On the Latency Bound of Deficit Round Robin

Transcription

1 Poceedngs of the Intenatonal Confeence on Compute Communcatons and Netwoks Mam, Floda, USA, Octobe 4 6, 22 On the Latency Bound of Defct Round Robn Sall S. Kanhee and Hash Sethu Depatment of ECE, Dexel Unvesty 34 Chestnut Steet, Phladelpha, PA E-mal: {sall,sethu}@ece.dexel.edu Abstact The emegng hgh-speed boadband packetswtched netwoks ae expected to smultaneously suppot a vaety of sevces wth dffeent Qualty-of-Sevce (QoS) equements ove the same physcal nfastuctue. Fa packet schedulng algothms used n swtches and outes play a ctcal ole n povdng these QoS guaantees. Defct Round Robn (DRR), a popula fa schedulng dscplne, s vey effcent wth an O() dequeung complexty. Usng the concept of Latency-Rate (LR) seves ntoduced by Stlads and Vama, we obtan an uppe bound on the latency of DRR and pove that ou bound s tght. Ou uppe bound s lowe than the pevously known uppe bound. Ths llustates that the DRR schedule has bette pefomance chaactestcs than pevously beleved, especally fo eal-tme applcatons whee the latency plays a ole n the sze of the playback buffe equed. I. INTRODUCTION A numbe of new applcatons such as vdeo-on-demand and dstance leanng ely on taffc schedulng algothms n the swtches and outes to guaantee pefomance bounds and meet the Qualty-of-Sevce (QoS) equements. Gven multple packets awatng tansmsson though an output lnk, the functon of the schedule s to detemne the exact sequence n whch these packets wll be tansmtted. Schedulng algothms can be boadly classfed nto two categoes soted-poty schedules and fame-based schedules. Sotedpoty schedules such as eghted Fa Queueng (FQ) [], Self-Clocked Fa Queueng (SCFQ) [2], Stat-tme Fa Queueng (SFQ) [3] and ost-case Fa eghted Fa Queueng (F 2 Q) [4] mantan a global vaable known as the vtual tme o the system potental functon. Ths vaable s used to compute the poty of each packet whch s known as the tmestamp. hle they acheve good faness and low latences, they ae not vey effcent due to the complexty nvolved n computng the vtual tme and the complexty of mantanng a soted lst of packets based on the tmestamps. On the othe hand, n fame-based schedules such as Defct Round Robn (DRR) [5] and Elastc Round Robn (ERR) [6], the schedule smply vsts all non-empty queues n a ound-obn ode. The sevce eceved by a flon a ound obn sevce Ths wok was suppoted n pat by the NSF CAREER Awad CCR and U.S. A Foce Contact F oppotunty s popotonal to ts fa shae of the bandwdth. These schedules do not have to mantan a global vtual tme functon and also do not have to pefom sotng among the packets. The ound obn sevce ode of the fame-based schedules educes the pe-packet wok complexty to O() wth espect to the numbe of flows, endeng them attactve fo mplementaton n outes and swtches. In ecent wok by Stlads and Vama [7, 8], the authos defne a geneal class of schedules, called Latency-Rate (LR) seves. The authos also develop and defne a noton of latency and detemne an uppe bound on the latency fo a numbe of schedules that belong to the class of LR seves. Ths noton of latency s based on the length of tme t takes a new flow to begn ecevng sevce at ts guaanteed ate and theefoe, s dectly elevant to the sze of the playback buffes equed n eal-tme steamng applcatons. DRR s one of the most popula fame-based fa schedulng dscplnes that s now employed n a numbe of eal envonments nvolvng fa schedulng, ncludng Csco outes and Mcosoft s ndows NT. It s shown n [8] that the DRR schedule belongs to the class of LR seves. Stlads and Vama epot an uppe bound on the latency of DRR [7], and ts devaton s detaled n [8]. In ths pape, we obtan a lowe value of the uppe bound on the latency of DRR and thus show that the schedule has bette pefomance chaactestcs than pevously beleved. e also show that ou uppe bound on the DRR latency s tght. The est of ths pape s oganzed as follows. In Secton 2, we befly ntoduce the DRR schedulng dscplne. In Secton 3, we pesent ou analyss of the latency bound of DRR. Secton 4 pesents a detaled compason between ou latency bound and the pevously known bound deved n [8]. Fnally, Secton 5 concludes the pape. II. DEFICIT ROUND ROBIN In ths secton, we pesent a bef ovevew of the DRR schedule, a detaled descpton of whch can be found n [5]. Consde an output lnk of tansmsson ate, access to whch s contolled by the DRR schedule. Let n be the total numbe of flows and let ρ be the eseved ate fo flo. Let ρ mn be the lowest of these eseved ates. Note that snce

2 all the flows shae the same output lnk, a necessay constant s that the sum of the eseved ates be no moe than the tansmsson ate of the output lnk. In ode that each flow eceves sevce popotonal to ts guaanteed sevce ate, the DRR schedule assgns a weght to each flow. The weght assgned to flo, s gven by, Note that n,. = ρ ρ mn () A flos sad to be actve dung a cetan tme nteval, f t always has packets awatng sevce dung ths nteval. The DRR schedule mantans a lnked lst of the actve flows, called the ActveLst. At the stat of an actve peod of a flow, the flos added to the tal of the ActveLst. A ound s defned as one ound obn teaton dung whch the DRR schedule seves all the flows that ae pesent n the ActveLst at the outset of the ound. Each actve flos assgned a quantum by the DRR schedule. The quantum allocated to a flos defned as the sevce that the flow should eceve dung each ound obn sevce oppotunty. Let Q epesent the quantum assgned to flo and let Q mn be the quantum assgned to the flow wth the lowest eseved ate. The quantum assgned to flo, Q s gven by Q mn. Thus, the quanta assgned to the flows ae n popoton of the eseved ates. In ode that the wok complexty of the DRR schedule s O(), t s necessay that Q mn should be geate than o equal to the sze of the lagest packet that may potentally ave dung the executon of the schedule. Note that dung some sevce oppotunty, a flow may not be able to tansmt a packet because dong so would cause the flow to exceed ts allocated quantum. The schedule mantans a pe-flow state, the defct count, whch ecods the dffeence between the amount of data actually sent thus fa, and the amount that should have been sent. Ths defct s added to the value of the quantum n the next ound as the amount of data the schedule should ty to schedule n the next ound. Thus, a flow that eceved vey lttle sevce n a cetan ound s gven an oppotunty to eceve moe sevce n the next ound. A fame s defned as the sum of the quanta allocated to all the actve flows n a DRR ound. Let F denote the sze n bts of a DRR fame. The uppe bound of the latency of DRR s deved n [8] as (3F 2φ )/, whee epesents the tansmsson ate of the output lnk. In the followng secton, we pove a tghte bound on the latency. Ou bound s tghte due to the use of a tghte uppe bound on the defct count of a flow and also due to the fact that we make a dstncton between the sze of the lagest packet that may potentally ave at a schedule and the sze of the lagest packet that actually aves dung an executon of the schedule. III. LATENCY BOUND OF DRR In devng an uppe bound on the latency of DRR, we boow the technque used n [8] based on the concept of LRseves fst poposed n [7]. In the followng, we povde some defntons and evew some othes that wll be useful n defnng the latency of guaanteed ate schedules and ou analyss of the latency bound. Defnton : Defne as the sum of the weghts of all actve flows that ae beng seved by the DRR schedule. Defnton 2: Defne m as the sze n bts of the lagest packet that s actually seved dung the executon of a schedulng algothm. Defne M as the sze n bts of the lagest packet that may potentally ave dung the executon of a schedulng algothm. Note that, M m. Defnton 3: An actve peod of a flos defned as the maxmal nteval of tme dung whch t has at least one packet awatng sevce o n sevce. Defnton 4: A busy peod of a flos defned as the maxmal tme nteval dung whch the flos actve f t seved exactly at ts eseved ate. The actve peod s dffeent fom the busy peod of a flow, n the sense that t eflects the actual behavo of the schedule snce the nstantaneous sevce offeed to the flow vaes accodng to the numbe of actve flows. Let Sent (t, t 2 ) epesent the amount of sevce eceved by flo dung the tme nteval (t, t 2 ). Let the tme nstant α be the stat of a busy peod fo flo. Let t > α be such that flo s contnuously busy dung the tme nteval (α, t). Defne S (α, t) as the numbe of bts belongng to packets n flo that ave afte tme α and ae scheduled dung the tme nteval (α, t). Note that dung ths tme nteval, the schedule may stll be sevng packets that aved dung a pevous busy peod, and hence S (α, t) s not necessaly the same as Sent (α, t). The eade s efeed to [7, 8] fo a detaled teatment of the dffeences between an actve peod and a busy peod. Defnton 5: Defne T as the set of all tme nstants at whch the schedule ends sevng one flow and begns sevng anothe. The set of all tme nstants at whch a schedule begns sevng flo s defned as T. Note that the set T s the unon of T fo all actve flows. Defnton 6: The latency of a flos defned as the mnmum non-negatve constant Θ that satsfes the followng fo all possble busy peods of the flow, S (α, t) max{, ρ (t α Θ )} (2) As defned n [7], a schedule whch satsfes Equaton (2) fo some non-negatve constant value of Θ s sad to belong to the class of Latency Rate (LR) seves. The above defnton captues the fact that the latency of a guaanteed-ate schedule should not meely be the tme t takes fo the fst packet of a flow to get scheduled, but should be a measue of the cumulatve tme that a flow has to wat untl t begns ecevng sevce at ts guaanteed ate.

3 Note that even though the defnton of the latency s based on flow busy peods, n pactce t s ease to analyze schedulng algothms based on the actve peod of a flow. Let τ be an nstant of tme when flo becomes actve. Let t > τ be some tme nstant such that the flos contnuously actve dung the tme nteval (τ, t). Let Θ be the smallest nonnegatve constant such that the followng equaton s satsfed fo all t, Sent (τ, t) max{, ρ (t τ Θ )} (3) Even though (τ, t) may not be a contnuously busy peod fo flo, t s poved n [7] that the latency, as defned by (2), s bounded by Θ. Ths allows one to detemne the latency bound of a schedule by consdeng only those peods dung whch a flos contnuously actve. Theoem : The DRR schedule belongs to the class of LR seves, wth an uppe bound on the latency, Θ fo flo, gven by: Θ ( ( )) ( )Q mn + (m ) + n 2 (4) whee n s the total numbe of actve flows and s the tansmsson ate of the output lnk. Poof: Snce the latency of an LR seve can be estmated based on ts behavo n the flow actve peod, we wll pove the theoem by showng that, Θ ( ( )) ( )Q mn + (m ) + n 2 Let flo become actve at tme nstant τ. In devng an uppe bound on the latency of DRR we consde a tme nteval (τ, t) dung whch flo s contnuously actve. Then, we obtan the lowe bound on the total sevce eceved by flo dung ths tme nteval. Lastly, we expess the lowe bound n the fom of Equaton (3) to deve the latency bound. In [9], n the context of devng the latency bound of Elastc Round Robn [6], t s poved that f the uppe bound of latency s met exactly dung the actve peod (τ, t), then the followng two condtons ae satsfed: ) τ T and 2) t T It can be easly vefed that these condtons ae applcable n the analyss of the latency bound of all ound obn schedules ncludng DRR. Let τ k be the tme nstant makng the stat of the k-th sevce oppotunty of flo. Note that τ k belongs to the set T. Fom the above, to detemne a tght uppe bound on the latency of the DRR schedule we need to only consde tme ntevals (τ, τ k ) fo all k. Fg. llustates the tme nteval unde consdeaton fo a gven k. Note that the tme nstant τ may o may not concde wth the end of a ound and the stat of the subsequent ound. Let k be the ound whch s n pogess at tme nstant τ o whch ends exactly at tme nstant τ. Let the tme nstant t h mak the end of ound (k + h ) and the stat of the subsequent ound. Let Sent (s) epesent the total data tansmtted fom flow n ound s of the DRR schedule. Also, let DC (s) epesent the defct count of flo followng ts sevce n ound s. It s poved n [5] that fo any flo n any ound s, DC (s) m (5) Sent (s) = Q mn + DC (s ) DC (s) (6) Note that the uppe bound of DC (s) s Q, as used n [8] n the devaton of the DRR latency, only f Q = M = m. In all othe stuatons, the uppe bound on the defct count as specfed by Equaton (5) s a tghte bound. As llustated n Fg., assume that the tme nstant when flo becomes actve concdes wth the tme nstant when some flow u s about to stat ts sevce oppotunty dung the k -th ound. Let G a denote the set of flows whch eceve sevce dung the tme nteval (τ, t ),.e., afte flo becomes actve. Smlaly, let G b denote the set of flows whch ae seved by the DRR schedule dung the tme nteval (t, τ ),.e., befoe flo becomes actve. Note that flo s not ncluded n ethe of these two sets snce flo wll eceve ts fst sevce oppotunty only n the (k + )-th ound. If the tme nstant τ concdes wth the tme nstant t, whch maks the end of the k -th ound and the stat of the (k +)-th ound, then the set G a wll be empty and all the n flows wll be ncluded n the set G b. Note that n ths case, flo wll be the last to eceve sevce n the (k + )-th ound and all subsequent ounds dung the tme nteval unde consdeaton. The fst step towads analyzng the latency bound nvolves obtanng an uppe bound on the sze of the tme nteval (τ, τ k ). Ths tme nteval can be splt nto the followng thee sub-ntevals: ) (τ, t ): Ths sub-nteval ncludes the pat of the k -th ound dung whch all the flows belongng to the set G a wll be seved by the DRR schedule. Summng Equaton (6) ove all these flows, t τ = {w j Q mn + j G a + DC j (k ) DC j (k )} (7) 2) (t, t k ): Ths sub-nteval ncludes k ounds of the DRR schedule statng at ound (k + ). Consde the tme nteval (t h, t h+ ) when ound (k + h) s n pogess. Summng Equaton (6) ove all n flows and snce s the sum of all the flow weghts, we have, t h+ t h = (Q mn) + n {DC j (k + h ) DC j (k + h)} j=

4 tme t t t 2 t k t k+ Round k Round k + Round k + k 2 u 2 u n 2 u n τ Flo becomes actve τ tme nteval unde consdeaton τ k Fg.. An llustaton of the tme nteval unde consdeaton Summng the above ove k ounds begnnng wth ound k +, t k t = (k )Q mn + n {DC j (k ) DC j (k + k )}(8) j= 3) (t k, τ k ): Ths sub-nteval ncludes the pat of the (k + k)-th ound dung whch all the flows belongng to the set G b wll be seved by the DRR schedule. Summng (6) ove all these flows, τ k t k = {w j Q mn j G b + DC j (k + k ) DC j (k + k)}(9) Combnng Equatons (7), (8) and (9) and snce s the sum of the weghts of all the n flows, we have, τ k τ = ( ) (k )Q w mn + Q mn + (DC j (k ) DC j (k + k )) j G a + (DC j (k ) DC j (k + k)) j G b + (DC (k ) DC (k + k )) () Now snce flo becomes actve dung ound k, ts defct count at the end of the k -th ound, DC (k ) s equal to zeo. Usng ths fact and the bounds on the defct count fom Equaton (5) n Equaton (), we have, τ k τ ( ) (k )Q w mn + Q mn (n )(m ) + DC (k + k ) Solvng fo (k ), (k ) Q mn (τ k τ ) (n )(m ) Q mn + DC (k + k ) () Q mn Note that dung the tme nteval unde consdeaton, (τ, τ k ), flo eceves sevce n (k ) ounds statng at ound (k + ). Hence, usng Equaton (6) ove these (k ) ounds of sevce fo flo, and snce the defct count of a newly actve flos, we get, Sent (τ, τ k ) = (k )Q mn DC (k + k ) (2) Usng () to substtute fo (k ) n (2), we get, Sent (τ, τ k ) (τ k w τ ) ( )Q mn (n )(m ) + DC (k + k ) DC (k + k ) (3) Now, snce the eseved ates ae popotonal to the weghts assgned to the flows as gven by (), and snce the sum of the eseved ates s no moe than the lnk ate, we have, ρ (4) Usng Equaton (4) n Equaton (3), we have, Sent (τ, τ k ) ρ (τ k τ ) ρ ( )(Q mn ) ρ ρ (n )(m ) ( w ) DC (k + k ) (5) Smplfyng futhe and notng that the latency bound eaches the uppe bound when DC (k + k ) equals (m ) we

5 get, Sent (τ, τ k ) max {, ρ ( ( τ k τ ( )Q mn ( )))} + (m ) + n 2 (6) As dscussed eale, flo wll expeence ts wost latency dung an nteval (τ, τ k ) fo some k. Theefoe, fom Equaton (6), the statement of the theoem s poved. e now poceed to show that the latency bound gven by Theoem s tght by llustatng a case when the bound s actually acheved. Assume that flo becomes busy at a cetan tme nstant τ, whch also concdes wth the stat of a cetan ound (k + ). Snce the othe flows n the ActveLst wll be seved fst, flo becomes backlogged nstantly and τ s also the stat of ts actve peod. Assume that fo any tme nstant t, t τ, a total of n flows, ncludng flo, ae actve. Let F epesent the set of all n flows. Also, assume that the summaton of the eseved ates of all the n flows equals the output lnk tansmsson ate,. Hence ρ = w. Snce flow became actve at tme τ, ts defct count at the stat of ound (k + ) s. Let the defct count of all the othe flows at the stat of ound (k +) be equal to (m ). Fom Equatons (5) and (6), a flow j can tansmt a maxmum of w j Q mn +(m ) bts dung a ound obn sevce oppotunty. In the wost case, befoe flo s seved by the DRR schedule, each of the othe (n ) flows wll eceve ths maxmum sevce. Hence, the cumulatve delay untl flo eceves sevce, X, s gven by, ( j F w j )(Q mn ) + (n )(m ) j X = = ( )(Q mn ) + (n )(m ) (7) Even though X epesents the tme fo whch flo has to wat untl ts fst packet s scheduled, Equaton (2) does not hold tue when X s substtuted as Θ. Ths s because n the tme nteval (τ, τ + X) flo has not yet stated ecevng sevce at ts guaanteed ate. e assume that the latency, Θ s gven by, Θ = X + Y (8) A plot of the sevce eceved by flo aganst tme s llustated n Fg. 2. Assume that the total sevce eceved by flow dung ts fst sevce oppotunty s Q mn (m ). Note that fom (5) and (6), ths equals the mnmum sevce that flow can eceve dung any sevce oppotunty. At the end of the (k +)-th ound, the defct count fo flo s (m ) wheeas the defct count fo all the othe flows s zeo. In the wost case, dung the (k + 2)-th ound, each flow j fom amongst the othe (n ) flows wll tansmt a maxmum of w j Q mn bts befoe flo eceves ts second sevce oppotunty. Dung ths sevce oppotunty, flo wll be able to tansmt at least a mnmum of Q mn bts, and wll thus stat ecevng sevce at ts guaanteed ate. Refeng to Fg. 2, we have, Y + Q mn m = Q mn m ρ ρ Now, snce ρ =, smplfyng futhe, we have, ( ) (m ) w Y = (9) Substtutng fo X and Y fom Equatons (7) and (9) n Equaton (8), t can be easly vefed that the latency bound s exactly met. IV. A DISCUSSION OF THE NE LATENCY BOUND In ths secton, we pesent a bef but detaled compason of the latency bound of DRR deved n Theoem wth the latency bound deved n [8]. Let Θ new epesent the latency bound deved n Theoem. Hence, Θ new = ( )Q mn + ( ) (m ) + n 2 (2) Let epesent the latency bound of DRR as deved n [8]. e have, = 3F 2Q (2) In the above equaton, F denotes the sze of a DRR fame whch s equal to Q mn, the summaton of the quanta of all the actve flows. Substtutng fo F n Equaton (2) and smplfyng we have, = ( )Q mn + (2 )Q mn (22) Note that both and Θ new ae epesented as a summaton of two tems of whch the fst tem s dentcal. Recall that Q mn M and that M m. Theefoe t can be easly vefed that Θ new s less than, demonstatng that the latency bound of DRR as poved n ths pape s a tghte bound. Also note that the expesson fo does not dstngush between m and M. In most netwoks ncludng the Intenet, the vast majoty of the packets n the taffc ae of much smalle sze than the maxmum possble sze [,]. Theefoe, snce m << M, we have m << Q mn. Thus, n stuatons wth small numbes of flows and whee m < M, Θ new can be much lowe than. In ode that the eade can fully appecate the dffeence, we povde a compason of these two latency bounds of DRR wthn the context of a pactcal example. Let us assume that the DRR schedule s contollng access to an output lnk wth

6 Sevce eceved by flo Q (m ) mn Q mn Q mn ρ Q mn (m ) τ + θ 2 τ X Y Q (m ) mn θ ρ τ τ tme Fg. 2. Illustaton of the latency bound 5 (a) m = M 5 (b) m = M/2 Θ new Θ new Latency Bound (n msec) 5 Latency Bound (n msec) Reseved Rate of flo (n Mbps) Reseved Rate of flo (n Mbps) Fg. 3. Compason of the latency bound ate, = 5 Mbps. Assume that M, the sze of the lagest packet that may potentally ave dung the executon of the DRR algothm s equal to 576 bytes. Also, let Q mn be equal to M. Let ρ mn be equal to. Mbps and let the numbe of flows, n be equal to. Also assume that the output lnk s completely utlzed,.e. n = ρ =. Note that ths mples that the sum of all the weghts s 5/. = 5. e compae the two latency bounds, and Θ new fo flo as a functon of ts eseved ate, ρ, fo two values of m: (a) m = M, (b) m = M/2. Fg. 3 llustates a plot of the latency bounds of flo fo both values of m. Note that expessons fo Θ new and depend on the sum of the weghts of all the flows but not on the dstbuton of the weghts among all the flows othe than weght. Theefoe, the weghts of flows othe than flow ae not dscussed n the context of ths llustaton. Fg. 3 eveals that Θ new s a tghte bound. V. CONCLUSION In ths pape, we have deved an uppe bound on the latency of Defct Round Robn (DRR), a popula fa schedulng dscplne that has found use n a numbe of commecal poducts ncludng Csco outes and Mcosoft s ndows NT. Ou bound s lowe than the pevously beleved uppe bound. e also show that ou uppe bound s a tght one. The latency expeenced by a flow captues the length of tme t takes a new flow to begn ecevng sevce at the guaanteed ate, and theefoe, t s dectly elevant to the sze of playback buffes needed fo eal-tme applcatons. Ths pape shows that the DRR schedule has bette pefomance chaactestcs,

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