Energy-Delay Tradeoffs in a Load-Balanced Router

Transcription

1 Energy-Delay Tradeoffs in a Load-Balanced Router Matthew Andrews Bell Labs, Murray Hill, NJ andrews@research.bell-labs.co Lisa Zhang Bell Labs, Murray Hill, NJ ylz@research.bell-labs.co arxiv: v [cs.ni] 3 Jan 203 Abstract The Load-Balanced Router architecture has received a lot of attention because it does not require centralized scheduling at the internal switch fabrics. In this paper we reexaine the architecture, otivated by its potential to turn off ultiple coponents and thereby conserve energy in the presence of low traffic. We perfor a detailed analysis of the queue and delay perforance of a Load-Balanced Router under a siple rando routing algorith. We calculate probabilistic bounds for queue size and delay, and show that the probabilities drop exponentially with increasing queue size or delay. We also deonstrate a tradeoff in energy consuption against the queue and delay perforance. I. INTRODUCTION The concept of a Load-Balanced Router was studied at length at the beginning of last decade. See for exaple [6], [7], [22], [2], [9], [8], [5]. In this wor we analyze various perforance aspects of a Load-Balanced Router, otivated by the potential energy saving enabled by this architecture. Energy efficiency in networing has recently attracted a large aount of attention. One of the ain ais in uch of this wor is captured by the slogan energy-follows-load, also nown as energy-proportionality. In other words, we wish to ae sure that the energy consued by a networing device atches the aount of traffic that the device needs to carry. This is in contrast to ore traditional architectures for which the device operates at full rate at all ties even if it is lightly loaded. Indeed, by a conservative estiate in a study conducted by the Departent of Energy in 2008, at least 40% of the total consuption by networ eleents such as switches and routers can be saved if energy proportionality is achieved. This translates to a saving of 24 billion Wh per year attributed to data networing []. A recent study [2] further confirs that the power consuption of soe stateof-the-art coercial routers stays within a sall percentage of the pea power profile regardless of traffic fluctuation, for exaple the significant daily variation in traffic load [26]. Various approaches have been proposed in order to achieve energy-proportionality. Speed scaling, also nown as rate adaptation, and powering down are two popular ethods for effectively atching energy consuption to traffic load. The forer refers to setting the processing speed of a networ eleent according to traffic load. It is typically assued that the energy consuption is superlinear with respect to the operating rate. The latter refers to turning off the eleent at certain ties and so it either operates at the full rate or zero rate. Both ethods are the subject of active research, though ost of the wor focuses on optiizing an individual eleent in isolation [6], [4], [29], [23], [3], [5], [7], [8], [25], [3]. A central question to both ethods is to set the speed so as to iniize energy usage while aintaining a desirable perforance, e.g. latency or throughput. The Load-Balanced Router architecture has the potential to handle the traffic in such a way that portions of the device can be turned off in response to lightly loaded traffic. We provide what we believe is the first detailed analysis of queue size and delay in a Load-Balanced Router, and we provide a tradeoff between energy consuption and queue size/delay. In order to describe our results in ore detail we now give a brief description of the Load-Balanced Router architecture. A. Motivation for Traditional Load-Balanced Architecture One of the ost fundaental goals of any router architecture is to achieve stability, which is soeties referred to as 00% throughput. In other words the router ais to process all the arriving traffic so long as no input and no output are inherently overloaded. The ey difficulty with doing this is that the arriving traffic ay be highly non-unifor, i.e. if A i is the arrival rate for traffic going fro input i to output, we will typically have A i A i for i i. Early wor on switching considered a crossbar architecture in which atchings between the inputs and the outputs are set up at every tie step. It was shown in [24] that the Maxiu Weight Matching algorith with weights equal to the baclog for each input-output pair can ensure stability. Subsequent papers looed at siplifications of this scheduler that could still achieve stability. However, a ajor drawbac of all these approaches is that they require a centralized scheduler with inforation about the baclogs of data on each input-output pair. A solution is the Load-Balanced Router that could ae use of randoized routing ideas first proposed by Valiant [28]. In the Load-Balanced Router there is a iddle stage placed between the input nodes and the output nodes. Each arriving pacet is routed to a iddle-stage node chosen at rando. After passing through the iddle stage each pacet is then forwarded to its designated output. In order to realize this architecture we place a switching fabric between the input stage and iddle stage and between the iddle stage and the output stage. The beauty of this design is that the rando routing ensures that for each of these switching fabrics no coplicated scheduling is needed. All we need to do is repeat

2 a unifor schedule in which each connection is served at least once [6]. B. Energy Consideration for Revisiting Load-Balanced Architecture Our otivation for revisiting the Load-Balanced Router architecture is that it provides an attractive fraewor for studying energy proportionality [2]. For exaple, the nuber of active nodes in the iddle stage can be reduced in the presence of light traffic, and increased with heavier traffic. The switch fabric between the input and iddle stage and between the iddle stage and the output can be functionally viewed as full eshes, as indicated in Figure. One possibility is to ipleent the esh with round-robin crossbars. For exaple, Keslassy s thesis [22] assues that the input, output and the iddle stage are all of size n and each fabric is an n n crossbar that operates in a tie-slotted fashion. At each tie slot t the first fabric connects input node i to iddle node i+t od n and the second fabric connects iddle node j to output nodej+t od n. In this ipleentation, if the nuber of active nodes in the iddle stage is reduced then the crossbar can either slowdown or be turned off periodically. Adjusting the size of the iddle stage or the speed of the switching fabric requires considerable technological and engineering challenges. In this note we do not ai to address these issues. Our focus is on analyzing queue size and delay given the active portion of the iddle stage, which leads to a tradeoff between power consuption and the size of the active iddle stage. C. Model and Definition We forally define the Load-Balanced Router architecture as follows. The router has n inputs and n outputs. We noralize the line rate such that it equals on each input and output lin. We also have a iddle stage lying in between the inputs and outputs that consists of nodes. The traditional Load-Balanced Router has = n. However, here we treat as a separate paraeter. In between the input and the iddle stage we have an n esh. Effectively each lin in this esh operates at rate α/ for a speedup of α. Siilarly, in between the iddle and the output stage we have an n esh with lin rate β/ for a speedup factor of β. Each of the two eshes is input-buffered, i.e. each input node has a separate buffer for each iddle node and each iddle node has a separate buffer for each output node. See Figure. Fro now on, we use i to index the input, j the iddle stage and the output. We refer to the pacets that wish to go fro input i to output as i pacets. Siilarly, lin connects input i and node j in the iddle stage and lin connects node j in the iddle stage and output. Buffer resp. at node i resp. j buffers pacets that are waiting to traverse lin resp.. The ey to possible energy savings is that we assue that not all nodes need to be active at periods of low load. We suppose that at any tie we can choose nodes to Input alpha/ Middle stage beta/ Output Fig.. A Load-Balanced Router architecture, where the nuber of nodes in the iddle stage can be different fro the nuber of inputs and outputs. be active in the iddle stage and that such a configuration requires power w for soe function w. When an i pacet p arrives we choose a rando iddle node j aong the active nodes in the iddle stage and place p in buffer at node. The lin operates continuously at rate α/ and transits pacets in the buffer in a FIFO anner. Siilarly, when p arrives at node j in the iddle stage, it is placed in the buffer. The lin continuously operates at rate β/ and transits pacets in the buffer in a FIFO anner. As we can see the scheduling for both stages requires no centralized intelligence and is extreely siple. Lastly we describe our traffic odel. As is coon in wor on scheduling in routers, we assue that pacets are of unit size or else are partitioned into cells of unit size. For any s,t let A i s,t be the aount of i traffic arriving at the router in the tie interval [s,t and let A i s,t = A is,t and A s,t = i A is,t. We assue that the i traffic, the input i traffic and the output traffic are σ i,r i, σ, and σ, ε constrained respectively for soe burst paraeters σ i and σ, for soe rate paraeters r i and for soe load paraeterε. In other words we assue that, A i s,t σ i +r i t s A i s,t σ+t s A s,t σ+ εt s. We rear that the arrival rates r i will typically vary over tie. Indeed, the energy savings that we hope to gain coe precisely fro the fact that we can atch the nuber of active coponents to the traffic. However, we assue that this happens over a slow tiescale and so we perfor our scheduling analysis as if the rates r i are fixed. Note that round robin is another possibility for choosing a iddle-stage node. However, a rando choice is ore robust against adversarial types of traffic arrivals. We do not go into details here.

3 D. Results In Section II we ae the siple stateent that r active iddle-stage nodes suffice for handling the traffic load where r i r i for all and r r i for all i. This in turn iplies that the energy required to serve the traffic over the long-ter is w r. In Sections III to IV-E we present a probabilistic analysis that bounds the queue sizes at the input and iddle stages and bounds the delay experienced by pacets as they travel fro the router input to the router output. We first bound the probability for a queue size to exceed a certain aount q, and the delay to exceed a certain aount of d, assuing a fixed sized nuber of iddle stages. An iportant feature of our bounds is that they decrease exponentially with q and d. For a fixed traffic load, we then derive a trade off between the queue size/delay perforance and the nuber of active iddlestage nodes which in turn gives a energy-delay and energy-queue tradeoff. Leaving energy iniization aside, we believe that this is the first detailed analysis of the delay and queue perforance of a Load-Balanced Router, which ay be interesting in its own right. In Section V we present soe nuerical exaples to validate our analytical findings. We note that our approach is different fro the traditional powerdown and rate adaptation techniques since we will be directing traffic in such a way that enables soe coponents to be off. In other words for the iddle stage nodes we are not trying to atch service rate to a traffic process that is exogenous. We are trying to atch the active iddle stage nodes to a traffic process that is under our control due to our ability to route within the router. We also rear that our bounds could be used to govern how any iddle nodes are active in a Load-Balanced Router without necessarily coputing all the bounds on the fly. We could instead precopute the bounds and create a siple loo-up table that deterines how any iddle stage nodes should be active based on easureents of the load at the inputs and outputs. II. THROUGHPUT ANALYSIS Recall that r i represents the current arrival rate of traffic that wishes to be routed fro input i to output. Let r i = r i and let r = i r i. Let r be such that r i r and r r for all i,. Lea. If we use iddle stage eleents then the router is not overloaded if r. Hence the power required to serve all the traffic in the long-ter is at ost w r. Proof: Follows fro the fact that if we turn on iddle stage eleents then for each iddle eleent j, j, the traffic rate that will be routed on lin will be at ost r/ which by assuption is at ost // = /. Since the capacity on the lin isα/ for soeα >, this iplies that the lin is not overloaded. A siilar arguent applies to each lin between the iddle and output stages III. OVERVIEW OF DELAY ANALYSIS Before diving into details of the delay analysis, we first provide a high level overview of our techniques. A. Relationship with Stochastic Networ Calculus We use a variant of networ calculus [0], [], [4] soeties referred to as stochastic networ calculus. The original for of networ calculus derives delay bounds by iposing upper bounds on the aount of traffic arriving at a node via arrival curves and lower bounds on the aount of traffic served by a node via service curves. By relating these two curves we can both obtain a bound on the delay suffered by data at a networ eleent and also characterize the arrival curves for the data at any downstrea nodes. However, in the traditional networ calculus all such bounds are required to hold with probability. In our context this will lead to extreely wea bounds since there is a non-zero probability that the router will send a large nuber of pacets to a single iddle-stage node, thus condening the all to extreely poor service. An alternative therefore is to use a stochastic networ calculus in which we only wish for bounds on service to hold with high probability. A detailed forulation of a stochastic networ calculus was outlined in a series of papers by Jiang and others [20], [9]. At a high level Jiang s approach obtains curves that bound the probability that the delay exceeds a certain aount at upstrea eleents, and then use these curves to bound the worst-case arrivals at downstrea eleents. However, we follow a slightly different approach since we are able to obtain better bounds by not directly utilizing the service curves at the input nodes to bound the arrivals at the iddle-stage nodes. We instead base our calculations at the iddle stage on the external arrivals of various ensebles of flows that ight then be tie-shifted due to delays at the input. This allows us to avoid handling coplicated convolutions of arrival and service curves. We elaborate further on this distinction later. B. Our Approach We divide our analysis into a series of pieces. a Bound the queue build-up at the input: Recall that between input i and iddle-stage node j we effectively have a lin with speed α/. Also recall that the input has a buffer especially dedicated to the traffic that wishes to go fro input i to iddle-stage node j. Suppose that at soe tie t this buffer has level q. Let s be the last tie that this buffer was epty. Note that during the tie interval [s,t the lin served data of total size αt s/. Therefore the total data that arrived for lin i during the tie interval [s,t is at least q +αt s/. Therefore the probability that lin i has a baclog of size q at tie t is upper bounded by the probability that for soe

4 s t, the aount of i data arriving at lin i during the interval [s,t is at least q + αt s/. However, recall that the traffic arriving to input i arrives at rate at ost r and each pacet is sent to a iddle-stage node chosen uniforly at rando. Hence, for fixed s and t, we can use a Chernoff bound to bound the probability that the aount of i data arriving during the interval [s,t is at least q+αt s/. It is easy to show that this probability decreases exponentially in s and so we can use a union bound to bound the probability that this occurs for any s t. b Bound the delay experienced at the input: Translating our bound on queue size into a bound on delay is siple. Since the transission rate on the lin is α/, the event that the head of line pacet for the lin at tie t has experienced delay d, iplies the event that at tie t d, the queue size for the lin was at least dα/. An upper bound on the probability of the latter event can be derived using the ethod described earlier. c Bound the queue build-up at iddle stage: We now give an overview of how we perfor the analysis at the head of the queue at iddle eleent j. This fors the crux of our analysis since we are in a ore coplicated situation due to the fact that the pacet arrivals at the iddle stage are affected by how they are served at the input. As before note that if the queue for lin has size q at tie t, then for soe tie s t, the arrivals for lin at iddlestage node j during the tie interval [s,t ust be at least q +βt s/. Now suppose that the oldest of this data arrived at the input at tie s d, and suppose in addition that this data arrived at iddle-stage node j on lin. We can therefore state that the total aount of data arriving at the syste in the tie interval [s d,t that is destined for lin is at least q +βt s/ and the delay experienced by data arriving at lin i at tie s d is at least d. Via union bounds we can calculate the probability that this occurs for any i,s,d. In particular, for d sall we can say that the probability that the arrivals exceed q+βt s/ is sall whereas if d is large then the probability that the lin i delay is at least d is sall. We ae three points about this analysis at the iddle stage. This is where our analysis deviates slightly fro the traditional ethodology of networ calculus. We do not calculate a service curve for the input and use that service curve to bound the arrivals at the iddle stage. Instead we analyze the delay behavior at the input and then use that calculation to bound the iddle stage queue size using an expression that still involves arrivals at the input. Our initial analysis aes a slight approxiating assuption in that for the i defined above, we treat the arriving traffic for lin and the delay behavior on lin as independent. In reality of course they will be correlated due to traffic that passes fro input i to output through iddle-stage node j. However, this approxiation will typically not have a large effect for larger values of since the i traffic only fors a / fraction of the total j. However, in order to get a ore accurate bound, in Section IV-E we present a ore detailed expression that deals with the correlation explicitly. Our analysis is based on Chernoff bounds. However, the for of the Chernoff bound that we use changes depending on whether the bound on data arrivals that we are considering for a particular lin is close to or far fro the expected nuber of data arrivals for that lin. This unfortunately leads to a soewhat involved case analysis. d Bound the delay experienced at the iddle stage: The conversion of the queue-size bound to a delay bound at the iddle stage can be done in the exact sae way as for the input. Now that we have an expression for the delay at both stages we can convert it into an expression for the end-to-end delay fro the inputs to the outputs. IV. ANALYTICAL BOUNDS ON DELAY We now present the details of our delay analysis. We rely heavily on the following Chernoff bounds [27]. In particular we use 2 to derive analytical bounds and we use to derive tighter bounds for nuerical siulation. Theore 2 Chernoff Bound. Let X,...,X n be independent binary rando variables, and let µ be an upper bound on the expectation E[ i X i]. For all δ > 0, Pr[ i X i +δµ] e δ +δ +δ e inδ2,δ µ/3 µ In what follows we soeties refer to µ as the aggregated ean and δ as the excess factor. For conciseness we shall often use the aggregated ean even though µ is strictly speaing a bound on the ean. A. Input stage analysis We begin by coputing the queue distribution at the head of lin, where i [,n] is an input and j [,] is in the iddle stage. As in Section II we assue that r i r and r r for all i, and we assue that iddle stage eleents are currently active. Initially, in order to eep the forulas anageable, we shall derive our forulas for the case in which r = and =. We shall also assue that the arriving traffic is sooth and so we do not have the burst ters σ i and σ. Later on, we shall show how to adapt the forulas when these assuptions do not hold. Let A i t be the binary rando variable that indicates whether a pacet with input i output is apped to iddle stage j at tie t. We use A i t,t 2 to denote t2 t=t A i t, the total arrival in the duration t,t 2 ]. Let t be the rando variable for the queue size at the head of lin at tie t. To copute Q i,j t, let us assue s < t Q i,j 2

5 be the last tie that the queue at is epty. ForQ i,j t to be larger than a value q, the arrival A i s,t over all [,n] ust be at least α t s +q since lin is operated at a rate α. Forally, Pr[Q t q] s tpr [ n = ] A i s,t α t s +q We now bound the right-hand-side of 3. Since A i t are independent binary variables, we apply the Chernoff bound to every ter in the suation. The following is the aggregated ean µ and the excess factor δ for the above probability. { µ = t s δ = α + q, t s since α t s +q = µ+δ. If α 2, we have δ and we use 2 to derive, Pr[Q t q] e q e t s α q If α < 2, we have δ when t s q 2 α. Otherwise δ <. We apply 2 as follows. Pr[Q t q] q 2 α e q 3 e t s α q 3 + α 2 + e 32 α q e α 2 t s= q 2 α e t s α 2 Note that both expressions are exponentially decreasing in q. For the tie being we shall proceed according to the case α 2 since this leads to ore anageable forulas. Later on, we shall indicate where to adapt the forulas when we are in a scenario where α < 2. Let D 5 t be the axiu delay that soe pacet has t. For this delay to be ore than d at tie t, soe pacet ust be in the queue at tie t d. Since lin operates at rate α in a FIFO anner, the queue at tie t d ust be at least dα/. Therefore, experienced at tie t in the queue Q Pr[D t d] Pr[Q t d dα/] e By a union bound, Pr[ i, D e t d] n dα dα. 6 Recall that the above analysis was perfored in the absence of the burst ter σ. If we do have bursty traffic then we can adjust the forulas by aing the following changes to the aggregated ean and the excess factor and propagating these changes through the resulting forulas. { µ = t s+σ δ = t sα +q σ t s+σ. B. Middle stage analysis We now copute the queue distribution at the head of lin, where j [,] is in the iddle stage and [,n] is an output. Let Q 2 t be defined siilarly as Q t. To 3 bound Q 2 at tie t, let s t be the last tie that the queue Q 2 was epty. For Q2 t to be larger than q, there ust be at least t sβ + q distinct pacets in Q2 during the tie period [s,t]. Further, let s d be the earliest tie one of these pacets arrived at an input, say i. This pacet ust experience a delay of at least d in Q. Therefore, d d Pr[Q 2 t q] [ Pr s t ne dα [ Pr s t i i ] A i s d,t t sβ +q Pr[ i, D s d] ] A i s d,t t sβ +q Note that the bound 6 on the delay distribution at the input stage is independent of the tie index. We can therefore ove Pr[ i, D s d] to outside the suation indexed by tie in the second inequality above. We proceed to bound ] s t Pr[ i A is d,t t sβ +q based on the following expressions for the expectation µ and the excess factor δ. { µ = t s+d δ = t sβ +q d, t s+d since +δµ = t sβ +q. There are two cases to consider, β 2 or β > 2. For β 2, we further consider the following subcases, depending on q d, the value of δ when t s = 0. Note that as t s increases, the value of δ approaches β. Case a: q q 2d d, and t s 2 β. In this case δ. Since δµ = t sβ +q d, bound 2 iplies 8. Case b: q q 2d d, and 2 β t s. In this case β δ, which iplies δ 2 µ β 2 µ. Bound 2 in turn iplies 9. Case 2: β q d, and for all values of t s. In this case, β δ, which is the sae situation as b and iplies 0 in the sae way. Case 3a: q d β, dβ+ 2q t s and dβ+ 2q β β 0. In this case β 2 δ β for all values of t s. Since δ 2 µ β 2 µ 4, bound 2 iplies.

6 Case 3b: q dβ+ 2q d β, and 0 < β t s. In this case β 2 δ β for t s dβ+ 2q β. Since δ 2 µ β 2 µ 4, bound 2 iplies 2. Case 3c: q dβ+ 2q d β, and t s < β. We trivially upper bound the probability by as in 3. a b 2 3a 3b 3c a b 2 3a 3b 3c Pr[Q 2 t q] s tpr[ A t sβ i s d,t +q] 7 d i 2 q d=0 q 2 d=0 q β d= q 2 2q β+ d= q β d= 2q β+ d= 2q β+ q 2d 2 β Pr[ i, D t s= q 2d 2 β t s= dβ+ 2q β n n ne α 6 q ne α 3β q ne 2α 3β+ q s d] e t sβ +q d 3 +8 e t s+d 3 β 2 +9 e t s+d 3 β 2 +0 e β 2 t s+d e β 2 t s+d e β 2 e β +ne α β e α 2q β + 2β β 2 q e β 2 2 e 6 β 2 q e β 2 2 e α 2q β+ e α e α β 2 e 4α+β 2 2 e 2α+ββ 6 2 2q β + e α 2q β+ Note that every ter above decreases exponentially with the queue size q. The case in which β > 2 is sipler. We oit the analysis for space consideration. Recall that all of the above forulas are for the case α 2. If α < 2 then we need to replace all the factors 4, e dα with 5 e dα + e dαα 2 2 α. e α 2 If output has a burst ter σ then as in the input stage we can reflect this by adjusting the aggregated ean and the excess factor to, { µ = t s+d+σ δ = t sβ +q d σ. t s+d+σ We conclude this section by bounding the delay distribution at the iddle stage. D 2 be the axiu delay that soe pacet has experienced at tie t in the queue Q 2 t. For this delay to be ore than d at tie t, soe pacet ust be in the queue at tie t d. Since lin operates in a FIFO anner, the queue at tie t d ust be at least dβ/. Hence Pr[D 2 t d] Pr[Q2 t d dβ/]. We stress again that all of the above forulas are for the case α 2. If α < 2 then we need to replace all the factors 4 with factors 5. C. Eventual end-to-end delay Now that we have delay bounds for the two stages of the router we can obtain a bound on the end-to-end delay distribution. In the following we let g,β q be a shorthand for the dβ+ 2q β 3 upper bound that we have derived on Pr[Q 2 t q] and f,α q a shorthand for our upper bound onpr[q t q]. Suppose that an i pacet is still traversing the router at tie e q 3 t, but it arrived at the router before tie t d. It is not hard + e β to see that either it is still waiting to traverse the input stage e 3 β 2 q at tie t d 2, or it arrived at the iddle stage by tie t d 2. + Hence the probability that the end-to-end delay is at least d e β 2 e α β 2 is at ost, e 3 β 2 q + Pr[D t d 2 d 2 ]+Pr[D2 t d 2 ] f,α dα + 2 +g,β dβ 2. D. Characterizing the tradeoff with the nuber of iddle eleents + In the above delay analysis we ade a nuber of siplifying assuptions to eep the notation anageable. For exaple we held = and r =. We now deonstrate that we can use the above forulas to handle the case of arbitrary and r. This in turn allows to characterize the tradeoff between energy consuption and end-to-end delay.

7 The ain idea is to scale tie so that the arrival rate at the inputs is scaled to. We also adjust the lin rates on the two stages of the esh. In particular, if we wish to analyze a syste with a given,, r, α and β, we define a new syste characterized by ˆ, ˆ, ˆ r, ˆα and ˆβ in which we set, ˆ = ˆ = ˆ r = ˆα = α ˆβ = β r r Then it is not hard to see that if we scale tie by a factor r, the new syste has exactly the sae behavior as the old one. However, we are now woring in a syste with ˆ = ˆ and r =. Hence we can apply the analysis that we have already derived. Our ain result is thus, Theore 3. If we run the router with iddle stage eleents then it uses energy w and the probability that the end-to-end delay is at least d is bounded by, f, dα α r 2 r +g, β r E. Dealing with dependence dβ 2 r. In our analysis of the iddle stage we iplicitly ade the siplifying assuption that i A i s d,t is independent fro D s. However, this is not strictly true since the arrivals for the path i will affect both i A i s d,t as well as D s. Since the i flow represents only a / fraction of the traffic that contributes to D s, this will typically have a negligible effect on the eventual results. However, if we required a true upper bound on the probability distribution of Q 2 we ust use the following adaptation of the forula, which for each î, conditions the event D îj s d on whether or not A î s d,t is greater. More forally, than 5t sβ n Pr[Q 2 t q] d s t î Pr[ î i î Pr[D îj F. Queues at output Pr[A î 5t sβ s d,t ]+ n A t sβ i s d,t s d A î 5t sβ s d,t ] n We conclude this section by explaining how the analysis can be extended if we also wish to bound the delay on the output lin fro the router. Let Q 3 t be the queue at the head of the output lin and recall that this lin has speed. To bound Q 3 t at tie t, let s t be the last tie that queue Q 3 was epty. For Q 3 to be larger than q, there ust be at least t s+q distinct pacets in Q 3 during the tie period [s,t]. Further, let s d be the earliest tie one of these pacets arrived at an input. Suppose that the path of this pacet is i. Then the pacet ust either be waiting in the queue Q at tie s d 2 queue Q 2 at tie s d 2 have experienced delay of d 2 or it ust be waiting in the. In the forer case the pacet ust in Q at tie s d 2 the latter case it ust have experienced delay of d 2 at tie s. Therefore, d d Pr[Q 3 t q] Pr A i s d,t t s+q and in in Q2 s t Pr[,D s d 2 d ]+ Pr[ j,d2 2 s d 2 ] nf,α dα 2 g,β dβ 2 ε d q +σ. In addition, as before, we can convert this bound to a bound on delay for the third stage. Let D 3 be the axiu delay that soe pacet has experienced at tie t in the queue Q 3 t. For this delay to be ore than d at tie t, soe pacet ust be in the queue at tie t d. Since lin operates in a FIFO anner, the queue at tiet d ust be at least d ε. Hence Pr[D 3 t d] Pr[Q3 t d d ε] d nf,α d α 2 g,β d β 2 ε d d ε+σ. V. NUMERICAL RESULTS In this section we show nuerical exaples of the queue bounds. In particular for these calculations we use forulas that are siilar to those derived in Section IV but we use Chernoff bounds of the for rather than 2 since the forer are tighter. Figure 2 plots the logarith of the probability of a iddlestage queue exceeding q pacets against the queue size q. For this instance, the router is The traffic is fully loaded for each of the inputs and outputs. We vary the speedup α = β in the range of 2,3,4 and 5. As we can see fro the figure the logarith of the probability decreases linearly with the queue size, which eans the probability +q 5t sβ decreases exponentially with the queue. As expected, we can ] also see that with increasing speedup, the probability of the n queue size exceeding q drops. Figure 3 deonstrates the tradeoff between the queue size and the nuber of active iddle-stage nodes. For this instance, the router is again The lin rate of the interconnect is set to /20. The traffic is fully loaded for each of the inputs and outputs. If we eep all = 80 nodes in the iddle stage active, we can see fro the botto curve of Figure 3 that the queue size is the sallest. However, this option is also the ost energy consuing as all 80 iddlestage nodes are ept active. For the other extree, we can activate 40 iddle-stage nodes, which is the ost energy efficient. However, we can see fro the botto curve of Figure 3 that the queue size is the sallest. The curves in between correspond to the interediate cases in which the nuber of active iddle-stage nodes are 50, 60 and 70.

8 logarith of probability alpha=2 alpha=3 alpha=4 alpha= queue size Fig. 2. Log of probability against queue size, for n = 20, = 80 and fully loaded traffic. Fro top to botto, the curves correspond to increasing speedup fro α = β = 2, 3,4 and 5. logarith of probability =40 =50 =60 =70 = queue size Fig. 3. Log of probability against queue size, for sae aount of total traffic but varying nuber of active iddle-stage nodes. Fro top to botto, the curves correspond to increasing nuber of active iddle-stage nodes fro = 40,50,60,70,80. The nuber of inputs and outputs is n = 20 and interconnect lin rate is /20. VI. CONCLUSION In this paper we revisit the Load-Balanced Router architecture, otivated by its potential of delivering energy proportionality for routers. We offer a detailed analysis on the queue lengths and pacet delays under a siple rando routing algorith which is robust against all adissible traffic. 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