Capacity Provisioning for Schedulers with Tiny Buffers

Transcription

1 Capacity Provisioning for Schedlers with Tiny Bffers Yashar Ghiassi-Farrokhfal and Jörg Liebeherr Department of Electrical and Compter Engineering University of Toronto Abstract Capacity and bffer sizes are critical design parameters in schedlers which mltiplex many flows. Previos stdies show that in an asymptotic regime, when the nmber of traffic flows goes to infinity, the choice of schedling algorithm does not have a big impact on performance. We raise the qestion whether or not the choice of schedling algorithm impacts the capacity and bffer sizing for moderate vales of e.g., few hndred. For Markov-modlated On-Off sorces and for finite, we show that the choice of schedling is inflential on 1 bffer overflow probability, 2 capacity provisioning, and 3 the viability of network decomposition in a non-asymptotic regime. This conclsion is drawn based on nmerical examples and by a comparison of the scaling properties of different schedling algorithms. In particlar, we show that the per-flow capacity converges to the per-flow long-term average rate of the arrivals with convergence speeds ranging from O log O 1 depending on the schedling algorithm. This speed of convergences of the reqired capacities for different schedlers to meet a target bffer overflow probability is perceptible even for moderate vales of in or nmerical examples. I. ITRODUCTIO Capacity and bffer provisioning in a schedler which mltiplexes mltiple traffic flows with stringent service demands is known to be challenging. In this joint design problem, bffer size can be decreased at the cost of increasing the capacity. Recently, several argments have been made in favor of small bffer sizes at packet switches: First, when the nmber of flows is large, adding small bffers sally satisfies the bffer overflow probability constraints [16]. Secondly, small bffers enable fast memory technologies sch as SRAM or alloptical bffering [7]. Finally, small bffers may mitigate traffic brstiness. For instance, the polynomially decreasing overflow probability of self-similar traffic trns to exponentially decreasing bffer overflow in schedlers with small bffers [15], [16]. Capacity and bffer provisioning to meet a target loss probability, have been stdied extensively in a many sorces asymptotic when the nmber of i.i.d. flows tends to infinity. Using large deviation theory, it has been shown that the steady state of the total backlog B exceeding a threshold b is asymptotically described by P B > b e I b, where I is called the asymptotic rate fnction taking different forms depending on the inpt traffic and bffer sizes [1], [19]. For instance, for a given tilization and a large class of On-Off sorces, asymptotic rate fnction for small bffers takes the form I b K b 1 + K 2 for some positive constants K 1 to and K 2 [16]. Albeit a single flow FIFO qee is assmed for most of the asymptotic backlog analyses, it is shown that the asymptotic reslts can be extended to general work-conserving schedlers [8], [22]. This sggests that for sfficiently large, capacity and bffer provisioning can be carried ot regardless of the schedling algorithm sed in the switches. The role of schedling on the bffer/capacity provisioning for a target bffer overflow probability for finite vales of has not been flly investigated. For a general work-conserving schedler and a class of Markov-modlated sorces [6], it is shown in [5] that O1 bffers are sfficient to satisfy a target overflow probability. In a mlti-flow FIFO schedler and Markov-modlated On-Off sorces, an O 1 per-flow capacity is shown to be sfficient to garantee a probabilistic end-to-end delay bond [5]. etworks where bffers are small enogh to limit traffic distortion, bt sfficiently large to reslt in small loss probability jstify a decomposition analysis, where each node in the network can be analyzed withot regards to other nodes in the network. The viability of network decomposition is considered in asymptotic [8], [17], [2], [22] and nmerically in nonasymptotic regimes [5], [6]. In this paper we aim to investigate whether or not schedling information is a key factor for the network analysis for finite vales of. We se the non-asymptotic probabilistic backlog bond from [1] for the class of -schedlers to compte the reqired capacity to satisfy a target overflow probability in a schedler with a given small non-zero bffer size. We show that a per-flow bffer size can be as small as O 1, which jstifies talking abot a tiny bffer regime. We employ etwork Calcls in or derivations [2], [3], [11]. In this paper, we stdy the impact of resorce provisioning and performance analysis for finite : We investigate how the per-flow backlog and otpt scale with the total nmber of flows for each choice of schedling algorithm. We qantify the reqired capacity to satisfy a predefined overflow probability when the bffer size is arbitrarily small. We show that the schedling algorithm determines the speed of the convergence of per-flow capacity to the long-term per-flow average rate. We stdy the viability of network decomposition in a non-asymptotic regime for different schedlers. Using nmerical examples, we show that network decomposition might be valid even in a non-asymptotic regime for some

2 a Scenario I A two-node network. Fig. 1: System model and network decomposition. b Scenario II Removing the pstream node from Scenario I. schedlers inclding FIFO. The rest of the paper is organized as follows. In the next section, we introdce or system model. In Sec. III we stdy the impact of schedling on the backlog bonds from [1] for a -schedler as the nmber of flows increases. Then, for a fixed and arbitrarily small bffer size, we formlate an optimization problem to derive the reqired capacity which satisfies a target overflow probability and explore the scaling of the per-flow capacity as a fnction of. Sec. IV is devoted to network decomposition of a network of -schedlers. We present nmerical reslts in Sec. V and we conclde the paper in Sec. VI. II. SYSTEM MODEL Fig. 1 illstrates the models we will consider in this paper. Fig. 1a Scenario I demonstrates a two-node network scenario. The pstream node is fed by inpt flows i.i.d. throgh flows A and n c = i.i.d. cross flows A c and has a total capacity c, where c is referred to as the per-flow capacity. After being served at the pstream node, the departres of the throgh flows from the pstream node D enter the downstream node with total capacity C d. The downstream node mltiplexes the throgh flows with n d c i.i.d. cross flows A d c. In Fig. 1b Scenario II, the pstream node is removed, and the throgh flows A enter the downstream node directly and are mltiplexed by cross flows A d c. We do not assme independence between throgh and cross flows. If the arrivals at a certain time exceed the capacity of the link, the exceeding arrivals are stored in a bffer to be served later. The bffer content at any time t is called the backlog B at that time. We assme that the bffer at each node is partitioned to separate throgh and cross flows arrivals. We denote by B B and BI d BII d, respectively, the backlog of the throgh flows total flows at the pstream node and the total backlog at the downstream node in Scenario I Scenario II. In a finite bffer qee, network resorces are provisioned so that the losses occr rarely, i.e., by choosing large enogh bffer/capacity. One of the main target design constraints is overflow probability defined as the likelihood that at a given time instant the inpt traffic finds the bffer fll. As many other papers in the literatre, we se the probability that the virtal backlog in an infinite bffer qee exceeds a threshold eqal to the bffer size. Compting the latter probability is less challenging than the overflow probability and is an pper bond on the overflow probability and is shown to be very close to the bffer overflow especially if is not small [9]. The backlog in an infinite bffer qee at any time t is the difference between the cmlative arrival A and departre processes D, i.e., Bt = At Dt. We stdy the impact of schedling on capacity provisioning in a single node in Sec. III, where the pstream node in Scenario I is sed as the model for the single node scenario. We examine network decomposition in Sec. IV by investigating whether or not the backlog in the downstream node of Scenario I is affected by the pstream node or it has similar statistical properties to those in the single node depicted in Scenario II. We se a continos time flid flow model, where At represents the total arrival from a process A in time interval [, t and As, t = At As. For any arrival process A, we define the long-term average rate ā as At ā = lim t t The per-flow long-term average rate of arrivals in the pstream link is defined as ā = n c ā c +nā, where ā and ā c are, respectively, the long-term average rates of the throgh and cross flows at that node. The per-flow link capacity mst satisfy c ā as stability condition. The tilization is defined as the ratio between the long-term average rate of the arrivals to the link capacity, e.g., the tilization of the pstream node is compted as U = ā c = ā c. We se the stochastic network calcls for the analyses which employs envelopes to describe the probabilistic pper bonds at each time interval. A non-decreasing fnction G is a statistical envelope [4] for process A with bonding fnction ε for all s, t with s t, if P As, t > Gt s; σ εσ, 1 for any σ, and εσ 1. A statistical envelope can be inferred immediately from the definition of a large class of traffic sorces known as the Exponential Bonded Brstiness EBB traffic arrivals [21]. An arrival process A.

3 is an EBB traffic with parameters M, ρ, α, represented by A M, ρ, α, if it satisfies that for any σ and any s t P As, t > ρt s + σ Me ασ. 2 A statistical sample path envelope G is a stricter envelope than the statistical envelope and is a non-decreasing fnction that satisfies the following at any time t [1]: P sp s t As, t Gt s; σ > εσ, 3 for any σ, and εσ 1. If A M, ρ, α then, for any > the following is a statistical envelope satisfying Eq. 3 Gt; σ = ρ + t + σ; εσ = Me 1 + ρ e ασ. 4 We assme that is fixed and does not scale with. This is a common assmption in most of the papers on network decomposition [5], [6], [8], [22]. In this paper, we assme that each traffic flow is a Markov-modlated On-Off MMOO sorce which is a common model for voice traffic [18]. An MMOO flow can be modelled by a two-state Markov chain with states On and Off. In the On state, traffic is generated with rate P, and no traffic is generated in the Off state. The sojorn time between On to Off and Off to On transitions are exponentially distribted with rates λ and µ, respectively. The average cycle time to retrn to the same state is T = λ+µ λµ. If A is an MMOO process with parameters, λ, µ, and P then, for any α, t : with [ sp E e αas,t+s] e αrαt, 5 s rα = 1 2α P α λ µ + P α µ + λ 2 + 4µλ. 6 ote that rα satisfying Eq. 5 is a special case of effective bandwidth [12] for process A. rα is non-decreasing in α and satisfies α : r = ā rα r = P, 7 where ā = P µ λ+µ is the average rate of A. Traffic sorces which satisfy Eq. 5 and in particlar, MMOO sorces are also sed in the papers stdying network decomposition in a non-asymptotic regime [5], [6]. Sppose that A is the aggregate of n independent flows each satisfying Eq. 5 with parameter r. Then applying the Chernoff bond shows that A is an EBB arrival in terms of Eq. 2 with 1, nrα, α. For simplicity of notation, we will se ρα = nrα and we drop α freqently in the paper. We assme that the schedling algorithm in the pstream node is a -schedler [14]. For any ordered pair of inpt flows i, j, there exists a constant i,j that determines the precedence between the arrivals from flow i and j. More specifically, if an arrival flow i arrives to a -schedler at time t, the arrivals from flow j have higher precedence if and only if they arrive at or before t + i,j Fig. 2. By the A i A j t t + i,j -schedler Fig. 2: -schedler algorithm. above definition, first in first ot FIFO, static priority SP, and earliest deadline first EDF are examples of -schedlers with FIFO: i,j = for any pair of flows i, j i,j = + if j has higher priority than i SP: i,j = if i has higher priority than j EDF: i,j = d i d j for any i, j where d j for any j in EDF is the a priori delay bond for flow j. We note that if the schedling information is not available, performance bonds mst be obtained by considering the schedling algorithm which leads to the worst-case bonds among all work-conserving schedlers. This benchmark is referred to as blind mltiplexing BMx and is eqivalent to a two-class SP schedler which gives the lower priority to that specific flow. III. SCALIG PROPERTIES OF BACKLOG AD CAPACITY In this section we stdy the impact of the choice of schedler on the backlog and capacity dimensioning at the pstream node depicted in Fig. 1a. Since we do not consider the downstream node in this section, we sometimes drop the sperscript to simplify notation. We se the following theorem from [1] which comptes a per-flow backlog bond for EBB traffic sorces in a -schedler. Theorem 1 Backlog bond for -schedlers [1]: Sppose that EBB throgh flows with aggregate parameters M, ρ, α satisfying Eq. 2 are mltiplexed with EBB cross flows with aggregate parameters M c, ρ c, α c in a -schedler with parameter,c = and capacity C. Define a vector σ = σ, σ c with arbitrary positive elements. For any C ρc ρ 2, define the following parameters θ = min σ c C ρ c, [σ c + ρ c + ] + C C, 8 bσ = σ + ρ + θ, 9 εσ = M e 1 + ρ e ασ + M c e 1 + ρ c e αcσc. 1 Then, the backlog of throgh flows B at any time t satisfies P B t > bσ εσ. 11 The above theorem can be sed for or system model in which throgh and cross flows are EBB processes, respectively, with A 1, r α, α and A c 1, n c r c α c, α c for any non-negative α and α c. To captre the scaling of the backlog behavior of the throgh flows, we

4 se the deterministic peak-rate envelope P for the throgh flows as a special case of the statistical sample path envelopes. To be more precise, we replace G t = ρ + t + σ and ε σ = M e1 + ρ e ασ in Eq. 4, respectively, with G t = P and ε =. This eliminates the first term in Eqs. 9-1 which correspond to the throgh flows statistical envelope bonding fnction and replaces ρ + with P in Eq. 9. The reslting backlog bond exhibits an exponential decay rate in for any np āc c ā c as shown below. Corollary 1: Sppose that any throgh and cross flow sorce in the pstream node in Fig. 1a satisfies Eq. 5, each throgh flow has a peak rate P, and np āc c ā c. Then, there exist constants α b > and K sch that for any σ P B t > σ Ke α bσ, 12 where K = O if and K = Oe β for some constant β if <. Proof: We consider B, separately, for and <. In both cases, the parameters are chosen sch that bσ from Eq. 9 with the above replacements is pper bonded by a fixed σ, where ε in Eq. 1 is shown to decay by e αbσ. : With the condition on in the corollary statement, we can always choose α c small enogh so that c np rcαc + r c α c. For any fixed σ, setting the parameters in Eq. 9 as follows, garantees that b σ α b = α cc ρ c α c c P ; σ c = α bσ α c. 13 Replacing the above parameters in Eq. 11, yields P B t > σ e 1 + ρ c e αbσ. 14 c Since α b = O1 from Eq. 13 and ρ c = r c, Eq. 12 holds with K = e 1 + ρc c = O. < : In this case, θ from Eq. 8 is always evalated to the second term. Choose α c small enogh so that c np rcαc + r c α c. For any fixed σ, selecting parameters in Eq. 9 as follows garantees that b < σ α b = α cc P ; σ c = α bσ α c ρ c + c. 15 By replacing these choices of parameters in Eqs. 8-11, we have P B t > σ e 1 + ρ c e αcρc+c e α bσ 16 c which implies that Eq. 12 holds with K = Oe β and β = n α cā c = Θ1. Corollary 1 shows that the decay rate of the backlog of the throgh flows varies with the choice of schedling algorithms. It also indicates that for a predefined overflow probability, the bffer size can be as small as O 1 sggesting that a tiny bffer is enogh for large vales of. A. Capacity provisioning for -schedlers Sppose that the throgh flows in the pstream node in Fig. 1a have a target overflow probability ε. Using Theorem 1, we can qantify the reqired capacity for a given bffer size B. The per-flow capacity mst satisfy the stability condition, i.e., c ρ +ρ c +2. Hence, we can set c = 1 ρ +ρ c +2+X for some positive slack variable X. Eqating b in Eq. 9 to B and solving the reslt for c, gives s another constraint on c with free parameters α, α c,. Combining both conditions on c and enforcing the target overflow probability constraint, we can write c = 1 inf ρ + ρ c X 17 α,α c,,x s.t. X 18 ε e 1 + ρ e ασ + e 1 + ρ c e αcσc 19 X + ρ + ρ c + 2 ρ α + min σ c + B σ ρ c α c + ρ α + 2 B σ, [σ c + ρ c α c + ] +. The above optimization problem can be sed as follows to find how the reqired capacity to meet a target bffer overflow probability scales with. Corollary 2 Per-flow capacity scaling properties: The per-flow capacity from Eq. 17 for MMOO inpt flows satisfies c ā = J, 21 log where J = O if and J = O log if <. Corollary 2 shows that if is sfficiently large, for any arbitrarily small bffer size, a target overflow probability can be satisfied even when the tilization is large. Proof: Replace the left-hand side of Eq. 2 with c to get c 1 ρ α + 22 B σ min σ c + B σ ρ c α c +, [σ c + ρ c α c + ] +. ρ α + In addition, trn the ineqality in Eq. 19 into an eqality and split ε eqally between the terms of the right-hand side of that eqation to get σ = 1 2e1 + ρ log ; σ c = 1 2e1 + ρ c log. α α c ε ε 23 Using the Taylor expansion of the rate r for an MMOO process from Eq. 6 for α 1 and since r c = ā c, we have r c α c = ā c + Oα c, or eqivalently ρ c α c = ā c + Oα c. 24

5 Combining the above eqation with Eq. 17, we get c ā αc = O + r α ā X, 25 where the last term is O 1 noting that X can be chosen n to be a constant with respect to and lim =. If α c = Θ, we find from Eqs , that σ c, ρ c = log O log. Ths, there is a choice of α c which can satisfy Eq. 22, and for sch an α c, we have c ā = O log from Eq. 25. If < then, choose σ c = ā c and find α c from Eq. 23 which is α c = O log. This choice of σ c relaxes the constraint in Eq. 22 by setting the second term in the minimm to zero. Combining all of the above reslts and from Eq. 25, we have c ā = O log. A tighter bond on the capacity than that in Corollary 2 for < can be obtained asymptotically, when is sfficiently large. Set α c to a constant w.r.t. to have σ c = Olog from Eq. 23. The second term in the minimm of Eq. 22 is zero for large enogh and from Eq. 25 we have c ā = O 1. IV. O-ASYMPTOTIC ETWORK DECOMPOSITIO FOR -SCHEDULERS In the previos section, we stdied the role of schedling algorithms in capacity provisioning for a target bffer overflow probability. In this section, we investigate how the viability of network decomposition can be affected by the choice of schedling algorithms. etwork decomposition if valid simplifies network analysis by eliminating the other nodes and analyzing each node in isolation Fig. 1. The viability of network decomposition can be jstified by showing that the random processes governing a mlti-node network converge to those in a single node network when the nmber of flows is large. In the literatre, two types of convergence are considered: 1- Convergence of D to A : The pstream node can be disregarded in the backlog analysis of the downstream node if the statistical properties of D and A are similar. Wischik [2] provides conditions nder which the otpt traffic D satisfies the same moment generating fnction as the inpt A. In a FIFO link which is fed by two classes of arrivals, throgh and cross flows, each consisting of independent, identical leaky-bcket arrivals, the random brstiness of the otpt of the throgh flows converges to the inpt deterministic leakybcket brstiness of the aggregate throgh flows [22]. Finally, for MMOO traffic flows, the EBB characteristics of the otpt process in an isolated FIFO qee is shown to converge to those of the inpt process exponentially fast in [5], [6]. 2- Convergence of BI d to Bd II : The convergence of the backlog process BI d in the two node scenario in Fig. 1a to the backlog process BII d in the single node scenario in Fig. 1b can be sed to show that pstream nodes have only a negligible impact. This provides an even stronger argment for decomposition than the convergence of D to A. Almost sre convergence is proved for reglated traffic and in probability for non-reglated traffic in [8]. In a non-asymptotic regime, the convergence of BI d to BII d is shown nmerically even for moderate vales of in [5], [6]. Finally, it is shown that the loss ratio at the downstream in Scenario I converges to that of Scenario II when [17]. We will consider both convergence criteria in or analysis of -schedlers. A. Otpt characterization We se backlog characterization from Corollary 1 to formlate an envelope for the departres D from the downstream node in Scenario I in Fig. 1a. Theorem 2 Otpt EBB characterization: Consider the pstream node in Fig. 1a. With the assmptions in Corollary 1, the aggregate departres of throgh flows D is an EBB process with D M ot, ρ, α ot, where α ot = M ot = 1 α α b and [M 1 + α ] α b αb [ α +α b K 1 + α ] α α b +α b. α 26 Proof: The departres of throgh flows from the pstream node D in a time interval [s, t satisfies Using Eq. 27, we can write D s, t A s, t + B s. 27 P D s, t > ρ t s + σ P A s, t + B s > ρ t s + σ 28 inf P A σ +σ b s, t > ρ t s + σ =σ + P B s > σ b 29 inf σ +σ b =σ M e ασ + Ke α bσ b 3 = M ot e αot σ, 31 where Eq. 27 is sed in the second line. The event in Eq. 28 is a sbset of the nion of the events in Eq. 29. Combining this with the nion bond yields Eq. 29. The next line applies the EBB characterizations of A and the backlog probabilistic bond from Corollary 1, and the last line ses Lemma 3 in [4]. Replacing K from Corollary 1 to the above theorem shows that if, the EBB parameters of the aggregate throgh flows is nchanged as it passes throgh the node, i.e., lim D M, ρ, α. This reslt relaxes the assmption on the independence between the throgh and cross flows in [5], [6] and the statistical independence of the throgh flows in any time window with the backlog stats at the start of that time window in [6].

6 B. on-asymptotic downstream node bffer overflow As shown in [8], to garantee the convergence of BI d to BII d as increases, it is sfficient to show that a sample path backlog bond for B decays to zero. To constrct a sample path for B, we se the following lemma: Lemma 1: Sppose that the pstream throgh flows in Fig. 1a has a peak-rate P and there is a non-decreasing fnction ε b sch that at any time t and for any σ >, then, for any and T, P B t > σ sp t T P B t > σ ε b σ, 32 T ε b σ P. 33 Proof: We discretize time by letting be a time nit. The difference between the backlog of throgh flows at a given time instant t and at the most recent discrete time slot cannot be larger than the total arrivals of the throgh flows at the pstream node in an interval of size. Ths, by defining T Z =,, 2,..., T 1 τs, we have sp B t max B s + A s, s + t T s T Z max B s + P. 34 s T Z With this reslt, we can write P sp B t > σ P max B s > σ P t T s T Z B s > σ P = s T Z P T ε b σ P, where the second line ses Boole s ineqality and the last line ses the fact that T Z has T elements. If T in Eq. 33 is a bsy period bond, Lemma 1 provides a sample path backlog bond. A bsy period of a workconserving link refers to a time dration in which the backlog is non-zero. A bsy period bond is formlated in [13] as follows. Sppose that process A is the arrival traffic to a link with capacity C and D is the corresponding departre process. Fix time t and define ˆx t to be the start of the bsy period containing t. That is ˆx t = sps t As Ds. 35 If t ˆx t T for any t then, T is a deterministic bsy period bond which redces the Reich s backlog eqation to Bt = sp At s, t Cs. 36 s T If G is a statistical sample path envelope for A with bonding fnction ε then, a probabilistic bsy period bond is given by T σ = infs Gs; σ Cs 37 in the sense that for any t P t ˆx t > T σ εσ. 38 We se the above method to compte probabilistic bsy periods on the downstream node for both scenarios in Fig. 1. We first stdy Scenario II. Assme that throgh and cross flows at the downstream node are EBB with parameters 1, ρ II, α II and 1, ρ II c, αc II. Here, we have simplified notation by setting ρ II = r α II and ρ II c = n c r c αc II. Inserting the EBB sample path envelopes from Eq. 4 in Eq. 37, a probabilistic bsy period bond can be obtained. If αc II, α, II and II are chosen sch that C d > ρ II +ρ II c and II Cd ρ II c ρii 2 then, for any σ σ T II σ = C d ρ II c ρ II 39 2II is a probabilistic bsy period bond for the node in Scenario II in the sense of Eq. 38 with bonding fnction ε TII σ = e 2 + ρii ρii c + II II e αii d σ, 4 where αd II = 1 α II c α II To compte a bsy period bond at the downstream node in Scenario I, we apply the pstream departre characterization from Theorem 2. Mltiplexing the reslting departre process with EBB cross flows at the downstream node with parameters 1, n d crcα I c, I αc, I a bsy period bond for Scenario I at the pstream network can be obtained similar to that of T II σ. Then, for any I Cd ρ I ρi c T I σ = 2, σ C d ρ I ρi c 2 I 41 is a bsy period bond in the sense of Eq. 38 with bonding fnction ε TI σ = e 1 + M ot + M ot ρ I I + ρi c I e αi d σ, α ot 1. Since M ot = O1, we have where αd I = 1 α I c ε TI σ = Oe αi d σ and ε TII σ = Oe αii d σ. The above bonds are sed in the next section to compare BI d with BII d. C. Almost sre network decomposition We stdied the network decomposition in the sense of the convergence of D to A in Theorem 2. In this section, we stdy the network decomposition in the sense of the convergence of BI d to BII d. The following theorem shows that BI d converges to BII d almost srely in the nmber of flows for all -schedlers. The rate of convergence is faster for negative vales of compare Lσ for and < in Theorem 3. In the specific case of FIFO = and for traffic sorces with bonded peak-rate satisfying Eq. 5, the following theorem strengthens the reslts of [5], [8] from convergence in probability to almost sre convergence. Theorem 3 a.s. convergence of BI d to BII d : Consider the scenarios depicted in Fig. 1 and keep the assmptions in Corollary 1. Then, there exists a constant α > and a nonnegative fnction L sch that for any σ P B d I t B d IIt > σ Lσe ασ, 43

7 where Lσ = O 2 if and Lσ = O 2 e β for some constant β if <. Proof: Denote by A d c the cross flows at the downstream node. Sppose that T I and T II are, respectively, the probabilistic bsy period bonds compted from Eqs. 41 and 39 and define T max = maxt I, T II. Then, for any σ, : P B d I t B d IIt > σ P sp T max A t, t + A d ct, t C d sp D t, t + A d ct, t C d > σ T max + P t ˆx I t > T max + P t ˆx II t > T max 44 P sp A t, t D t, t > σ T max + P t ˆx I t > T max + P t ˆx II t > T max 45 P sp B t B t > σ 46 T max + P t ˆx I t > T I σ + P t ˆx II t > T II σ P sp B t > σ 47 T max + P t ˆx I t > T I σ + P t ˆx II t > T II σ T max ε b σ P + ε TI σ + ε TII σ, 48 where ˆx I t and ˆx II t are the starts of the bsy periods containing t at the downstream node, respectively, in Scenarios I and II. For the first ineqality, we se Eq. 36 and the fact that P X P X Y + P Y for any events X and Y, and P Y = 1 P Y. Eq. 45 ses the fact that sp X sp Y sp X Y for any X and Y. The last line is an application of Lemma 1 and the bsy period bonds. ε b is formlated in Eq. 14 for, and in Eq. 16 for <. Moreover, ε TI and ε TII are formlated, respectively, in Eqs. 42 and 4. Proper choices of the parameters in Eq. 48 completes the proof as follows. The last two terms are Oe αminσ, where α min = minα I, α II. Set = T max for some υ > to ensre 1+υ that P decays to zero as increases. Inserting these choices in Eq. 48 shows that for any υ > Eq. 43 holds with Lσ = O 2+υ if and Lσ = O 2+υ e β for some constant β if <. The theorem is then proved by noting that υ can be arbitrarily close to zero. The above reslts imply that in addition to statistical mltiplexing, the choice of schedling algorithm is an important factor affecting the viability of network decomposition. Althogh a network can be decomposed in a many sorces asymptotic regime for any work-conserving schedler as also shown in [8] and [22], the viability of decomposition for moderate vales of is a matter of the schedling algorithm. V. UMERICAL EXAMPLES In this section we examine or analytical reslts in nmerical examples for scenarios in Fig. 1. Each throgh and Otpt envelope Kbit FIFO Or bond BMx Or bond FIFO [7] BMx [7] FIFO [6] = 1, = 1 = 1, = Time ms Fig. 3: Comparison of the otpt envelopes in a FIFO or BMx link with = 1, 1 throgh flows, = 1, 1 Mbps, U = 9%, and ε = 1 6. Each flow is an MMOO process with P = 1.5 Mbps and T = 1 ms. Envelope Kbit Inpt Otpt 1 Otpt = 1 Otpt = Otpt = 1 = 1 = Time ms Fig. 4: Comparison of the inpt and otpt envelopes in a - schedler with = 1, 1 throgh flows, C = 1 Mbps, U = 9% = 6, and ε = 1 6. Each flow is an MMOO process with P = 1.5 Mbps and T = 1 ms. cross flow is an MMOO sorce with parameters λ = 1ms 1, µ =.11ms 1 which has an average rate of ā =.15 Mbps, and an average cycle time of T = 1 ms. The violation probabilities of the bonds in all examples are set to ε = 1 6. We do not evalate the free parameters to those from the previos sections. Instead we nmerically optimize or formlations over the free parameters α and. A. Otpt characterization We first present or work for a single node. We consider the pstream node in Fig. 1a. In Fig. 3, we compare the otpt bond from Eq. 29 with the existing otpt envelopes in [5], [6]. ote that or model does not reqire the independence assmption between throgh and cross flows in [5], [6]. In addition, the FIFO otpt envelope in [6] is obtained by frther assming that the throgh flows arrivals in any time window are independent of the backlog at the start of that time window. The traffic mix of throgh and cross traffic is set by fixing the

8 Relative error B II d BI d /BII d n = = 1 = 1 = 1 BMx Δ = 1 Δ = 1 Δ = Δ = 1 Per flow capacity Mbps 1.5 = = 1 Peak rate Average rate Δ 1 Δ = Δ = Total nmber of flows Fig. 5: Comparison of B I d with B II d by compting an pper bond on the relative difference between them as a fnction of with = 1, U = 9%, and ε = 1 6. Each flow is an MMOO process with P = 1.5 Mbps and T = 1 ms. = Total nmber of flows Fig. 6: The reqired per-flow capacity as a fnction of the total nmber of flows to meet a target overflow probability ε = 1 6, with = 1 throgh flow and cross flows. Each flow is an MMOO process with P = 1.5 Mbps and T = 1 ms. ratio n 1. In particlar, we se parameters = 1, = 1 and = 1, = 1. We find a statistical envelope Eq. 1 for the departres of throgh flows by taking the point-wise minimm of mltiple EBB characterizations sing different vales of α. We compare the otpt envelopes from Eq. 3 with those from [5] and [6]. The graph shows that even thogh or model has fewer independence assmptions, or otpt envelopes are still comparable and in fact, often tighter than those in [5], [6]. In particlar, only the corresponding envelope to the FIFO schedler from [6] is smaller than or FIFO envelope for all time intervals. In Fig. 4 we compare the inpt/otpt envelopes by fixing C = 1 Mbps and U = 9% = 6 and compting the inpt and otpt envelopes for = 1 and = 1 for different schedlers. The plot shows that for a set of schedlers inclding FIFO, the otpt envelope is comparable to the inpt envelope even for the moderate vale of = 6 sed in this example. However, for some other schedlers 1 the otpt envelope is mch larger than the inpt envelope, sggesting that for moderate vales of, the network decomposition might be valid for some schedlers and invalid for some others. = 1 B. Decomposing a network of -schedlers In this example, we examine or analytical reslts on network decomposition by comparing the downstream backlog statistics in the downstream node in Scenario I with the backlog statistics in Scenario II in Fig. 1. We set the perflow capacity sch that the tilization at both pstream and downstream nodes are fixed to 9% independent of. We compare Bd I with BII d by compting the probabilistic bond on Bd I from Theorem 1 and the probabilistic pper bond on Bd II BI d from Eq. 48. We plot the normalized difference BII d BI d in Fig. 5. Bd II Per flow capacity Mbps = 1 = 1 = 1 Peak rate Average rate Δ 1 Δ = Δ = Bffer size Kbit Fig. 7: The reqired per-flow capacity as a fnction of throgh flow bffer size when the bffer overflow is bonded by ε = 1 6, with = 1 throgh flow and cross flows. Each flow is an MMOO process with P = 1.5 Mbps and T = 1 ms. We se Eqs to compte T max = maxt I σ, T II σ. We set ε TI, ε TII ε = 2 +2 and T max ε b σ P to 2 ε 2 +2, where ε b is formlated in Eq. 11 and = Tmax. Then, the backlog bond from 2 Eq. 9 with σ = σc α = σ and = α 2 c = α is an pper bond on Bd II BI d. We se Eq. 11 to obtain the pper bond on B II d. Fig. 5 shows that if is not sfficiently large, schedling algorithm is a key factor on the accracy of estimating Bd I by Bd II. For instance, when = 1, for a set of schedlers 1 inclding FIFO, the normalized error of estimating Bd I by BII d is less than 1% when is only few hndred while, this is happening for BMx when is as large as 1 4. BII d BI d Bd II

9 C. Capacity provisioning In Fig. 6 we compte the reqired per-flow capacity for the pstream node in Fig. 1a to satisfy a target overflow probability ε sing the optimization problem in Eq. 17. We have also inclded the per-flow peak-rate and average rate as benchmark vales for the per-flow capacity. We fix the throgh flows bffer threshold to b = 1.5 Kbits which is eqal to the traffic generated in 1 ms from each flow in the On state. Then, for the choices of = 1 and = 1, we compte the reqired per-flow capacity c as a fnction of. Fig. 6 exhibits the considerable impact of schedling on the rate of convergence of the per-flow capacity to the average rate. For some schedlers inclding FIFO the convergence happens when is as small as few hndred, sggesting that a tilization close to 1 is achievable for those schedlers for moderate vales of and small bffer sizes. However, for some other schedlers 1 this happens only when is mch larger. By increasing the ratio of the throgh flows in the traffic mix, the reqired per-flow capacity increases sbstantially. In Fig. 7 we let = 1 and = 1 and compte the reqired per-flow capacity for throgh flow bffer sizes ranging from.15 Kbit to 15 Kbit. As shown in the figre, the reqired per-flow capacity decreases with a mch faster pace as bffer size increases for small bffer sizes compared to large bffers. This plot shows that the impact of adding a small bffer on the bffer overflow decrease is sbstantially affected by the schedling algorithm. VI. COCLUSIOS Recent stdies showed that the capacity and bffer provisioning in a schedler in a many sorces asymptotic regime is insensitive to the schedling algorithm. In this paper, we have investigated the impact of schedling on capacity provisioning when the bffer size is small. Using a non-asymptotic perflow backlog bond formlation which applies to a large set of schedlers -schedlers and assming MMOO traffic sorces, we showed that the schedling algorithm determines the decay rate of the bffer overflow probability sbstantially for moderate vales of. Then, by fixing the bffer threshold to an arbitrary small vale, we derived the reqired per-flow capacity which can satisfy a predefined overflow probability. We showed that the difference between the per-flow capacity log and per-flow long-term average rate ranges from O to O log and even O 1 when is large enogh depending on the schedling algorithm. Combining the observations from the nmerical examples and the comparison of the scaling properties for different schedlers indicates that for moderate vales of, having the schedling information is a key factor in capacity provisioning. We have also considered the viability of network decomposition in a non-asymptotic regime for different schedlers. merical reslts show that for a class of schedlers inclding FIFO, the convergence can happen even when is as small as few hndred. We analytically showed that the qee statistics of the downstream node in a two-node scenario converge to those of the simplified scenario when the pstream node is removed. The mode of convergence is almost sre for all schedlers, bt the rate of convergence depends on the schedling algorithm. REFERECES [1] R. R. Bahadr and R. R. Rao. On deviations of the sample path mean. The Annals of Mathematical Statistics, 314: , 196. [2] J. Y. Le Bodec and P. Thiran. etwork Calcls. Springer Verlag, Lectre otes in Compter Science, LCS 25, 21. [3] C. S. Chang. Performance garantees in commnication networks. Springer Verlag, 2. [4] F. Cic, A. Brchard, and J. Liebeherr. Scaling properties of statistical end-to-end bonds in the network calcls. 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