SRC Research. Report. An Efficient Matching Algorithm for a High-Throughput, Low-Latency Data Switch. Thomas L. Rodeheffer and James B.

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1 Novemer 5, 998 SRC Researh Report 6 An Effiient Mathing Algorithm for a High-Throughput, Low-Lateny Data Swith Thomas L Rodeheffer and James B Saxe Systems Researh Center 30 Lytton Avenue Palo Alto, CA

2 An Effiient Mathing Algorithm for a High-Throughput, Low-Lateny Data Swith Thomas L Rodeheffer and James B Saxe Novemer 5, 998

3 Copyright Compaq Computer Corporation 998 This work may not e opied or reprodued in whole or in part for any ommerial purpose Permission to opy in whole or in part without payment of fee is granted for nonprofit eduational and researh purposes provided that all suh whole or partial opies inlude the following: a notie that suh opying is y permission of the Systems Researh Center of Compaq Computer Corporation in Palo Alto, California; an aknowledgement of the authors and individual ontriutors to the work; and all appliale portions of the opyright notie Copying, reproduing, or repulishing for any other purpose shall require a liense with payment of fee to the Systems Researh Center All rights reserved

4 Astrat This paper fouses on two desired properties of ell-ased swithes for digital data networks: () data ells should not e detained inside the swith any longer than neessary (the work-onserving property) and () data ells that have een in the swith longer (older ells) should have priority over younger ells (the order-onserving property) A well-known, ut expensive design of a work- and order-onserving swith is the outputqueued swith A different swith design is the speedup rossar swith, in whih input uffers are onneted to output uffers through a rossar that runs at a multiple (alled the speedup) of the external ell rate A mathing algorithm determines whih ells are forwarded through the rossar at any given time Previous work has proposed a mathing algorithm alled the lowest output oupany first algorithm (LOOFA) It is known that a LOOFA swith with speedup at least is work-onserving We propose a refinement of LOOFA alled the lowest output oupany and timestamp first algorithm (LOOTFA) The main result of this paper is that a LOOTFA rossar swith is work- and order-onserving provided that the speedup is at least 3 We prove this result and onsider some generalizations

5 Contents Introdution Formal model of a rossar speedup swith5 Slot struture5 Basi notational onventions 6 3 State variales6 4 Cell input or output suset notation6 5 Conflit notation 7 6 Cell ordering notation 7 7 The initial state7 8 An inhale phase8 9 A phase 8 0 An exhale phase8 3 The LOOTFA swith 8 3 Output oupany, oo 9 3 Output oupany ordering, < oo() 9 33 Timestamp ordering, < t 9 34 Transfer time ordering, < x 9 35 The LOOTFA mathing ondition and w()9 36 The LOOTFA exhale onditions 4 The LOOTFA theorem 4 Earliest failing exhale phase, e 4 The failing ell, f 43 Relevant ells, R 44 Earliest inhale of a relevant ell, h3 45 Least important relevant ell, lir 3 46 Output uffer trailing ells, OBT 4 47 Potential, p 4 48 Lower ound on potential at time h+5 49 Effet of an inhale phase5 40 Effet of an R- phase 6 4 Effet of a nonr- phase 6 4 Effet of an exhale phase 7 43 Lower ound on potential at time e 8 44 The potential at time e 8 5 Generalizations 9

6 5 Generalized time of evaluation, w()9 5 Generalized phase arrangement 0 53 Generalized timestamp ordering, < t 0 6 Computing a LOOTFA math 6 The gloal minimum greedy algorithm 6 The per-input minimum greedy algorithm3 Aknowledgments4 Referenes 4

7 Introdution A ell-ased swith proesses fixed-sized hunks of data alled ells, whih arrive at swith inputs, pass through the swith proper, and depart from swith outputs Eah ell ontains an identifiation of the single output to whih it is destined For onveniene, we assume that the swith has the same numer, N, of inputs and outputs and we assume that eah input and output has the same apaity in ells per seond This apaity is alled the ell rate, and its reiproal, the ell time We assume that all ativities of the swith are synhronized to slots, eah of whih lasts one ell time Figure illustrates a ell-ased swith ells input input swith output output input N output N slot Figure : An N N ell-ased swith Although any realisti implementation would make extensive use of pipelining, for onveniene we model the ativity in the swith during eah slot as a sequene of phases: an inhale phase, during whih at most one ell from eah input is aepted into the swith; a numer of phases, during whih ells move around inside the swith; and an exhale phase, during whih the swith emits at most one ell onto eah output See Figure Aepting a ell during the inhale phase an e onsidered as the ookkeeping neessary to aount for a ell that arrived during the previous slot, and emitting a ell during the exhale phase an e onsidered as the ookkeeping neessary to aount for a ell that will depart during the following slot These ookkeeping ativities are overed y the pipeline delay and take no real time in an implementation The swith must ontain uffer memory to hold temporary exesses of ells that result from short-term flutuations in the arrival rate of ells destined to a given output For example, multiple ells destined for the same output ould e inhaled into the swith during the same slot, and the swith would have to hold these ells while the output exhaled them one y one Mehanisms to prevent uffer overflow suh as flow-ontrol ak-pressure or rate reservation are important ut eyond the sope of this paper We also ignore the rate- or phase-mathing uffer at eah input that is typially used to ring arriving ells into synhrony with the slot time of the swith In this paper we fous on two desired ehaviors of a ell-ased swith: () ells should not needlessly sit in uffers and () ells that have een in the swith longer (older ells) should have priority over younger ells

8 slot slot slot pipelined ativities in a swith implementation ell arriving at input ell departing at output real time phases of ativity in our model of a swith inhale exhale model time Figure : Model of the ativities in a swith during a slot The lateny of a ell is the numer of slot oundaries etween its inhale and its exhale The first desired ehavior an e stated formally as: the total lateny over all ells is as small as possile This is equivalent to the ondition that eah output always exhales some ell whenever there are any ells in the swith destined for that output A swith that ehaves in this manner is alled work-onserving Whenever the swith ontains multiple ells destined to the same output, the total lateny is unaffeted y the order in whih the ells are exhaled Given the hoie, it seems good to give older ells priority over younger ells Stated formally, we desire that eah time an output exhales a ell, there are no older ells in the swith destined for that output A swith that ehaves in this manner is alled order-onserving In Setion 53 we revisit the notion of order-onserving in a more general ontext A ell-ased swith that is oth work- and order-onserving should rightly e alled ideal, ut a more ommon term is the eponymous output-queued To avoid onfusion we refer to the ehavior as ideal and the well-known implementation, desried in the next paragraph, as output-queued The well-known implementation of an ideal ell-ased swith is the output-queued swith, in whih the swith takes ells diretly into uffers loal to eah output, as shown in Figure 3 Assuming eah non-empty output unit always exhales one of its oldest ells, this design is learly work- and order-onserving, hene ideal Unfortunately it also is expensive Beause all inputs ould simultaneously inhale ells destined to the same output, the onnetion into eah output unit must have a apaity of N times the ell rate: either N times wider (as in Figure 3), N times faster, or some omination None of these alternatives sales well as N inreases

9 output units input output input output input N output N Figure 3: An N N output-queued swith Another ell-ased swith design is the rossar speedup swith, whih is illustrated in Figure 4 This swith ontains input units, output units, and a rossar interonnet Cells are uffered at the input units and at the output units The ations during eah slot onsist of an inhale phase, S (the speedup) phases, and an exhale phase During the inhale phase, eah input unit inhales at most one ell and uffers it During eah phase, the rossar moves ells from input units to output units, sujet to the restritions that no more than one ell an e removed from any input unit and no more than one ell an e delivered to any output unit During the exhale phase, eah output unit removes at most one ell from its uffer and exhales it input units output units input rossar output input output input 3 output N Figure 4: A rossar speedup swith Sine eah onnetion etween the rossar and an input or output unit is required to at most one ell per phase, of whih there are S per slot, eah suh onnetion requires a andwidth of only S times the ell rate Eah phase proeeds in two parts: first a mathing algorithm selets whih ells in the input units to (the math), and then the seleted ells are red We say that the ells in the input units ompete for inlusion in the math No pair of in- 3

10 luded ells an onflit, either y sharing the same input (whih would e an input onflit) or sharing the same output (whih would e an output onflit) The mathing algorithm typially produes a maximal math, in whih no additional ell an e inluded eause eah non-inluded ell has a onflit with some inluded ell Sine exatly the inluded ells are red, we also all them the red ells In the types of rossar speedup swith we investigate, some ordering of ells is used to determine whih ells are more important and thus win the ompetition Different mathing algorithms use different orderings Typially, eah input unit uffers its ells in a separate queue for eah output, as shown in Figure 5 Although illustrated as separate queues, a linked-list implementation is typial, and the usual name for these strutures is virtual output queues This design requires that the oldest ell in eah queue always e a most important ell in that queue Hene the oldest ell an always e inluded in a math in preferene to any younger ell in its queue, and in fat the younger ells need not even e onsidered input units output units input rossar output input output input N output N Figure 5: A rossar speedup swith with (virtual) output queues If the mathing algorithm an e designed so that for eah output, some ell destined to that output (if any exist) is always present in the output unit at the eginning of the exhale phase, then the rossar speedup swith will e work-onserving Krishna et al [] have developed a mathing algorithm alled the lowest output oupany first algorithm (LOOFA) that ahieves this property provided that the speedup S is at least The oupany of an output is the numer of ells urrently uffered in the output unit In LOOFA, a ell destined to an output with lower oupany is more important than a ell destined to an output with higher oupany Intuitively, an output unit ontaining fewer ells will need another ell sooner than an output unit ontaining more ells and hene ells destined to the lower oupany output should e more important If the mathing algorithm an e designed so that for eah output, an oldest ell destined to that output (if any exist) is always present in the output unit at the eginning of the exhale phase, then the rossar speedup swith will e order-onserving in addition to eing work-onserving that is, it will e ideal Prahakar and MKeown [] have developed a mathing algorithm alled the most urgent ell first algorithm (MUCFA) that 4

11 ahieves this property provided that the speedup S is at least 4 In their design, the swith shedules an exhale slot to eah ell as it is inhaled, using the next availale (not-yetsheduled) exhale slot for the ell s destined output Lower-numered inputs get priority when the swith simultaneously inhales multiple ells destined to the same output A ell s urgeny is the numer of slot oundaries remaining until its sheduled exhale In MUCFA, a ell with lower urgeny is more important than a ell with higher urgeny Clearly suh a swith is ideal if it exhales eah ell when its urgeny is zero Both LOOFA and MUCFA use mathing algorithms that guarantee that eah non-inluded ell has a onflit with some inluded ell that is at least as important, aording to their respetive definitions of importane, as the non-inluded ell As a onsequene, their mathes are maximal Sine LOOFA takes no aount of ells ages, there is learly no guarantee that it is order-onserving However, the slight modifiation of resolving ties in output oupany y favoring older ells produes an ideal swith provided that the speedup S is at least 3 We all this refinement the lowest output oupany and timestamp first algorithm (LOOTFA) The fat that a LOOTFA swith with S 3 is ideal is our main result Formal model of a rossar speedup swith In this setion we present our notation and a formal model of a rossar speedup swith The formal model defines the state of the swith and the allowale hanges in this state that an happen during eah phase In a LOOFTA swith, the mather and the output seletors further onstrain the ehavior In any speifi exeution history, the sequene of input data also onstrains the ehavior The formal model has two parameters, N and S: N the numer of inputs of the swith; also the numer of outputs S the rossar speedup fator Slot struture Time is divided into slots Eah slot onsists of an inhale phase, S phases, and an exhale phase We lael phase oundaries with onseutive integers starting with 0 The phase eginning at oundary is alled phase See Figure 6 In Setion 5 we onsider a more general phase arrangement 5

12 inhale exhale inhale exhale inhale exhale inhale exhale inhale time phase 0 phase phase phase 3 phase 4 phase 5 phase 6 phase 7 phase 8 phase 9 phase 0 phase phase phase 3 phase 4 phase 5 phase 6 phase 7 phase 8 phase 9 phase 0 phase oundary slot slot slot slot Figure 6: Example slot struture and phase oundary laels (S=3) Basi notational onventions We use the following notational onventions: i o h x e i() o() h() an input, i N an output, o N (the eginning of) an inhale phase (the eginning of) a phase (the eginning of) an exhale phase (the eginning of) any phase a ell ell s input ell s destined output ell s inhalation phase: is inhaled during phase h() 3 State variales The model has the following state variales: IB OB the set of ells in any input unit at time the set of ells in any output unit at time 4 Cell input or output suset notation Given an aritrary set C of ells, we use the following susript notation for identifying susets onsisting of those ells with a given input or output (regardless of whether the ells are present in the swith at any given time): C i =i { C : i() = i} those ells in C with input i C i i { C : i() i} those ells in C not with input i C o =o { C : o() = o} those ells in C destined to output o C o o { C : o() o} those ells in C not destined to output o 6

13 Here are three examples of this notation: IB =, { IB i() = i}, i = i, o o { IB i() = i o() = o}, o o { OB o() = o} i i IB = OB = = : ells in input unit i at time = : ells in IB, i= i destined to output o = : ells in output unit o at time 5 Conflit notation Cells that share an input or an output are in onflit and annot oth e red in the same phase We use the following notation for the relation of two ells in onflit: i ( ) = i( ) o( ) = o( ) input or output onflit ~ 6 Cell ordering notation We distinguish different ell orderings using susripts: < y preedes (is more important than) aording to ordering y < z preedes aording to z = z ties aording to z z preedes or ties aording to z In all of the orderings we use in this paper, two ells tie if and only if neither preedes the other, and furthermore, as suggested y our notation, tying is an equivalene relation We use the notation < y, z to designate the ordering derived from < y with ties roken y < z : y, z < y = y < z = y, z = y = z < ( ) preedes aording to y then z ties aording to y then z Next we give the initial state of the swith and the allowale hanges in the state during inhale,, and exhale phases 7 The initial state Initially there are no ells in the swith IB0 = 0 OB0 = 0 the input uffer initially is empty the output uffer initially is empty 7

14 8 An inhale phase For any inhale phase, there exists a set of inhaled ells H suh that: OB + = OB the output uffer does not hange IB+ = IB H inhaled ells arrive in the input uffer H =i i :, i eah input inhales at most one ell () H : h = inhalation time is orret 9 A phase For any phase, there exists a set of red ells X suh that: X IB a suset of the input uffer IB+ = IB X red ells depart from the input uffer OB+ = OB X red ells arrive in the output uffer X =i i :, i at most one red ell for eah input X =o o :, o at most one red ell for eah output The set of red ells X is the set of ells inluded in the mathing for phase In a LOOTFA swith, X also satisfies an additional ondition given in Setion 35 0 An exhale phase For any exhale phase, there exists a set of exhaled ells E suh that: IB + = IB the input uffer does not hange E exhale a suset of the output uffer OB + = OB E exhaled ells depart from the output uffer OB E =o o :, o eah output exhales at most one ell In a LOOTFA swith, E also satisfies additional onditions given in Setion 36 3 The LOOTFA swith In this setion we present the additional onditions that a rossar speedup swith must satisfy in order to e a LOOTFA swith and we develop onepts speifi to the LOOTFA swith 8

15 3 Output oupany, oo We define the output oupany oo () of a ell at time as the numer of ells in s destined output unit at time Formally, () o o() oo OB, = 3 Output oupany ordering, < oo() Given any two ells,, we say that preedes aording to the output oupany ordering at time, written < oo(), iff at time, the output oupany of is less than the output oupany of Formally, < () oo ( ) oo ( ) oo < 33 Timestamp ordering, < t Given any two ells,, we say that preedes aording to the timestamp ordering, written < t, if and only if is inhaled efore Formally, ( ) h( ) < t < h The timestamp ordering indiates whih ells are older than others In Setion 53 we onsider alternative definitions of the timestamp ordering 34 Transfer time ordering, < x Given any two ells,, we say that preedes aording to the asi time ordering, written < x, if and only if is red efore We onsider that a ell that is atually red is red efore a ell that is never red Formally, ( x : X x < x ) ( x X ) < x x : X x x : x We resolve ties in < x aritrarily to produe the total ordering < x, alled the time ordering Note that the time ordering is a property of an exeution history of the swith, and is not in general availale from the swith state at any moment in time The time is not used in the implementation of the swith, ut only in our analysis of its ehavior We use < x in the definition of the least important relevant ell in Setion 45 The oraular nature of < x enales us to pik the ell that an exeution history in fat treats as less important in the event of a tie in the mathing ondition 35 The LOOTFA mathing ondition and w() Like LOOFA and MUCFA, in eah phase LOOTFA requires that eah non-inluded ell have a onflit with some inluded ell that is at least as important Roughly 9

16 speaking, LOOTFA uses a definition of importane that favors ells with lower output oupanies, reaking ties in favor of ells with earlier timestamps A sutlety arises at this point Whereas a ell s timestamp never hanges, a ell s output oupany an hange over time In partiular, after any phase, the relative output oupanies of the ells surviving in the input uffer may e different from what they were at the eginning of the phase Sine rapidly onstruting a math is ruial to the performane of the swith, an implementation would most likely pipeline this proess as muh as possile Reevaluating the relative importane of surviving ells on every phase seems like it would e othersome It turns out to e suffiient for the phase to onstrut its mathing ased on output oupanies as they were at the end of the most reent inhale phase This has the onsequene that the relative importane of surviving ells does not hange during the phases in the same slot, whih seems like a property that ould e exploited in a pipelined implementation We define the funtion w() of time as the time at the end of the most reent inhale phase efore Formally, 0 w() = w ( ) if = 0 if phase is an inhale phase otherwise In Setion 5 we onsider alternative definitions of w (Note that sine the inhale phase does not affet output oupanies, we ould equivalently use the initial output oupanies as of the eginning of the urrent slot Krishna et al [] disovered that all of the phases in the same slot ould use initial output oupanies when they proved that an S LOOFA swith was work-onserving) Now we an define the LOOTFA mathing ondition For every phase, a LOOTFA swith satisfies the following ondition in addition to the phase onditions in Setion 9: IB X : X : ~ ( w() ) t oo, That is, for eah ell in the input uffer that is not inluded in the math, there exists some onfliting, inluded ell that is at least as important as, where a ell is more important than another if it has a lower output oupany at time w () or, in the event of a tie, if it has an earlier timestamp Sine is red while remains in the input uffer, we neessarily have < oo ( w() ), t, x We say that is red in preferene to 0

17 36 The LOOTFA exhale onditions For every exhale phase, a LOOTFA swith satisfies the following onditions in addition to the exhale phase onditions in Setion 0: o : OB o o > 0 E, o=, = o > 0 () t OB work-onserving E : OB,o = o : OB order-onserving That is, eah non-empty output o always exhales a ell, and the ell it exhales preedes or ties aording to the timestamp ordering all ells in the output uffer destined to o 4 The LOOTFA theorem We now ome to our main result Theorem (LOOTFA): A LOOTFA swith with speedup S 3 is ideal The rest of Setion 4 is devoted to a proof of this theorem We assume an exeution history that is a ounterexample, define a numer of attriutes (e, f, R, h, lir, OBT, p, H, X, and E) of this exeution history, and finally arrive at a ontradition 4 Earliest failing exhale phase, e Reall from Setion that a swith is ideal if and only if it is oth work-onserving and order-onserving To e work-onserving, the swith must ensure that whenever there are any ells in the swith destined to output o at the eginning of an exhale phase, output o exhales some ell during phase To e order-onserving, the swith must ensure that whenever an output o exhales some ell, there are no ells in the swith destined to output o that preede aording to the timestamp ordering Formally, a swith is ideal if, in every exeution history, the following onditions oth hold for every exhale phase : ( IB OB ) > 0 E 0 o= o o o E ( IB OB ) () = o :, > work-onserving : : order-onserving o = o t We say that an exhale phase fails if it violates one or oth of the aove onditions (For example, if at the eginning of an exhale phase, a rossar speedup swith has a ell destined to o in its input uffer ut no ells destined to o in its output uffer, then exhale phase is sure to fail) In our assumed ounterexample exeution history, there must e some exhale phase that fails We define e to e the earliest suh failing exhale phase 4 The failing ell, f In order for exhale phase e to fail, there must e some ell ( ) IB e OB e in the swith suh that either () no ell is exhaled on output o () (whih would violate work-onserving) or () a ell is exhaled on output o () that preedes aording to the time-

18 stamp ordering (whih would violate order-onserving) We pik one suh ell and all it f, the failing ell We laim that at time e, ell f must e in the input uffer and it must preede all ells in output o ( f) aording to the timestamp ordering This laim follows from the LOOTFA exhale onditions of Setion 36 Formally, we first prove that OBe,o = o( f) : f < t Assuming the ontrary, there exists a ell OB e, o= o( f) suh that t f Then from the OB work-onserving ondition, output o ( f) must exhale some ell, and from the OB order-onserving ondition, we have t, whene t f This ontradits the definition of f, so our statement is proved Sine f does not preede itself aording to the timestamp ordering, f annot e in OB e, o= o( f), and therefore f OBe By definition f ( IB e OB e ), so we have f IBe This ompletes the proof of our laim In summary, we have f IB e, and OBe,o = o( f) : f < t The rest of the proof proeeds as follows We define a set of relevant ells, whih are those ells sharing the same input as f that ontriute to allowing f to survive in the input uffer until the earliest failing phase e We define the least important relevant ell at time and prove a property of its output oupany We examine the output uffer trailing ells, whih are those ells in the output unit o ( f) that are preeded y f aording to the timestamp ordering Then we define a potential at time as a linear omination of various salient quantities in the swith state at time We estalish a lower ound on the potential at the inhalation of the first relevant ell, push this ound forward phase y phase, and thus otain a lower ound at time e Finally we diretly ompute the potential at time e and otain a value that violates the lower ound, thus showing a ontradition 43 Relevant ells, R We define a ell to e relevant if: () = f or () shares the same input as f and is red in preferene to some relevant ell during some phase < e Reall from Setion 35 that a ell is said to e red in preferene to a ell during phase if and only if is red, survives in the input uffer, and onflit, and is at least as important as ; formally, X IB X ~ oo, ( w() ) t

19 Intuitively, the relevant ells are f and ells that, diretly or indiretly, delay the of f y means of input onflits We define R as the set of all relevant ells For any time we define R as the set all of relevant ells present in the input uffer at time Formally, R R IB A phase during whih some relevant ell is red we all an R- phase A phase during whih no relevant ell is red we all a nonr phase 44 Earliest inhale of a relevant ell, h Eah relevant ell R has an inhalation phase h () We define h to e the earliest inhalation phase of any relevant ell Formally, () h min h R Sine R is non-empty ( f R ), h is well-defined We laim that for any time in the range h < e, we have R > 0 Clearly R h+ > 0, sine the swith has just inhaled a relevant ell and has not yet had a hane to it An R- phase < e s a relevant ell R, ut sine annot e f (eause f is not red efore e), must e red in preferene to some other relevant ell R, and onsequently we have R+ No other phase an remove a relevant ell from the input uffer, so the laim is proved 45 Least important relevant ell, lir For any time in the range h < e, we define the least important relevant ell lir at time as the maximum element of R aording to < oo ( w() ), t, x That is, lir R R : oo, lir ( w() ), t x Sine R > 0 and < x is total, the least important relevant ell exists and is unique Note that the least important relevant ell is defined in terms of the output oupany ordering as it is at time w (), whih, not surprisingly, is the output oupany ordering used in the LOOTFA mathing ondition We now prove two useful lemmas aout the least important relevant ell Note that these lemmas relate to the assumed ounterexample exeution history with respet to whih e, R, h, and lir are defined Lemma (lir survival): For any phase in the range h < < e, we have lir R + 3

20 Proof: By definition lir R If is an inhale phase, an exhale phase, or a phase that does not lir, then lir survives in the input uffer at time +, and onsequently lir R + It remains to onsider the ase in whih is a phase and lir X In this ase, we must have lir f, sine f is not red efore e From the definition of relevane, lir must e red in preferene to some other relevant ell R, whih means that lir is at least as important as, that is lir oo( w() ), t Sine lir is red efore, we have lir < x But this gives us lir < oo w, t, x, whih ontradits the definition of lir This ompletes the proof ( ()) Lemma (lir output oupany): For any phase in the range h < < e, we have ( lir ) oo ( lir ) oo Proof: Intuitively, either the hoie of lir + is ased on output oupanies at time + or else lir + = lir By definition, lir + is the maximum element of R + under < oo ( w( +) ), t, x Sine we have lir R + y the previous lemma, it follows that lir oo( w( + )), t, x lir+ and hene lir oo( w( + )) lir+ If w ( + ) = + then we are done Otherwise, y the definition of w (see Setion 35), w ( + ) = w( ) and phase annot inhale any ells Sine lir + annot have een inhaled during phase, it must have een in the input uffer at time, and onsequently lir+ R By definition, lir is the maximum element of R under < oo ( w() ), t, x, so it follows that lir + oo( w() ), t, x lir But w ( + ) = w( ), so we have lir + oo( w( +) ), t, x lir We now have lir and lir + eah at least as important as the other aording to < oo ( w( +) ), t, x Sine this ordering is total, it follows that lir + = lir and we are done 46 Output uffer trailing ells, OBT For any time in the range h < e, we define the output uffer trailing ells OBT at time as the set of those ells in output unit o ( f) that are preeded y f aording to the timestamp ordering Formally, OBT { OB f },o = o : < t ( f) 47 Potential, p For any time in the range h < e, we define the potential p at time y the following magi formula: ( lir ) OBT R p oo 4

21 We estalish a lower ound on the potential at time h +, analyze the hanges in potential with eah phase, and show that the resulting lower ound on potential at time e ontradits the atual potential at time e 48 Lower ound on potential at time h+ To ound the potential at time h + we ound the omponents in its definition oo ( lir ) 0 h+ h+ h+ = 0 An output oupany annot e negative OBT Consider any ell in output unit ( f) h+ = OB h +, o= o o at time h +, that is, ( f) Cell must e red during some earlier x < h +, and sine phase h is an inhale phase, we phase () have x () < h Cell must e inhaled efore it is red, hene h () < x() < h Sine h is the inhalation time of the earliest relevant ell and f is relevant, we have h h( f) and thus h < x < h f Hene from the definition of the timestamp () () ( ) ordering we have t f So is not an output uffer trailing ell R At time h + the swith has just inhaled the earliest relevant ell Comining the omponents, we have p ( lirh+ ) OBTh+ R h+ = ooh+ h+ Next we onsider the effets of eah phase as advanes from h + to e 49 Effet of an inhale phase To ound the hange in potential during an inhale phase, we ound the hanges of the omponents oo ( lir ) oo ( lir ) + + The output uffer is unhanged y an inhale phase, so we have oo + ( lir ) = oo ( lir ) Comining this with the lir output oupany lemma (Setion 45) we get oo + ( lir+ ) oo+ ( lir )= oo ( lir ) OBT + = OBT The output uffer is unhanged y an inhale phase R Input ( f) R + + Comining the omponents, we have p i an inhale at most one ell ( lir+ ) OBT + R + = oo+ + 5

22 oo ( lir ) OBT ( R +) ( lir ) OBT R = oo = p 40 Effet of an R- phase To ound the hange in potential during an R- phase, we ound the hanges of the omponents oo+ ( lir+ ) oo( lir ) No ells an depart the output uffer, so oo ( lir ) oo ( lir ) + Comining this with the lir output oupany lemma (Setion 45) oo lir oo lir oo lir we get ( ) ( ) ( ) OBT + OBT + There might e a new output uffer trailing ell, ut there an e at most one R = R + Comining the omponents, we have p Exatly one relevant ell is red ( lir+ ) OBT + R + = oo+ + oo ( lir ) ( OBT + ) ( R ) ( lir ) OBT R + = oo = p + 4 Effet of a nonr- phase To ound the hange in potential during a nonr- phase, we ound the hanges of the omponents oo ( lir ) oo ( lir ) Sine lir is relevant, lir is not red during phase Therefore from the LOOTFA mathing ondition (Setion 35) there must e some ell red in preferene to lir Sine any ell red in preferene to lir and sharing input ( lir ) i( f) i = would y definition e relevant, and sine no relevant ell is red during a nonr- phase, there must e some ell red in preferene to lir that shares output + lir = oo lir + o ( lir ) Therefore oo ( ) ( ) Comining this 6

23 with the lir output oupany lemma (Setion 45) we get oo lir oo lir = oo lir ( ) ( ) ( ) OBT + = OBT Sine f is relevant, f is not red during phase Therefore from the LOOTFA mathing ondition (Setion 35) there must e some ell red in preferene to f Sine any ell red in preferene to f and sharing input i ( f) would y definition e relevant, and sine no relevant ell is red during a nonr phase, there must e some ell red in preferene to f that shares output o ( f) Let e suh a ell Sine is red in preferene to f, we have oo ( w() ), t f Sine o () = o( f), we have = ( ()) f oo w and hene t f Therefore is not an output uffer trailing ell Sine at most one ell an e red to any given output during a single phase, is the only ell red to output o ( f) during phase So no output uffer trailing ells are red during phase R + = R No relevant ell is red Comining the omponents, we have p ( lir+ ) OBT + R + = oo+ + ( oo( lir) + ) OBT R ( lir ) OBT R + = oo = p Effet of an exhale phase To ound the hange in potential during an exhale phase, we ound the hanges of the omponents oo+ ( lir+ ) oo( lir ) Sine output o ( lir ) an exhale at most one ell, we have oo ( lir ) oo ( lir ) Comining this with the lir output oupany lemma (Setion 45) we get oo + ( lir+ ) oo+ ( lir ) oo ( lir ) OBT + = OBT Sine < e and exhale phase e is assumed to e the earliest phase in whih the swith fails, output o ( f) annot exhale any memer of OBT R + = R The input uffer is unhanged 7

24 Comining the omponents, we have p ( lir+ ) OBT + R + = oo+ + ( oo( lir) ) OBT R ( lir ) OBT R = oo = p 43 Lower ound on potential at time e To summarize the preeding setions, the effet on p of the phases etween time h + and time e is as follows: eah inhale phase dereases p y at most, eah phase (regardless of whether R- or nonr-) inreases p y at least, and eah exhale phase dereases p y at most Let H the numer of inhale phases etween time h + and time e, X the numer of phases etween time h + and time e, and E the numer of exhale phases etween time h + and time e (Note that E does not inlude the failing phase, whih starts at time e) Then formally, p e p h + H + X E Oserve from the slot struture (Setion ) that in any interval starting at the end of an inhale phase and ending at the start of an exhale phase, there are more phases than twie the numer of inhale phases plus the numer of exhale phases In partiular, we have X > H + E Comining the aove with the lower ound p (Setion 48), we have p e p h + H + X E > p h+ 44 The potential at time e Now let us ompute oo ( lir ) oo ( f) e e = e p e diretly h+ From the definition of R, eah relevant ell exept f is red during some phase < e Hene we have R e = { f } and thus lir e = f 8

25 ( f) OBTe = ooe Sine y assumption exhale phase e fails, all ells in output unit o ( f) at time e must e preeded y f aording to < oo ( w() e )t,, that is, they are all output uffer trailing ells R = R e = { f} e Comining the omponents, we have p e e ( lire ) OBTe Re = oo ( f) oo ( ) = oo f e e =, whih ontradits the lower ound otained in Setion 43 Hene the assumption of a ounterexample arrives at a ontradition, and the LOOTFA theorem is proved 5 Generalizations For larity, we have presented LOOTFA with a numer of onrete assumptions that are not stritly required y our proof In this setion we show some generalizations 5 Generalized time of evaluation, w() When phase deides whih ells to inlude in the math, it evaluates the importane of ells ased on output oupanies as of time w () Our original definition of w (Setion 35) auses eah phase to onstrut its math ased on output oupanies as they are at the end of the most reent inhale phase However, the only plae in whih we use properties of w is the proof of the lir output oupany lemma in Setion 45, whih sueeds if w satisfies the following onditions for eah phase : w ( + ) = w( ) w( + ) = + IB > 0 w( + ) = IB + + same as previous, or eome urrent eome urrent if any inhaled ells The asi idea is that the swith maintains a reord of output oupanies ased on whih it onstruts the math for a phase and, from time to time, the swith updates this reord to the urrent output oupanies A LOOTFA swith must update its output oupanies at the end of every inhale phase that atually inhales a ell, and it an also update whenever onvenient () For example, onsider the definition w, whih satisfies the generalized onditions Under this definition, every phase onstruts its math ased on urrent output oupanies As explained in Setion 35, this would likely e othersome to implement However, the resulting swith would e ideal provided S 3 9

26 5 Generalized phase arrangement Our original phase arrangement was a sequene of slots eah onsisting of an inhale phase, S phases, and an exhale phase (Setion ) However, the only plae in whih we use properties of the phase arrangement is the ounting argument in Setion 43 This argument requires that X > H + E, for any interval starting at the end of an inhale phase and ending at the start of an exhale phase, where H the numer of inhale phases in the interval, X the numer of phases in the interval, and E the numer of exhale phases in the interval In addition to our original slot struture, there are many other ways of arranging phases that satisfy this ondition For example, we an arrange the phases into multislots, in whih eah phase of a slot repeats n times, as illustrated in Figure 7 inhale inhale exhale exhale inhale inhale exhale exhale inhale inhale time oundary multislot multislot Figure 7: Example multislot struture (n=, S=3) Sine all of the phases in the same multislot are allowed to use the same output oupanies, a multislot implementation ould proaly e pipelined more extensively than a single slot design This would ahieve higher throughput at the ost of higher lateny in real time due to inreased disrepany etween real time and model time 53 Generalized timestamp ordering, < t Our original timestamp ordering given in Setion 33 said that a ell inhaled efore another must preede the other aording to the timestamp ordering However, the only plae in whih we use properties of the timestamp ordering is the lower ound alulation in Setion 48, whih sueeds if < t satisfies the following timestamp ondition for any ells, and any phase x: h ( ) x < h( ) t < That is, if the inhalation phases of two ells are separated y an intervening phase, the earlier ell must preede or tie the later ell aording to the timestamp ordering 0

27 The timestamp ontrols how a LOOTFA swith orders ells destined to the same output Provided that the speedup S 3, during eah exhale phase eah output exhales a ell that preedes or ties, aording to the timestamp order, all ells in the swith destined to that output (if any) That is, for every exhale phase, we have ( IB OB ) > 0 E 0 o= o o o E ( IB OB ) () = o :, > work-onserving : : order-onserving o = o t We ould say that the timestamp ordering defines the meaning of older and younger and therefore the meaning of order-onserving Perhaps it would e etter to all the ondition timestamp-onserving For example, our original timestamp ordering orders ells aording to their inhalation phases An output of an S 3 LOOTFA swith using this timestamp exhales ells in order of inhalation phase, ut an aritrary order applies to ells inhaled during the same phase For a seond example, onsider the timestamp ordering whih orders ells aording to their inhalation phases and reaks ties y ordering ells aording to their input numer Sine an input an inhale at most one ell per phase, this is a total order An output of an S 3 LOOTFA swith using this timestamp exhales ells stritly aording to this total order For a third example, onsider a multislot swith (Setion 5) in whih the timestamp ordering orders ells aording to their inhalation multislot, ut aritrarily orders ells that are inhaled during the repeated inhale phases of the same multislot An output of an S 3 LOOTFA swith using this timestamp exhales ells aording to the timestamp ordering, whih may or may not e useful For a fourth example, onsider the trivial timestamp ordering, in whih all ells tie The effet of suh a definition is to remove from the LOOTFA swith all onsideration of the age of a ell, thus reduing it to a LOOFA swith In this ase the order-onserving suess ondition is trivially satisfied and the only interesting property is work-onserving We an slightly modify our proof of the LOOTFA theorem to otain a proof of the fat that an S LOOFA swith is work-onserving Assuming a failing exhale phase e, using the same definitions of f, R, h, lir, H, X, and E, and using the modified potential p at time defined y p ( lir ) R oo, we an show that p h+, p + p for eah inhale phase, p + p + for eah phase, p + p for eah exhale phase, p p H + X E p, and e h+ > h+

28 p =, e whih is a ontradition, hene proving that there an e no exhale phase that fails (Note that under a timestamp ordering in whih all ells tie, there are never any output uffer trailing ells) Krishna et al [] proved that an S LOOFA swith is work-onserving Our result here provides an alternate proof 6 Computing a LOOTFA math The ovious algorithm to ompute a LOOTFA math is to repeatedly pik the most important non-onflited ell until no more ells an e piked We all this the gloal minimum greedy algorithm It turns out that if the timestamp is the same for all ells (as in the fourth example of Setion 53), then the math an also e omputed y visiting the input units in an aritrary order piking the most important non-onflited ell in eah 6 The gloal minimum greedy algorithm Given two sets C and C of ells, we define the set C ~ C as those ells in C that do not onflit with any ell in C Formally, ~ C C : C : ~ The gloal minimum greedy algorithm omputes a LOOTFA math at time with an iterative sequene X, 0, X,,, X, Z starting with X,0 = {} and produing X, z+ from X, z y adding a most-important (that is, minimal) element of IB ~ X, z aording to < oo ( w() )t, There may e ties, in whih ase the hoie etween the tied minimal elements is aritrary When IB ~ X, z is empty, we delare that Z = z and the algorithm is done The result X is X, Z Eah step is alled a round Oserve that Z N C { } Clearly any result of the gloal minimum greedy algorithm satisfies the formal model phase requirements (Setion 9), X IB, X =i i :, i, and o : X, o =o To show that the result satisfies the LOOTFA mathing ondition (Setion 35), IB X : X : ~, ( w() ) t oo, we introdue the following invariant, whih we laim holds at any round z: ( X : ~ ) ( IB X ) IB :, z oo ~, z ( w () ), t

29 For round z = 0, X, 0 is empty, so IB ~ X,0 = IB and therefore IB ~ X, 0 for all IB Assuming the invariant holds at some round z < Z, we now show it holds at round z + Consider any ell IB We must show that ( X ~ ) ( IB X ), z+ : oo ~, z+ ( w() ), t Case : Suppose X, z : ~ oo w() X X, z+, z ( ) t, Then we are done, eause Case : Otherwise, y the invariant for round z, we have unique ell in X, z+ X, z IB ~ X, z Let e the Case a: Suppose ~ (whih inludes the ase = ) Beause the gloal minimum greedy algorithm hooses a most-important not-yet-onflited ell on eah round, we have oo ( w() ), t Thus we have X, z+ and ~ and oo ( w() ), t, and we are done Case : On the other hand, if does not onflit with, we have IB ~ X, z+, and we are done This ompletes the proof of the invariant By the termination ondition, IB ~ X, Z = {}, the invariant at the end of the final round redues to IB : X :, ( w() ) t, Z ~ oo, from whih the LOOTFA mathing ondition follows immediately It also turns out that any legal LOOTFA math X an e produed y some run of the gloal minimum greedy algorithm, y hoosing the elements of X in non-dereasing order aording to < oo ( w() )t, Hene the possile results of the gloal minimum greedy algorithm are preisely the mathes that are allowed y the definition of a LOOTFA swith Note that for eah pair of input i and output o an implementation an ignore any ells in IB, i = i, o= o exept for any single minimal element aording to < t If the timestamps for ells of input i and output o advane monotonially with inhalation time (as in our initial LOOTFA timestamp ordering in Setion 33), a virtual output queue will always have a most important ell at the front 6 The per-input minimum greedy algorithm We onsider the ase of the trivial timestamp ordering, in whih all ells tie This redues the LOOTFA swith to a LOOFA swith Oserve that if an inluded ell has o = o oo oo and hene an output onflit with, that is, ( ) ( ), then we have ( ) ( ) = = oo ( w() ), t Beause of this property, a LOOFA math an e omputed y the per-input minimum greedy algorithm, in whih eah round z onsists of two parts: first hoos- 3

30 ing an input i aritrarily, sujet to the onstraint that IB, i = i ~ X, z is non-empty; and then produing X, z+ from X, z y adding a minimal element of IB, i = i ~ X, z aording < to w() oo ( )t, Regardless of the order in whih the inputs are onsidered, the per-input minimum greedy algorithm produes a math that satisfies the LOOTFA mathing ondition The proof uses the same invariant as used in the previous setion, ( X : ~ ) ( IB X ) IB :, z oo ~, z ( w () ), t The only differene ours in the proof within Case a that is at least as important as, that is, oo ( w() ), t If and share the same input, then the result follows from the fat that the per-input minimum greedy algorithm hooses a most-important not-yet-onflited ell from a hosen input Otherwise, and must share the same output, from whih it follows that = oo ( w() ), t, as explained aove The per-input minimum greedy algorithm has a parallel implementation in whih eah input extends a id to the destined output of its most important non-onflited ell, and then eah output that gets a id aepts one and rejets the others Additional id-aept rounds handle rejeted inputs Although typially only a few rounds are neessary, in the worst ase N rounds are required Krishna et al [] use this implementation of the perinput minimum greedy algorithm Unfortunately, the per-input minimum greedy algorithm does not work for a non-trivial timestamp ordering, whih in general is the ase in a LOOTFA swith Aknowledgments The authors would like to thank Greg Nelson for his advie on larifying our notation and Sharon Perl for many helpful omments on an earlier draft Referenes P Krishna, N S Patel, A Charny, R Simoe On the Speedup Required for Work- Conserving Crossar Swithes Aepted for IWQoS-98 B Prahakar and N MKeown On the Speedup Required for Comined Input and Output Queued Swithing Computer Systems La, Tehnial Report CSL-TR , Stanford University 4