Bandwidth Allocation for Best Effort Traffic to Achieve 100% Throughput

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1 Bndwidth Alloction for Best Effort Trffic to Achieve 100% Throughput Msoumeh Krimi, Zhuo Sun, nd Deng Pn Florid Interntionl University, Mimi, FL E-mils: {mkri001, zsun003, Abstrct Generlized Processor Shring (GPS) is powerful model nd there re mny prcticl scheduling lgorithms tht cn perfectly emulte it. GPS is widely used s n idel firness model to schedule pckets for gurnteed performnce trffic. However, there hs not been wy for GPS to properly hndle the best effort trffic. In this pper, we propose bndwidth lloction scheme for GPS clled Queue Length Proportionl (QLP) in crossbr switches without speedup. QLP dynmiclly obtins fesible bndwidth mtrix to schedule best effort flows. In QLP, the mount of service tht ech flow receives is proportionl to the length of its bcklogged queue. We nlyticlly prove tht QLP is strongly stble nd hence provides 100% throughput for ny dmissible trffic, no mtter whether the trffic distribution is uniform or non-uniform. Moreover, we show tht QLP is fesible, which mens the llocted bndwidth does not exceed the vilble cpcity. We lso discuss how to trck the queue length in GPS. Finlly, we perform simultions to verify the theoreticl results nd to mesure the performnce of QLP. I. INTODUCTION Generlized Processor Shring (GPS) hs long been known s simple nd powerful fluid model for trffic scheduling [1]. All fir queueing lgorithms ultimtely emulte the GPS idel model becuse it chieves perfect firness for pcket scheduling [2], [3], [4]. GPS is theoreticl fluid model nd divides the vilble bndwidth into logiclly independent chnnels. Thus, trffic of ech flow is smoothly trnsmitted through its own exclusive chnnel from the input port to the output port. GPS is widely used s n idel firness model to schedule pckets for gurnteed performnce trffic. However, there hs not been wy for GPS to properly hndle the best effort trffic. The objective of this pper is to ddress the pproprite scheduling of best effort trffic in order to be employed in crossbr switches. Crossbr switches hve received significnt ttention due to the non-blocking cpbility nd lrge bndwidth utiliztion in comprison with bus bsed switches [6], [7]. The chllenge of bndwidth lloction in crossbr switch is how to efficiently shre the vilble cpcity of ech input port nd output port. Simple proportionl bndwidth lloction for shred link is not proper for crossbr switches [8], [9] becuse flows of switch re subject to two bndwidth constrints: the vilble bndwidth t both the input port nd output port of the flow. The scheme should be efficient to fully utilize the vilble bndwidth, nd should be fesible in order to be pplied in prctice. Our motivtion for this work rises from the fct tht in crossbr switches we need to dynmiclly obtin n dmissible bndwidth mtrix to properly hndle best effort trffic. Gurnteed performnce flows reserve resources for n llocted trnsmission rte [10]. However, best effort flows try to mke the best use of the vilble trnsmission cpcity but hve no gurntee to the qulity of service [7]. As cn be seen, the bndwidth lloction scheme plys severl importnt roles in gurnteeing the high performnce of switch [11]. First, the scheme helps to determine the trffic dmission policy nd buffer mngement strtegy. Second, n efficient scheme mkes it possible for switch to chieve 100% throughput. Third, the scheme is used s the scheduling criterion by fir scheduling lgorithms. There re mny fir scheduling lgorithms designed for bndwidth lloction in different crossbr switch rchitectures [2], [3], [12], [13], [14], [15]. However, most of them focused on providing qulity of service for gurnteed performnce trffic nd there is reltively less work on how to better support best effort trffic. In this pper, we present bndwidth lloction scheme for GPS clled Queue Length Proportionl (QLP) to properly hndle best effort trffic in crossbr switches without speedup. In QLP, the mount of service tht ech flow receives, or its dedicted bndwidth, is proportionl to the length of its bcklogged queue. QLP essentilly fvors the queues with the gretest occupncy nd thus ssists the crossbr switch to be more work-conserving. We conduct theoreticl nlysis to prove tht QLP is strongly stble nd therefore chieves 100% throughput for ny dmissible trffic, no mtter whether the trffic distribution is uniform or non-uniform. We lso show tht QLP is fesible, which mens the llocted bndwidth does not exceed the vilble cpcity. Furthermore, we discuss how to trck the queue length is GPS. Lstly, we conduct simultions to verify the nlyticl results nd to mesure the performnce of QLP. The orgniztion of this pper is s follows. In Section II, we present the bstrct switch model nd the QLP bndwidth lloction scheme. In Section III, we theoreticlly nlyze the stbility nd throughput of QLP nd then provide some discussions. We show the simultion results in Section IV. Finlly, we conclude the pper in Section V. II. QUEUE LENGTH POPOTIONAL (QLP) SCHEME In this section, we present our bndwidth lloction scheme. First, we briefly explin the switch model. Then, we describe the QLP bndwidth lloction lgorithm for best effort trffic.

2 A. The Abstrct Switch Model The considered switch rchitecture includes N input ports nd N output ports, connected by crossbr with no internl speedup. Let In i denote the i th input port nd Out j denote the j th output port. The vilble bndwidth of ech input port nd output port nd lso the crossbr is. Define the trffic from In i destined to Out j to be flow F ij. Use ij (t) to represent the llocted bndwidth of F ij t time t. Denote the queue of pckets t In i destined to Out j s Q ij. B. Algorithm Description In this subsection we present our Queue Length Proportionl (QLP) bndwidth lloction scheme nd investigte its fesibility property. As mentioned erlier, simple proportionl bndwidth lloction policy of GPS does not pply to switches [8], [9], [11]. For GPS server works t fixed bndwidth, the φ rte of j will gurntee the performnce of ech flow, j φj where φ j is the weight of ech flow [1]. However in contrst with single server, flows of switch re subject to two bndwidth constrints: the vilble bndwidth t both the input nd output port of the flow. Nive bndwidth lloction t the output port my mke the flows violte the bndwidth constrints t their input ports, nd vice vers. QLP dynmiclly ssigns the bndwidth to ech best effort flow proportionl to the bcklogged queue length. We use queue length Q ij (t) s dynmic weight of ech flow F ij nd define Q j (t) = i Q ij(t) to be the number of bits queued t ll input ports directed to prticulr Out j t time t, nd Q i (t) = j Q ij(t) to be the number of bits queued t prticulr In i destined to different output ports t time t. We lso denote the llocted bndwidth of flow F ij respecting to the constrint of ech input port nd output port s i.in(t) nd j.out (t), ccordingly. eclling the GPS fluid model, trffic of ech flow cn smoothly strem from the input port to the output port through its own exclusive chnnel, without buffering in the middle, s illustrted in Figure 1. Thus, by considering the bndwidth constrints t both the input port nd output port of ech flow F ij we hve ij.in (t) = Q ij(t) Q i (t) = Q ij(t) j Q ij(t) (1) ij.out (t) = Q ij(t) Q j (t) = Q ij(t) i Q ij(t) (2) In order to void bndwidth violtion t the input port nd the output port, we consider the overll llocted bndwidth ij (t) of F ij to be the minimum [8] between two clculted bndwidths s follows ij(t) = min{ij.in (t), ij.out (t)} (3) = min{ Q ij(t) Q i (t), Q ij(t) } (4) Q j (t) Q ij (t) = mx{q i (t), Q j (t)} (5) As cn be seen, QLP bndwidth lloction scheme essentilly fvors the queues with the gretest occupncy. It trets ll Input 1 Input i 1 = j 1 j Input i = j ij N = j Nj Fig F11 Output F 1 i1 F N1 1 = i i1 F Nj F 1 j F 1 N Output F j Output ij F in F NN j = N = i ij i in GPS idel fluid model used for scheduling in crossbr switch. queues firly nd ssists the crossbr switch to be more workconserving [20]. One of the dvntges of our scheme is its simplicity. It does not require ny sorting or re-sorting process. Using the clculted informtion of the queue lengths, QLP just needs to find the mximum between the ggregted queue length of flows witing in prticulr input port nd flows directed to specific output port. Now we discuss the importnt properties of QLP, fesibility nd stbility. A bndwidth lloction scheme should efficiently utilize the vilble bndwidth while mintining fesibility. Definition 1: The llocted bndwidth ij (t) of F ij will be fesible if no over-subscription hppens t ny input port or output port. In other words, the ggregted ssigned rte of flows does not exceed the vilble cpcity, i.e. i nd j (6) For esy presenttion, ssume normlized bndwidth; = 1. Now, we show tht our scheme is fesible. According to the QLP description we cn write i = ij(t) = j=1 i=1 j=1 i=1 Q ij (t) mx{ j Q ij(t), i Q ij(t)} j = ij(t) Q ij (t) = mx{ j Q ij(t), i Q ij(t)} Since N j=1 Q ij(t) mx{ j Q ij(t), i Q ij(t)} nd N i=1 Q ij(t) mx{ j Q ij(t), i Q ij(t)}, we hve i 1 nd j 1 (7) which indictes QLP is fesible. We study the stbility of our scheme in the next section. III. PEFOMANCE ANALYSIS A. Stbility nd Throughput In this subection, we theoreticlly prove tht QLP is strongly stble for ny dmissible trffic which implies tht QLP chieves 100% throughput. We first dopt the following definitions presented in [23]. The switch size is shown by P = N N = N 2.

3 Definition 2: U is the set of vector Y = (Y 1,..., Y P ) nd Y +P, such tht Y (i+jn) 1 j = 0, 1,..., (N 1) (8) i=1 N 1 Y (i+jn) 1 i = 1,..., N (9) j=0 Definition 3: Y is the Eucliden norm of vector Y, i.e., Y = y y2 i... + y2 P = P i=1 (y2 i ). Definition 4: Ŷ is the normlized vector prllel to Y, given tht Y 0, i.e., Ŷ = Y Y = Ỹ Ỹ. Definition 5: Ỹ is the mximl vector prllel to Y, given tht Y 0 nd k, i.e., Ỹ = mx k ky. Definition 6: Ψ Y is the symmetric mtrix ssocited with the projection opertor long the direction of Ŷ, given tht Y 0, i.e., Ψ Y = Ŷ T Ŷ. Property 1: Given tht Y 0, we hve Y Ψ Y = Y, which is strightforwrd result of definition 6. The following vribles re used to represent the sttus of the crossbr switch. Their initil vlues re ssumed to be zero t time t = 0. Q ij (t) is the number of bits buffered in Q ij t time t, belong to flow F ij. Q(t) is the vector of queue lengths t time t, i.e., Q(t) = (Q 11,..., Q ij,..., Q NN ). A ij (t) is the number of bits rriving t Q ij up to time t. A(t) is the vector of the number of rrivls t time t, i.e., A(t) = (A 11,..., A ij,..., A NN ). D ij (t) is the number of bits deprting from Q ij up to time t. D(t) is the vector of the number of deprtures t time t, i.e., D(t) = (D 11,..., D ij,..., D NN ). Assume tht the number of rriving bits A ij (t) stisfies the Strong Lw of Lrge Numbers (SLLN), i.e. A ij (t) lim = λ ij (10) t t where λ ij is the rrivl rte of Q ij. Consider the incoming trffic is dmissible, which mens tht no over-subscription t ny input ports or output ports, i.e., for ll λ ij 0 we hve i, λ ij 1, nd j, λ ij 1 (11) j i Correspondingly, Λ is the vector of the verge rrivl rtes, i.e., Λ = (λ 11,..., λ ij,..., λ NN ), nd due to the dmissible trffic we hve E[A(t)] = Λ. Similr to tht in [25][18], the evolution eqution of the switch for the intervl [t, t + 1] is described s follows Q(t + 1) = Q(t) + A(t) D(t) (12) Before investigting the stbility of our scheme, we first clculte the verge deprture rte s the following lemm. Lemm 1: For crossbr switch without speedup, the verge deprture rte in QLP bndwidth lloction scheme is proportionl to the queue length, i.e. E[D(t)] = Q(t) (13) Proof: According to the previous Section, the dedicted portion of the vilble bndwidth for ech queue Q ij t time t, cn be described s weight fctor w ij (t). It mens tht for flow F ij we cn hve { 0, if Qij (t) = 0 w ij (t) = Q ij(t) mx{ j Qij(t), Qij(t)}, otherwise (14) i which is positive when there is n offered lod Q ij (t) > 0. Consequently, ij (t), the llocted bndwidth of flow F ij Q t time t will be ij (t) mx{ j Qij(t), i Qij(t)}. Obviously, the deprture rte of ech flow will be equl to its llocted bndwidth s follows. D ij (t) = ij(t) = Q ij (t) mx{ j Q ij(t), i Q ij(t)} = Q ij (t) which is proportionl to its queue length t time t. By considering the normlized vilble bndwidth = 1, we hve D ij (t) = Q ij (t), nd thus E[D(t)] = Q(t). In order to prove 100% throughput of our scheme, we introduce the following definition nd lemm from [23][25]. Definition 7: A system of queues is strongly stble if lim sup E[ Q(t) ] < (15) t which implies 100% throughput nd bounded dely gurntee. Lemm 2: Given system of queues whose evolution is described by DTMC with stte vector S(t) N M, nd whose stte spce H is subset of the Crtesin product of denumerble stte spce H Q nd finite stte spce H K, then, if lower bounded function V (Q(t)), clled Lypunov function, V : N P cn be found such tht E[V (Q t+1 ) S(t)] < S(t) nd there exists ɛ +, B + such tht S(t) : Q(t) > B E[V (Q(t + 1)) V (Q(t)) S(t)] < ɛ Q(t) (16) then the system of queues is strongly stble. It mens tht the queue length does not grow infinitely which implies 100% throughput nd bounded verge dely. Proof: We need to show lim t sup E[ Q(t) ] <. For the detiled proof, see theorem 2 in [23]. Now, we present the min theorem for the stbility of QLP. Theorem 1: For crossbr switch without speedup, the QLP bndwidth lloction scheme is strongly stble for ny dmissible trffic, i.e., it chieves 100% throughput. Proof: According to lemm 2 nd eqution (16), we cn define qudrtic Lypunov function V (t) = Q(t)ZQ T (t) s tht in [24][25], if there exists symmetric copositive mtrix Z P P. Similrly, we cn consider Z = I ρψ Λ (17) where I is n identity mtrix, ρ such tht 0 ρ 1, Λ = E[A(t)] is the vector of the verge rrivl rte, nd by definition 6 we hve Ψ Λ = ˆΛ T ˆΛ. It is esy to prove tht Z is positive (semi)definite. We lso ssume tht the stte vector S(t) = Q(t).

4 Now, we need to prove tht for some ɛ +, B + (B is lrge enough), ρ such tht for Q(t) > B, we hve E[Q(t + 1)ZQ T (t + 1) Q(t)ZQ T (t) Q(t)] < ɛ Q(t) For nottionl convenience, we define the timing index s subscript, e.g., Q t+1 is equivlent to Q(t + 1). Therefore E[Q t+1 ZQ T t+1 Q t ZQ T t Q t ] < ɛ Q t (18) By substituting Q t+1 nd Q T t+1 from the evolution eqution (12) into the left hnd side of (18) we hve E[Q t+1 ZQ T t+1 Q t ZQ T t Q t ] = (19) E[(Q t + A t D t )Z(Q T t + A T t D T t ) Q t ZQ T t Q t ] = E[2(A t D t )Q T t + (A t D t )(A t D t ) T Q t ] (20) For simplicity, we find the limit of (20) when Q t tends to infinity, s follows. E[2(A t D t )ZQ T t + (A t D t )Z(A t D t ) T Q t ] lim Q t Q t As cn be seen, the terms (A t D t ) nd (A t D t ) T re bounded since the number of rrivls nd deprtures in time intervl [t, t + 1] re bounded. Also, we know tht Z is positive (semi)definite mtrix. It mens tht, the limit of E[(A t D t )(A t D t ) T Q t ] Q t when Q t is 0. As result, the remining prt of the eqution will be E[2(A t D t )ZQ T t Q t ] lim Q t Q t By definition 4 nd knowing tht Q T t = Q t, we obtin lim Q t By using (17) in (23), we hve E[2(A t D t )Z( ˆQ T t Q t ) Q t ] Q t (21) = (22) E[2(A t D t )Z ˆQ T t Q t ] (23) 2E[(A t D t )(I ρψ Λ ) ˆQ T t Q t ] (24) By definition 6 nd property 1, (24) cn be written s follows 2(Λ ˆQ T t E[D t ] ˆQ T t ρλ ˆQ T t + ρe[d t ]Ψ Λ ˆQT t ) By replcing the result of lemm 1 for E[D t ], we obtin 2(Λ ˆQ T t (1 ρ) Q t ˆQT t + ρ Q t Ψ Λ ˆQT t ) (25) As cn be seen, (25) is function of ρ nd ˆQ t, i.e., f(ρ, ˆQ t ) = 2(Λ ˆQ T t (1 ρ) Q t ˆQT t + ρ Q t Ψ Λ ˆQT t ) (26) In order to proof the stbility, we need to show tht for the entire domin of ˆQ t, there exists ρ such tht (26) is lwys less thn finite negtive constnt, i.e., Since ˆQ(t) = ρ, ˆQ t Q(t) Q(t) = : f(ρ, ˆQ t ) < ɛ Q(t) is normlized vector, ij Q2 ij (t) for given ρ, the domin of f(ρ, ˆQ t ) is the surfce of the unit sphere such tht ˆQt +P. On the other hnd, for given vector ˆQ t, vrition of f(ρ, ˆQ t ) is liner versus the sclr ρ. Knowing tht 0 ρ 1, we nlyze two cses s below, for ll vlues of ˆQ t. Cse 1. when ρ = 1: f(1, ˆQ t ) = 2( Q t ˆQT t + Q t Ψ Λ ˆQT t ). If ˆQ t is in prllel with Λ, we will hve ˆQ t = ˆΛ, by definition 6 we cn write Ψ Λ = ˆQ T t ˆQ t. Thus f(1, ˆQ t ) = 2( Q t ˆQT t + Q t ˆQT t ˆQt ˆQT t ) = 2 Q t ( ˆQ T t + ˆQ T t ) = 0. If ˆQ t is not in prllel with Λ, we cn find Q t ˆQT t > Q t Ψ Λ ˆQT t. It leds to hve negtive vlue for f(1, ˆQ t ). As result, for ll vlues of ˆQ t we obtin f(ρ, ˆQ t ) ρ=1 = f(1, ˆQ t ) 0. Cse 2. when 0 ρ < 1, or in other words: 1 ρ 1 < 0. To investigte this cse, we cn write f s follows f(ρ, ˆQ t ) = f(ρ, ˆQ t ) ρ=1 +(ρ 1) ρ f(ρ, ˆQ t ), where the prtil derivtive of f cn be found s ρ f(ρ, ˆQ t ) = 2( Λ ˆQ T t + Q t Ψ Λ ˆQT t ). If ˆQ t is in prllel with Λ, i.e., ˆQ t = ˆΛ nd thus Ψ Λ = ˆQ T ˆQ t t. We cn write the prtil derivtive of f s ρ f(ρ, ˆQ t ) = 2( ˆQ t ˆQT t + Q t ˆQT t ˆQt ˆQT t ), which is strictly positive, i.e., ρ f(ρ, ˆQ t ) > 0. Since in cse one we obtined f(1, ˆQ t ) 0, fter comprison with f(ρ, ˆQ t ) = f(1, ˆQ t )+(ρ 1) ρ f(ρ, ˆQ t ) for 0 ρ < 1, we find tht f(ρ, ˆQ t ) < 0. If ˆQ t is not in prllel with Λ, we will hve Λ ˆQ T t < Q t Ψ Λ ˆQT t nd therefore, ρ f(ρ, ˆQ t ) will be strictly positive. Similrly, it cn be shown tht for 0 ρ < 1 we hve f(ρ, ˆQ t ) < 0. Hence, QLP is lwys stble for crossbr switch without speedup for ny dmissible trffic, no mtter whether the trffic distribution is uniform or non-uniform. It mens tht the queue length t input buffers does not grow infinity nd there exists finite upper bound B < t which bcklogged queues will settle. B. Discussions In this subsection, we discuss how to find out the queue length of ech flow in GPS. For trcking the queue length of flow F ij, similr to eqution (12) we cn hve n evolution eqution for queue Q ij during intervl [t 1, t 2 ] s follows Q ij (t 2 ) = Q ij (t 1 ) + A ij (t 2, t 1 ) D ij (t 2, t 1 ) (27) where Q ij (t 1 ) is the remining bcklogged queue of F ij t time t 1. We know tht D ij (t 2, t 1 ), the number of deprted bits from prticulr flow F ij during intervl [t 1, t 2 ] in GPS, cn be obtined s D ij (t 2, t 1 ) = t2 t 1 ij(t)dt (28) Assume tht the llocted bndwidth of F ij is fixed to constnt ij during intervl [t 1, t 2 ], i.e., ij (t) = ij, eqution (28) cn be clculted s t2 D ij (t 2, t 1 ) = ijdt = ij (t 2 t 1 ) (29) t 1 Thus Q ij (t 2 ) = Q ij (t 1 ) + A ij (t 2, t 1 ) (ij (t 2 t 1 )) (30) As cn be seen, the length of ech queue Q ij cn be found during ny intervl [t 1, t 2 ].

5 IV. SIMULATION ESULTS In this section, we crry out simultions to verify the theoreticl results in Section III, nd to evlute the performnce of QLP. We consider crossbr switch. Ech input nd output hs bndwidth of = 1 Gbps, nd the crossbr hs speedup of one. We set the pcket length to be distributed between 40 nd 1,500 bytes [26]. For the destintion of the pckets, we consider both uniform trffic nd non-uniform trffic pttern. For uniform trffic, the destintion of new incoming pcket is uniformly distributed mong ll the output ports, i.e., λ ij = η/n, where η is the effective lod nd N is the switch size. The η tkes one of the 10 possible vlues of [0.1,1] with step of 0.1 For non-uniform trffic, we use the sme model s tht in [27]. The trffic rrivl rte λ ij is defined by i, j nd n { unblnced probbility w s follows. ( ) w + 1 w λ ij (t) = N, if i = j 1 w N, if i j In this cse, the η is fixed to 1 nd w tkes one of the 11 possible vlues of [0,1] with step of 0.1. When w = 0, the trffic rrivl is uniformly distributed mong the outputs, i.e., λ ij (t) = /N. Otherwise, the incoming pckets t In i re more directed to Out j rther thn the other outputs, which is clled the hotspot destintion. A specil cse will hppen when w = 1, i.e., λ ii (t) =. To constrin the burstiness of flow F ij, we consider leky bucket model (η λ ij, σ ij ), where σ ij is the burst size of F ij [7]. We set σ ij of every flow to fixed vlue of 10,000 bytes, nd the burst my rrive t ny time during simultion run. To evlute the performnce of our scheme, We compre our simultion dt with Loclized Independent Pcket Scheduling (LIPS), in which n input port or output port mkes scheduling decisions solely bsed on the stte informtion of its locl crosspoint buffers [6]. We consider four different LIPS implementtion versions with different rbitrtion rules s follows: FP (Fixed Priority) ssigns fixed priority order to ll the virtul queues of the sme input to the sme output nd lwys picks the cndidte with the highest priority; D (ndom) mkes the rbitrtion on rndom bsis; (ound obin) lterntively chooses eligible cndidtes in round robin mnner to void strvtion; OPF (Oldest Pcket First) uses the pcket rrivl time s the rbitrtion criterion, i.e., the pcket rriving erlier hs higher priority. Now, we investigte the results on the throughput nd the verge dely. A. Throughput To verify theorem 1, we present the simultion dt to show tht our scheme chieves 100% throughput. Figure 2() displys the throughput under uniform trffic. It cn be seen tht, the throughput of different schemes grows consistently with the effective lod, nd finlly reches 100% when the effective lod becomes 1, only FP slightly decreses when the lod is the mximum. Figure 2(b) shows the throughput under non-uniform trffic. It clerly reflects the non-monotonic performnce vritions of other methods. As expected, QLP significntly yields higher throughput thn the other schemes when the speedup is one. The other schemes Throughput Throughput Switch, Uniform Trffic Lod () Switch, Non-uniform Trffic Unblnced Probbility Fig. 2. Throughput of QLP. () With different lods (b) With different unblnced probbilities. chieve the lowest throughput when the unblnced probbility is round 0.5 nd then, throughput is grdully improved until the unblnced probbility becomes 1. In fct t this point, ll the pckets of In i go to Out j nd no scheduling is necessry. The results confirm tht, QLP chieves 100% throughput nd outperforms the other four methods when the speedup is one. B. Averge Dely Next, we study the dely performnce of QLP. We mesure the totl time tht pcket stys in the switch. It is the intervl from the time tht the lst bit of pcket rrives t its input to the time tht the lst bit of the pcket is sent to the output. We plot the verge dely of different schemes in logrithmic scle nd the verge dely is mesured in seconds. Figure 3() displys the verge dely under uniform trffic. As cn be seen, the dely grows grdully when the effective lod increses, nd jumps when the effective lod becomes 1. Surprisingly, for the effective lods greter thn 0.8, the QLP outperforms the other four schemes. Figure 3(b) depicts the verge dely under non-uniform trffic. As expected, QLP shows n optimistic behvior in comprison with the other four methods nd the dely difference is more thn one decde. The verge dely of other schemes increses with the unblnced probbility nd reches to the mximum when the unblnced probbility is round 0.5. Then, the verge dely (b)

6 Averge Dely (Logrithmic Scle) Averge Dely (Logrithmic Scle) Switch, Uniform Trffic Lod () Switch, Non-uniform Trffic Unblnced Probbility (b) Fig. 3. Averge Dely of QLP. () With different lods (b) With different unblnced probbilities. of ll schemes drops when the unblnced probbility is equl to 1, becuse t this point ll trffic of n input is destined to the sme output nd no switching is necessry. It is observed tht, different unblnced probbilities do not significntly ffect the verge dely of QLP, which demonstrtes tht our scheme works well under non-uniform trffic. V. CONCLUSIONS In this pper, we hve presented the Queue Length Proportionl (QLP) bndwidth lloction scheme for GPS to schedule best effort trffic in crossbr switches without speedup. In QLP, the mount of service tht ech flow receives is proportionl to the length of its bcklogged queue. QLP essentilly fvors the queues with the gretest occupncy nd thus ssists the crossbr switch to be more work-conserving. 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