM algorith under general adissible traffic rather than restricting to the Bernoulli i.i.d. case. We first show that the PB-MWM is stable even for any for of regenerative adissible traffic, with the tie of regeneration being finite in ean(note: Bernoulli i.i.d. traffic is special case of regenerative traffic). We obtain this result using a different proof technique, which sees to be soewhat sipler. Next we consider general adissible traffic with Strong Law of Large Nubers property. We show that there exists a counter exaple for which the PB-MWM is not stable. In general, this counter exaple shows that any scheduling algorith that tries to schedule in worconserving fashion or in axial-sense every tie, will not be stable. This counterexaple suggests a fundaental difference between pacet-based and cellbased scheduling algoriths. Hence in general to obtain stability we need to design a different type of pacetbased scheduling algorith. We propose a new class of algoriths which is called waiting algoriths. In particular, we show that waiting odification of PB- MWM is stable for any adissible traffic with bounded pacet lengths using fluid technique siilar to [6] (note: in general ean pacet length should be bounded). The structure of this paper is as follows. In section II, we describe the input-queued switch architecture, the cell-based axiu weight atching (MWM) algorith and the fluid odel for the switch briefly. In Section III, the pacet-based switching algoriths are defined. The canonical extension of MWM algorith for the pacet-based scenario is defined. In section IV, we prove the stability of the pacet-based MWM for regenerative adissible traffic extending the proof of [7]. In section V we present the counter-exaple that otivates the classification of pacet-based algorith into two classes of waiting and non-waiting groups. In section VI, we introduce a siple waiting algorith, which is proved to be stable using fluid techniques. Finally in section VII we conclude the paper. II. INPUT-QUEUED SWITCH In this section we describe the odel of an Input Queued (IQ) switch that is the ain architecture studied in this paper. Figure 1 shows the logical structure of an IQ switch. Although it is not necessary, we assue that the switch has the sae nuber of input and output ports denoted by N. In fact, in practical designs, generally one input and output interface reside on the sae line card, thus the nuber of inputs and outputs is the sae. We assue that the tie is slotted and at each tie slot, at ost one data unit (of nown fixed size) can arrive to each input port. We call this data unit a cell. Cells arriving at input i and destined for output j are stored at input in a FIFO buffer called virtual output queue (VOQ), denoted here by VOQ. This queue separation avoids the loss of throughput due to the head-of-line blocing proble. The cross-bar fabric is assued to be eory-less. We say that a switch has speed up S, if at each tie slot at ost S cells can be reoved fro each input and at ost S cells can be transferred to each output. Input 1 Input N VOQ 11 VOQ 1N VOQ N1 VOQ NN Switching Fabric Figure 1. An Input Queued switch. Output 1 Output N The scheduling algorith decides which cells should be transferred between the inputs and outputs of the switch at every tie slot, i.e., it selects a atching between inputs and outputs in such a way that no input (respectively, outpu ay be atched to ore than one output (respectively, inpu. We say a scheduling algorith is "wor conserving" or "axial", if an input is never left un-atched when it has a pacet for an unatched output. We represent a atching by an N N atrix =[ ] where if input i is connected to output j, we have =1, otherwise =0. The set of all possible atchings is denoted by M. Let A (n) denote the nuber of cells that have arrived at input i destined for output j up to tie n. We adopt the convention that A (0)=0. We assue that the arrival processes A(n)=[A (n)] satisfy the Strong Law of Large Nubers (SLLN), that is, A li = λ i, j = 1,..., N a.s. (1) n n We call λ the arrival rate at VOQ. This assuption on the arrival process is very ild.

3 Definition : We say the arrival processes with arrival rate atrix Λ = [λ ], is adissible if the above condition holds and no input or output is overloaded, in other words, N λ 1 j = 1,..., N i= 1 N (2) λ 1 i = 1,..., N. (3) j= 1 Let D (n) show the nuber of departures fro VOQ up to tie n. Again let D (0)=0 and D(n)=[D (n)] Definition : A switch operating under a atching algorith is called stable (rate stable) if, with probability one, D = λ n li i, j = 1,..., N, (4) n for any adissible arrival process A (n) with rate λ. We say the traffic is i.i.d. if the arrival process is such that, (a) the arrivals to different input ports are independent, and (b) the arrival to the sae input port at different tie slots are also independent. We would lie to note that, a general adissible traffic, satisfying SLLN as above, does not need to have independence. Let Z (n) show the nuber of cells in VOQ at tie n, including any arrival at tie n, then the atrix Z(n)=[Z (n)] shows the queue occupancy at tie n. For any atching M the weight W (n) of the atching at tie n is defined as; where < A,B > = the sae size. W =<, Z( n) >, (5) i, j A B for two atrices A and B of A. Maxiu Weight Matching (MWM) Algorith At each tie slot, MWM algorith will select the atching with the axiu weight aong all atchings in M. If there are ultiple MWMs, one of the is selected arbitrarily. We denote the axiu weight atching and its corresponding weight at tie n by ( n ) and W ( n ) respectively. That is, ( ) = arg ax W, (6) n M W ( = M n) = ax W W. (7) In [1][3], it was shown that under any adissible Bernoulli i.i.d. traffic, MWM algorith is stable. In [6] using the fluid odel analysis it was shown that MWM is stable for any adissible traffic satisfying (1). The notion of stability(rate stability) in [6] is weaer than the notion of stability used in [1]-[3], but in [6] the stability is proved for a larger class of arrival traffic. In this paper also we adopt notion of rate stability as in (4). B. Fluid Model and Switch Dynaics This section describes the fluid odel of a discrete tie switch. For any M, let T (n) represent the cuulative aount of tie that the atching has been used up to tie n under the scheduling algorith used. We assue that T (0)=0. Note that T (n) is a nondecreasing function with respect to n. For a discrete-tie switch the following three equations govern the dynaics of the switch: D Z = A D, (8) ( T T ( n 1 ) = D ( n 1) + 1 Z > 0} ) M {, (9) T ( n = n. (10) ) M The first equation siply states that the nuber of cells in VOQ equals the total nuber of arrivals inus the total nuber of departures. The second equation obtains the nuber of departure by considering all the atchings that can connect the input i to output j. The third equation siply states that at each tie slot, exactly one of the possible atchings is used. In [6], the fluid odel of a discrete-tie switch was introduced. We will use this fluid odel in this paper without presenting any proofs or justification. An interested reader can refer to [6] for an elaborate exposition to this topic. Fro [6], the continuous equations governing the dynaics of the fluid odel of switch described above are as follows: for every i,j=1,,n D = Z = λ t D, (11) M M T ( t ) if Z > 0, (12) T ( t ) = t, (13) where the functions Z ( t ), D ( t ), and T ( t ) are called the fluid liits and are obtained fro the discrete rando processes Z (n), D (n), and T (. For

4 exaple the Z ( t ) is obtained as follows. First, a continuous version Zˆ of the discrete function Z (n) is created; Z ˆ = Z ( t ) + Z ( t + 1) Z( t ) ( t t ). (14) ( ) Then the fluid liit is obtained as follows; Z ( t ) Zˆ ( r = li. (15) r r All other fluid liit functions are obtained in a siilar anner, that is, the tie is scaled by r and the function is re-noralized by dividing by r. III. PACKET-BASED SWITCHING We described the structure of a cell-switch in the previous section, now we can define how a pacet-based switch perfors. Pacets with different sizes can arrive to the switch. However we assue that the fabric wors on fixed-size data units (cells). So in each tie slot only one cell can be sent to each output. Thus, the received pacets ust be segented into an integer nuber of cells. For siplicity, without loss of generality we will assue that each pacet is ade up of an integer nuber of cells. We constrain the scheduling algorith to deliver contiguously all the cells obtained fro the segentation of the sae pacet, i.e., at the output they are not interleaved by the cells fro another input port. More forally we can define a pacet-based scheduling algorith as follows: Definition: A pacet-based scheduling algorith is a scheduling algorith such that once it starts transitting the first cell of a pacet to an output port, it continues the transission until the whole pacet is copletely received at the corresponding output port. With this constraint of scheduling algorith being non-preeptive on pacets avoids the proble of segentation at input ports and reassebly of cells at output ports in a switch. In any cell-based switching syste, different cells of the sae pacet ay observe different delay values before leaving the syste. It is reasonable to assue that the delay seen by the user is sae as the delay observed by the last cell of any pacet. Therefore a scheduling algorith that transfers the last cell of a pacet with larger delay perfors poorly, even if it perfors well on all other cells of the pacet. Most of the nown cell-based scheduling are not aware of the existence of pacets, and therefore there is a chance that a pacet-based scheduling algorith which is aware of the entity of a pacet can use this inforation to do a better scheduling (in the sense of the waiting delay observed by the users). Siilar reasoning was given in [7] by authors in the favor of pacet scheduling. It is easy to convert a nown cell-based algorith into a pacet-based one. Let us consider any cell-based scheduling algorith X (e.g. MWM, axial atching, etc.). We can easily convert X into a pacet-based algorith as follows: At each tie slot, we divide the input-output ports into two disjoint sets: Busy ports: the set of input-output ports which have been atched to each other in the previous tie slot and are still in the iddle of sending a pacet. Free ports: the set of input-output ports which either have no pacets to send, or just finished sending a pacet. The scheduling algorith PB-X eeps the atching already used by busy ports and finds a new (sub- )atching for free ports using the cell-based scheduling algorith X. Initially all the ports are assued to be free. In [7] Marsan et al. considered pacet-based scheduling algoriths in the way defined as above. They described the odel of a pacet-based scheduling algorith, and highlighted the effect of considering pacet entity in designing the scheduling algoriths. They proved that the PB-MWM is stable for any adissible Bernoulli i.i.d. traffic. With the help of siulation results, they showed that pacet-based scheduling algoriths could outperfor a cell-based algorith for certain cases. In the next section we give a different proof for stability of PB-MWM using the fluid odel technique. This proof proves the stability of PB- MWM for a ore general class of arrival process. IV. PB-MWM STABILITY In this section we consider the stability of the pacetbased MWM algorith. First we define soe notations used in the proofs. Definition: A atching (n) used at tie n is called -iperfect atching, if = ( n ). (16) In other words, is -iperfect if it is equal to the axiu weight atching tie slots ago. Obviously, any axiu weight atching is a 0- iperfect atching at the tie it is chosen by the scheduler. The following Lea states a very siple but iportant property of -iperfect atchings. Lea 1: The weight of a -iperfect atching is at ost 2N different fro the weight of the axiu

5 weighted (0-iprefec atching at any given tie slot, i.e., if (n) is a -iperfect atching with weight W (n), used at tie slot n, then; W W 2N. (17) that the pacet lengths are bounded. For this section, we assue that arrival process is stationary. Let p T ( denote stationary probability of event {T=t}. Let W(n) denote the weight of atching obtained by scheduling algorith SA at tie n. Then, Proof: For any atching ' let W '( n ) be its weight at tie n and W (n) be its weight at tie n, then the following inequalities hold under any scheduling algorith: W n ) N W W ( n ) + N. (18) '( ' ' Thus; E{ W Z } t= 0 p T = 0 [ W 2Nt] = W ( n ) 2 N tp T ( t ). (22) This is true because of the following siple reason: during tie slots, at ost cells can arrive (depar at an input port, which in turn can increase (decrease) queue size at any input port by at ost. There are N input ports, and hence the net weight can increase (decrease) by at ost N. We now that (n) is -iperfect, thus, ( n) = ( n ), i.e., at tie n, has the largest weight aong all possible atchings; M W n n ) W ( n ). (19) ( )( ' ' Thus if we select = we have; W n ) W ( n ). (20) ( Furtherore, using the above inequities we can write; W + N W ( n ). (21) Cobining the last three inequalities, we obtain the result stated in lea 1. QED. Let us consider the following scheduling algorith, which we denote as SA: 1. Let s (n) be the schedule used by SA at tie n. 2. At tie n+1, if all ports are free then s ( n + 1) = ( n + 1) else s ( n + 1) = s( n). Let T be the tie between two successive occurrences of event that all ports becoe free. Note that the atching obtained by algorith SA is at worst T-iperfect, by definition. Now T is a rando variable which depends on the arrival process and the pacetlengths. For siplicity without loss of generality assue E{ W Z } W 2NE{ T}. (23) Now if E{T} is finite we say that the traffic pattern is regenerative. In other words, it has property that on average it requires a finite aount of tie to reach the state where all the input and output ports are free. Note that this property is related to the traffic pattern and not with the scheduling algorith SA. It is already shown in [7] that if the traffic is fored by variable length pacets with independent rando size with finite average and variance, and adissible Bernoulli i.i.d. traffic the average value of T is bounded. In general, traffic need not be Bernoulli i.i.d. to obtain this property and there is uch larger class of distribution under which this property holds true. For such regenerative traffic, we prove the stability of SA algorith. The following is a ey lea, which states a ore general result about stability. Lea 2: A scheduling algorith is rate-stable for any adissible traffic with property (1) if the average value of the weight of the atching it uses at each tie slot, is at ost away fro the axiu weight by a bounded constant value, i.e., if E{ W Z } W B, (24) then the algorith is stable. Proof: Let Z( t ) = [ Z ] and D = [ D ], then consider the Lyapunov function L( as; 2 L =< Z, Z( >= Z. (25) It was shown in [6] that for MWM L & 0 if any Z 0. This in turn iplies that if Z (0) = 0 then Z = 0 for all t. This proves the rate-stability of the switch. Hence, our interest is in showing L & 0 if any i, j

6 Z 0 under the scheduling algorith in consideration with property that the expected weight of the atching used is at ost a bounded constant away fro the weight of MWM. Consider all t such that the fluid quantities are differentiable and hence L & ( is well defined. By definition where W ( t ) = Z,. L( = 2 Now fro (12) we have; D t Z ( ), Z(, Z( D( = 2 Λ, Z( t = t Λ D ( ) 2 Z ( ), 2 Z(,. (26) = = Let us define (n) as the difference between the weight of the MWM and weight of the atching obtained by scheduling algorith at tie n. We now that E[ ( n)] is bounded by soe constant B which does not depend on n. Further (n) is a positive rando variable. Hence (n) is bounded alost surely. Thus, on the fluid liit scale we obtain that; M M Z, T T W, (27) ˆ ( r Bˆ ( = li li = 0. (28) r r r r atching. If we show the set of atchings used by the scheduling algorith by M, we have; D( Z(, = M T W T t = W t ( ) ( ). (30) M Note that although M M but since M is the set of atchings used by the scheduler, we can odify (13) to; M T = t. (31) changing the M to M. Now cobining this result with (30) we obtain; Thus the derivative of D( Z ( t ), = W ( t ). (32) L( is; L = 2 Z, Λ 2W ( t ). (33) By Biroff-von Neuann s Theore for the doubly sub-stochastic (adissible) traffic Λ, we obtain: Λ γ, γ 1 <. (34) By definition of the axiu weight atching; Z, ( t W ). (35) Thus, in the fluid scale the weight of the MWM and the weight of the atching used by scheduler will be the sae, i.e., W W ( t = ). (29) Therefore the algorith will only use the atchings that have the sae weight as the axiu weight Hence fro (33) and (34) we obtain; L 2 γ 1 W ( t ). (36) Hence if any Z > 0 then L & < 0. The rest of the proof for the rate-stability follows fro [6]. QED.

7 Now cobining this lea with the above result, we conclude that the algorith SA is stable. Note that, the above proof does not require the bounded pacet lengths, but requires only independent pacet lengths with bounded ean. Theore 3: The SA algorith is stable under regenerative adissible input traffic. We would lie to note that, under PB-MWM algorith the tie between successive occurrences of event when all ports becoe free will also have required property as in Lea 2 under Bernoulli i.i.d. traffic for independent pacet lengths with bounded ean. Hence the stability for PB-MWM will follow fro Lea 2. We would lie to note that as proved for SA, the PB- MWM algorith is stable for a ore general class of traffic. In the next section we show that there are adissible traffic patterns for which the algorith is unstable. V. PACKET-BASED ALGORITHM CLASSIFICATION It is proved in [6] that cell-based MWM algorith has strong stability property that it is stable as long as the input traffic is adissible with property (1). It does not require any other condition on distribution. In previous section we proved the stability for PB- MWM(and SA) for adissible traffic with additional condition that it should be regenerative. The question arises is: whether the PB-MWM(or SA) is stable for any adissible traffic under only (1). In this section, we show that it is not stable using a siple counter-exaple. A 11 A 12 Tie =, 2 =. (37) When the first pacet arrives to the switch, the PB- MWM uses parallel atching ( 1), and then the scheduler is forced to eep the sae atching for 3 tie slots till the pacet finishes. Before this pacet is finished, a pacet of length 2 coes to input 1 and it is scheduled for output 1 under scheduling algorith. In this way, under this traffic pattern, it is easy to see that whenever one input port becoes free, the other input port is busy serving a pacet. Hence both input ports are never free together. This forces the scheduling algorith to use the parallel schedule all the tie. Therefore none of the pacets arriving at VOQ 12 and VOQ 21 will ever get the chance to depart. Thus, the switch is unstable. Note that cell-based MWM will be able to handle this traffic. The counter-exaple described above also shows that any wor-conserving or axial algorith is not stable for that particular traffic pattern. This otivates us to classify the pacet-scheduling algorith in the following two classes: 1. Wor-conserving(non-waiting) algoriths: under these algoriths an input is never left un-atched when it has a pacet for any of unatched output. 2. Waiting algoriths: these algoriths are not always wor-conserving, that is, they do wait for an infinite nuber of tie slots. The above counter-exaple suggests the following general result about the wor-conserving (axial) algoriths. Theore 4: There is no wor-conserving scheduling algorith that is stable under any adissible traffic with condition (1). A 21 A 22 Figure 2. Traffic Pattern. Consider a switch operating under PB-MWM (or SA) with input traffic pattern as shown in Figure 2. A (i,j=1,2) shows the arrival to VOQ. The traffic pattern is periodic with period of length 10. Note that no input or output is overloaded. In fact, λ 1, 1 = 0.8, λ1,2 = 0.1, λ 2, 1 = 0. 1, and λ 2, 2 = The switch can use one of the two possible atchings, naely 1 which is called the parallel atching and 2 the cross atching, i.e., Note that even if an algorith is waiting for finite nuber of steps it becoes wor-conserving after that and hence fro above counter-exaple it becoes unstable. This requires that if algorith wants to be stable then it ust wait for infinitely any tie steps. VI. A GENERALLY STABLE WAITING PACKET-BASED ALGORITHM. In this section we describe a waiting algorith. We will show that the waiting -MWM algorith will achieve 100% throughput for any adissible traffic pattern and in particular, it will be stable for the traffic pattern described in previous section for which PB-

8 MWM or any pacet-based wor-conserving scheduling policy was unstable. The waiting algoriths are otivated fro the counter-exaple described in previous section for worconserving algoriths. The ain proble is that the wor-conserving algorith greedily atches the ports whenever possible, forcing it to always eep the parallel atching in the counter-exaple of figure 2. One way to overcoe this proble is the following: when a pacet gets served do not schedule the freed ports till all ports becoe free and schedule according to a full MWM schedule. The waiting, synchronizes the weight of schedule to the weight of MWM schedule. Hence if waiting is done frequently enough then the weight of schedule is always not ore than a bounded constant away fro MWM, by reasoning siilar to Lea 1. However note that waiting eans that during the waiting period soe ports are losing bandwidth. Hence if waiting is done too aggressively then the algorith can not utilize full bandwidth. These observations lead to the following waiting algorith which we denote as PBwMWM. PB-WMWM The switch runs at speedup (1+ ) for arbitrarily sall positive constant >0. Let the axiu length of any pacet be L (this assuption can be relaxed to ean pacet length being finite which will be described later). Divide the tie into period of length ( L / ) units. Thus tie is considered as [ 0, L / ], [ L / + 1,2 L / ], and so on. Scheduling decisions are ade only when any of scheduled pacets finishes its service and corresponding ports get epty. Let one or ore pacets get served at tie n [ L / + 1,( + 1) L / ]. Consider the following cases: n [ L + 1,( + 1) L L ] : use usual PB (1 + ) MWM to atch the free nodes as before. Otherwise wait on all the pacets till all scheduled pacets get over and all ports are free. After that, use full MWM to re-schedule all the ports and serve. Note that the above algorith at ost loses bandwidth of L per every ( L / ) tie slots. That is it loses bandwidth of per tie slot at ost. The algorith runs at ( 1+ ) speedup in order to ae up for this lost bandwidth. We state the following theore about stability of PB-wMWM: Theore 5: The PB-wMWM algorith is stable(rate stable) under any adissible traffic with property (1) at at speedup ( 1+ ) for any > 0. Proof: Note that the way algorith is defined, every ( L / ) tie the weight of atching is sae as weight of axiu weight atching. Thus any tie the algorith is at worst ( L / ) -iperfect. Hence by Lea 1, the weight of the atching is at ost B = 2NL / away fro MWM. The fraction of tie the algorith idles on any of the ports is bounded above by L (1 + ) L =. (38) 1+ Under speedup ( 1+ ) assuing the algorith is scheduling all the tie, the equation (13) changes to T = (1 + t. (39) ) M But in our algorith, since it is waiting, the above equation ay not be true. Fro above discussion, at worst fraction of the bandwidth is lost in waiting. That is, at least ( 1+ )(1 ) = 1, (40) 1+ is the effective speedup obtained. Thus, the equation (13) of the fluid odel changes to In other words, M M T ( t ) t T 1 (41) (42) Now the arguents siilar to ones used in proof of Lea 2, yield the desired result that PB-wMWM is stable. QED. The above algorith PB-wMWM, assues the pacet lengths to be bounded and bound L is nown. But in reality the L ight not be nown. Further we do not require the pacet lengths to be bounded, but only ean pacet length should be bounded. To address this issue we odify the PB-wMWM algorith as follows,

9 MODIFIED PB-WMWM (PB -WMWM) Initially start with the MWM algorith and start waiting iediately. Copute the axiu aount of idling done by any port. When the waiting starts, there are soe unfinished pacets. Note that the axiu waiting done by any port is at ost the axiu length of any pacet that was under schedule. Let L e (1) represent the axiu length of pacets under schedule. Set M ( 1) = Le (1) / and do the PB-MWM for M (1) tie slots and then start waiting after that. Now let L e (2) be the axiu length of the pacets under schedule when the waiting starts at the end of M (1) tie slots. Siilarly define M 2) = L (2) /. ( e Continue this process recursively over tie. In general we obtain the following recursive expression: Le ( l) M ( l) =. (43) The two ain properties required in the proof of Theore 5 are: (a) The effective speed is at least 1, and (b) the weight of schedule used by algorith is at ost bounded constant away fro the weight of MWM. In the above algorith, let s copute these two quantities as follows: (a) The effective bandwidth lost: In the l total idling per port is at ost for tie while the length of period is th L e period the ( l) /(1 + ) = Le ( l) Le ( l) Le ( l) P( l) M ( l) + = (44) + Thus the fraction of bandwidth lost is at ost: L e ( l) (1 + ) P( l) 1 = (45) Note that this bandwidth loss is sae as the loss in PB-wMWM coputed in proof of Theore 5. (b) The difference between the weight of MWM and the schedule will be at ost M ( l)(1 + ) that is L e (l)(1+ ). (46) Given that L e (l) has bounded ean and pacet lengths are independent we will obtain that the above quantity is bounded alost surely as required in Lea 2. Fro above discussion we obtain the following Theore: Theore 6: The PB -wmwm is rate-stable for any adissible traffic with property (1) and independent pacet lengths with bounded ean. A. Note on speedup In PB-wMWM and PB -wmwm we required speedup ( 1+ ) for arbitrary > 0. If we now that the γ = (1 α) (47) in (34), then in PB-wMWM (PB -wmwm) we can use < α / 2 in defining different quantities and have speedup of 1 only to achieve stability. VII. CONCLUSIONS In this paper we considered the pacet-scheduling algoriths for IQ switch architecture. The result of [7] showed that odification of cell-based MWM for pacet scheduling yields 100% throughput for any adissible Bernoulli i.i.d. traffic with independent pacet lengths of bounded ean. We generalized this result for soe what broader class of arrival traffic pattern. We showed that there exists adissible traffic pattern for which no worconserving or axial algorith is stable. To overcoe this proble we proposed a new class of waiting algoriths. Under the waiting algorith the switch becoes stable for any adissible traffic. This was proved using fluid liit technique. It is interesting to note that the wor-conservation for pacet scheduling is not always beneficial in this sense, unlie cell-based scheduling. This suggests that scheduling pacet-based is quite different fro the cell-based scheduling. ACKNOWLEDGMENT The authors than B. Prabhaar and N. McKeown for suggesting the proble and discussions.

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