Latency-Rate Servers: A General. Model for Analysis of Trac. Dimitrios Stiliadis. Anujan Varma UCSC-CRL July 18, 1995.

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1 Latency-Rate Seves: A Geneal Model fo Analyss of Tac Schedulng Algothms Dmtos Stlads Anujan Vama UCSC-CRL July 18, 1995 Baskn Cente fo Compute Engneeng & Infomaton Scences Unvesty of Calfona, Santa Cuz Santa Cuz, CA USA abstact In ths pape, we develop a geneal model, called Latency-Rate seves (LR-seves), fo the analyss of tac schedulng algothms n boadband packet netwoks. The behavo of an LR schedule s detemned by two paametes the latency and the allocated ate. We show that seveal well-known schedulng algothms, such as Weghted Fa Queueng, VtualClock, Self-Clocked Fa Queueng, Weghted Round Robn, and Dect Round Robn, belong to the class of LR-seves. We deve tght uppe bounds on the end-toend delay, ntenal bustness, and bue equements of ndvdual sessons n an abtay netwok of LR-seves n tems of the latences of the ndvdual schedules n the netwok, when the sesson tac s shaped by a leaky bucket. Thus, the theoy of LR-seves enables computaton of tght uppe-bounds on end-to-end delay and bue equements n a netwok of seves n whch the seves on a path may not all use the same schedulng algothm. We also dene a self-contaned appoach to evaluate the faness of LR-seves and use t to compae the faness of many well-known schedulng algothms. Keywods: Tac schedulng, ATM swtch schedulng, fa queueng, end-to-end delay bounds, faness. Ths eseach s suppoted by the NSF Young Investgato Awad No. MIP

2 1 Intoducton Boadband packet netwoks ae cuently enablng the ntegaton of tac wth a wde ange of chaactestcs wthn a sngle communcaton netwok. Deent types of tac have sgncantly deent qualty-of-sevce (QoS) equements [1]. Rgd eal-tme applcatons eque a guaanteed poton of the lnk bandwdth, as well as bounded end-to-end delay, low delay jtte, and low packet loss ate. Adaptve applcatons can adjust the behavo to the cuent netwok state, as long as they can eceve a specc poton of the lnk bandwdth ove a peod of tme; that s, they can eceve the guaanteed bandwdth ove an aveagng peod longe than that of gd eal-tme applcatons. Fnally, most of the data tac s tansmtted n \best-eot" mode, equng no bandwdth guaantees. The nstantaneous bandwdth left ove afte allocatng the bandwdth to eal-tme ows can be used to tansmt packets belongng to best-eot tac. Povdng QoS guaantees n a packet netwok eques the use of tac schedulng algothms n the swtches (o outes). The functon of a schedulng algothm s to select, fo each outgong lnk of the swtch, the packet to be tansmtted n the next cycle fom the avalable packets belongng to the ows shang the output lnk. Implementaton of the algothm may be n hadwae o softwae. Because of the small cell-sze n an ATM (Asynchonous Tansfe Mode) netwok, the schedulng algothm must usually be mplemented n hadwae n an ATM swtch. In a packet netwok wth lage packet-szes, such as the cuent Intenet, the algothm can be mplemented n softwae. Seveal sevce dscplnes ae known n the lteatue fo bandwdth allocaton and tansmsson schedulng n output-bueed swtches. FIFO schedulng s pehaps the smplest to mplement, but does not povde any solaton between ndvdual sessons that s necessay to acheve detemnstc bandwdth guaantees. Seveal sevce dscplnes ae known n the lteatue fo povdng bandwdth guaantees to ndvdual sessons n output-bueed swtches [2, 3, 4, 5, 6, 7, 8, 9, 10]. Many of these algothms ae also capable of povdng detemnstc delay guaantees when the bustness of the sesson tac s bounded (fo example, shaped by a leaky bucket). In geneal, schedules can be chaactezed as wok-consevng o non-wok-consevng. A schedule s wok-consevng f the seve s neve dle when a packet s bueed n the system. A non-wok-consevng seve may eman dle even f thee ae avalable packets to tansmt. A seve may, fo example, postpone the tansmsson of a packet when t expects a hghe-poty packet to ave soon, even though t s cuently dle. When the tansmsson tme of a packet s shot, as s typcally the case n an ATM netwok, howeve, such a polcy s seldom justed. Nonwok-consevng algothms ae also used to contol delay jtte by delayng packets that ave ealy [11]. Wok-consevng seves always have lowe aveage delays than non-wok-consevng seves. Examples of wok-consevng schedules nclude Genealzed Pocesso Shang (GPS) [4], Weghted Fa Queueng [3], VtualClock [2], Delay-Ealest-Due-Date (Delay-EDD) [6], Weghted Round Robn [7], and Dect Round Robn [8]. On the othe hand, Heachcal-Round-Robn (HRR) [9], Stop-and-Go queueng [10], and Jtte-Ealest-Due-Date [11] ae non-wok-consevng schedules. Anothe classcaton of schedules s based on the ntenal stuctue [12]. Accodng to ths classcaton thee ae two man achtectues: soted-poty and fame-based. In a sotedpoty schedule, thee s a global vaable usually efeed to as the vtual tme assocated 1

3 wth each outgong lnk of the swtch. Each tme a packet aves o gets sevced, ths vaable s updated. A tmestamp, computed as a functon of ths vaable, s assocated wth each packet n the system. Packets ae soted based on the tmestamps, and ae tansmtted n that ode. VtualClock, Weghted Fa Queueng, and Delay-EDD follow ths achtectue. In a fame-based schedule, tme s splt nto fames of xed o vaable length. Resevatons of sessons ae made n tems of the maxmum amount of tac the sesson s allowed to tansmt dung a fame peod. Heachcal Round Robn and Stop-and-Go Queueng ae fame-based schedules that use a constant fame sze. As a esult, the seve may eman dle f sessons tansmt less tac than the esevatons ove the duaton of a fame. In contast, Weghted Round Robn and Dect Round Robn schedules allow the fame sze to vay wthn a maxmum. Thus, f the tac fom a sesson s less than ts esevaton, a new fame can be stated ealy. Theefoe, both of these schedules ae wok-consevng. Whle thee s a sgncant volume of wok on the analyss of vaous tac schedulng algothms, most of these studes apply only to a patcula schedulng algothm. Lttle wok has been epoted on analyzng the chaactestcs of the sevce oeed to ndvdual sessons n a netwok of seves whee the schedules on the path of the sesson may use deent schedulng algothms. Snce futue netwoks ae unlkely to be homogeneous n the type of schedulng algothms employed by the ndvdual swtches (outes), a geneal model fo the analyss of schedulng algothms wll be a valuable tool n the desgn and analyss of such netwoks. Ou basc objectve n ths pape s to develop such a geneal model to study the wost-case behavo of ndvdual sessons n a netwok of schedules whee the schedules n the netwok may employ a boad ange of schedulng algothms. Such an appoach wll enable us to calculate tght bounds on the end-to-end delay of ndvdual sessons and the bue szes needed to suppot them n an abtay netwok of schedules. Ou basc appoach conssts n denng a geneal class of schedules, called Latency-Rate seves, o smply LR-seves. The theoy of LR seves povdes a means to descbe the wostcase behavo of a boad ange of schedulng algothms n a smple and elegant manne. Fo a schedulng algothm to belong to ths class, t s only equed that the aveage ate of sevce oeed by the schedule to a busy sesson, ove evey nteval statng at tme fom the begnnng of the busy peod, s at least equal to ts eseved ate. The paamete s called the latency of the schedule. All the wok-consevng schedules known to us, ncludng Weghed Fa Queueng (o PGPS), VtualClock, SCFQ, Weghted Round Robn, and Dect Round Robn, exhbt ths popety and can theefoe be modeled as LR-seves. The behavo of an LR schedule s detemned by two paametes the latency and the allocated ate. The latency of an LR-seve s the wost-case delay seen by the st packet of the busy peod of a sesson, that s, a packet avng when the sesson's queue s empty. The latency of a patcula schedulng algothm may depend on ts ntenal paametes, ts tansmsson ate on the outgong lnk, and the allocated ates of vaous sessons. Howeve, we show that the maxmum end-to-end delay expeenced by a packet n a netwok of schedules can be calculated fom only the latences of the ndvdual schedules on the path of the sesson, and the tac paametes of the sesson that geneated the packet. Snce the maxmum delay n a schedule nceases dectly n popoton to ts latency, the model bngs out the sgncance of usng low-latency schedules to acheve low end-to-end delays. Lkewse, uppe bounds on the queue sze and bustness of 2

4 ndvdual sessons at any pont wthn the netwok can be obtaned dectly fom the latences of the schedules. We also show how the latency paamete can be computed fo a gven schedulng algothm by devng the latences of seveal well-known schedules. Ou appoach n modelng the wost-case behavo of schedulng algothms wth espect to an end-to-end sesson s elated to the wok of Cuz [13, 14], Zhang [15], and Paekh and Gallage [4, 16]. Cuz [13, 14] analyzed the end-to-end delay, bue equements, and ntenal netwok bustness of sessons n an abtay topology netwok whee all souces ae leaky-bucket contolled. Whle the objectves of ou analyss ae smla, thee ae thee majo deences between the appoaches taken: Fst, the class of schedulng algothms we study ae capable of povdng bandwdth guaantees to ndvdual sessons. Theefoe, we can deve detemnstc end-to-end delay guaantees that ae ndependent of the behavo of othe sessons. Second, we do not study ndvdual schedules n solaton and accumulate the sesson delays as n Cuz's wok, but nstead model the behavo of the chan of schedules on the path of the connecton as a whole. Thd, we estmate the latency paametes fo the ndvdual schedules tghtly, takng nto account the ntenal stuctue. Thus, ou appoach, n geneal, povdes much tghte end-to-end delay bounds fo ndvdual sessons. Paekh and Gallage analyzed the wost-case behavo of sessons n a netwok of GPS schedules [4, 16] and deved uppe bounds on end-to-end delay and ntenal bustness of sessons. Howeve, the analyss apples to a homogeneous netwok consstng of only GPS schedules. Ou analyss accommodates a boad ange of schedulng algothms and the ablty to combne the schedules n abtay ways n a netwok. Zhang [15] deved end-to-end delay bounds fo a class of non-wok-consevng schedulng algothms when tac s e-shaped at each node of the netwok. Ths allows the delays of ndvdual schedules on the path to be accumulated n a smple manne. Ou appoach des fom ths wok n that we consde the boade class of wok-consevng schedules n ou analyss, and we do not assume any tac e-shapng mechansms wthn the netwok. Anothe model fo delay-analyss based on a class of guaanteed-ate seves was pesented n [17]. The man poblem of ths model, howeve, s that t s closely coupled wth tme-stamp based algothms; the analyss of schedulng algothms based on a deent achtectue s not staghtfowad. The LR-class povdes a moe natual appoach fo analyzng the wost-case behavo of tac-schedulng algothms, ndependent of the schedule achtectue. Fnally, Golestan ecently pesented a delay analyss of a class of fa-queueng algothms ncludng Self-Clocked Fa Queueng [18]. Howeve, ths analyss does not apply to unfa algothms lke VtualClock. In addton to the delay analyss, we also study the faness chaactestcs of LR-schedules. The faness analyss was motvated by Golestan's wok [5], whee a self-contaned appoach fo faness was dened. Ths appoach s based on compang the nomalzed sevce oeed to any two connectons that ae contnuously backlogged ove an nteval of tme. We wll analyze many well-known schedulng algothms belongng to the LR class usng ths appoach. The est of ths pape s oganzed as follows: In Secton 2, we dscuss the necessay popetes that a schedulng algothm must satsfy fo applcaton n a eal netwok. In Secton 3, we dene the class of Latency-Rate (LR) seves and deve uppe bounds on the end-to-end delays, bue equements, and tac bustness of ndvdual sessons n an abtay-topology netwok of LR seves when the sesson tac s shaped by a leaky bucket. In Secton 4 we pove that seveal 3

5 well-known schedulng algothms, such as VtualClock, Packet-by-Packet Genealzed Pocesso Shang (PGPS), Self-Clocked Fa Queueng (SCFQ), Weghted-Round-Robn (WRR), and Dect- Round-Robn (DRR) all belong to the class of LR seves, and deve the latences. In Secton 5, we deve a slghtly mpoved uppe bound on end-to-end delay n a netwok of LR seves by boundng the sevce of the last seve on the path moe tghtly. In Secton 6, we analyze the faness popetes of these algothms. Fnally, Secton 7 pesents some conclusons and dectons fo futue wok. Appendx A contans the poofs of many lemmas and theoems pesented n the pape. 2 A Common Famewok fo Schedulng Algothms Schedulng algothms fo output-bueed swtches have been classed based on two ctea: wok-consevaton and ntenal achtectue. Nethe of these classcatons povdes us wth a common famewok that wll allow evaluaton of the elatve pefomance n eal netwoks. In ths secton we dscuss the thee mpotant attbutes of schedulng algothms that ae most mpotant n the applcaton n eal netwoks. These ae () delay behavo, () faness, and () mplementaton complexty. We wll, theefoe, compae schedules along these thee dmensons. 2.1 End-to-End Delay Guaantees The algothm must povde end-to-end delay guaantees fo ndvdual sessons, wthout seveely unde-utlzng the netwok esouces. In ode to povde a detemnstc delay bound, t s necessay to bound the bustness of the sesson at the nput of the netwok. The most common appoach fo boundng the bustness of nput tac s by shapng though a leaky bucket [19]. Seveal pevous studes have used ths tac model [4, 13, 14, 16]. We use the same tac model n ou devatons of end-to-end sesson delays and assume that the tac of sesson s smoothed though a leaky bucket wth paametes ( ; ), whee s the maxmum bustness and s the aveage aval ate. In devng the end-to-end delay bound fo a patcula sesson, howeve, we do not make any assumptons about the tac fom the est of the sessons shang the same lnks of the netwok. In addton to mnmzng the end-to-end delay n a netwok of seves, the delay behavo of an deal algothm ncludes the followng attbutes: 1. Insenstvty to tac pattens of othe sessons: Ideally, the end-to-end delay guaantees fo a sesson should not depend on the behavo of othe sessons. Ths s a measue of the level of solaton povded by the schedule to ndvdual sessons. Note that solaton s necessay even when polcng mechansms ae used to shape all the ows at the enty pont of the netwok, as the ows may accumulate bustness wthn the netwok. 2. Delay bounds that ae ndependent of the numbe of sessons shang the outgong lnk: Ths s necessay f the algothm s to be used n swtches suppotng a lage numbe of ows. 3. Ablty to contol the delay bound of a sesson by contollng only ts bandwdth esevaton: Ths popety of the algothm povdes sgncant exblty n tadng o sesson delays wth the bandwdth allocatons. 4

6 Note that the thee attbutes ae elated. We wll show late that the wost-case delay behavo of a sesson can de geatly n two deent schedules wth dentcal bandwdth esevatons. 2.2 Faness Sgncant dscepances may exst n the sevce povded to deent sessons ove the shot tem among schedulng algothms. Some schedules may penalze sessons fo sevce eceved n excess of the esevatons at an eale tme. Thus, a backlogged sesson may be staved untl othes eceve an equvalent amount of nomalzed sevce, leadng to shot-tem unfaness. Theefoe, two schedulng algothms capable of povdng the same delay guaantee to a sesson may exhbt vastly deent faness behavos. Whle thee s no common accepted method fo estmatng the faness of a schedulng algothm, t s easy to dene faness n an nfomal manne. In geneal, we would lke the system to always seve connectons popotonal to the esevatons and dstbute the unused bandwdth left behnd by dle sessons equally among the actve ones. In addton, sessons should not be penalzed fo excess bandwdth they eceved whle othe sessons wee dle. Followng Golestan's wok [5], we dene the faness paamete of a schedulng algothm as the maxmum deence between the nomalzed sevce eceved by two backlogged connectons ove an nteval n whch both ae contnuously backlogged. Based only on the end-to-end delay bounds and faness popetes, Genealzed-Pocesso- Shang (GPS) s an deal schedulng dscplne [4]. GPS multplexng s dened wth espect to a ud-model, whee packets ae consdeed to be nntely dvsble. The shae of bandwdth eseved by sesson s epesented by a eal numbe. Let B(; t) be the set of connectons that ae backlogged n the nteval (; t]. If s the ate of the seve, the sevce W (; t) oeed to a connecton that belongs n B(; t) s popotonal to. That s, W (; t) Pj2B(;t) j (t? ): The mnmum sevce that a connecton can eceve n any nteval of tme s P Vj=1 (t? ); j whee V s the maxmum numbe of connectons that can be backlogged n the seve at the same tme. Thus, GPS seves each backlogged sesson wth a mnmum ate equal to ts eseved ate at each nstant; n addton, the excess bandwdth avalable fom sessons not usng the esevatons s dstbuted among all the backlogged connectons at each nstant n popoton to the ndvdual esevatons. Ths esults n pefect solaton, deal faness, and low end-to-end sesson delays. The VtualClock algothm, n contast, does not bound the deence n sevce eceved by two backlogged sessons ove an nteval that s smalle than the backlogged peod. Ths s the esult of the schedule pefomng an aveagng pocess on the ate of sevce povded to ndvdual sessons. In VtualClock, the aveagng nteval can be abtaly long. The GPS schedule, on the othe hand, occupes the opposte exteme whee no memoy of past bandwdth usage of sessons s mantaned. Note that, accodng to ou denton of faness, some amount of 5

7 shot-tem unfaness between sessons s nevtable n any packet-level schedule, snce each packet must be sevced exclusvely. In pactce, we can only eque that the deence n nomalzed sevce eceved by two sessons be bounded by a constant. 2.3 Implementaton Complexty Fnally, schedules de geatly n the mplementaton complexty. The schedulng algothm may need to be mplemented n hadwae n a hgh-speed netwok. In addton, t s desable to have the tme-complexty of the algothm not depend on the numbe of actve connectons n the schedule. If V s the maxmum numbe of connectons that may shae an output lnk, the mplementaton of a schedule based on the soted-poty achtectue nvolves thee man steps fo pocessng each cell [12]: 1. Calculaton of the tmestamp: The PGPS schedule has the hghest complexty n ths espect, snce a GPS schedule must be smulated n paallel n ode to update the vtual tme. Ths smulaton may esult n a pocess ovehead of O(V ) pe packet tansmsson n the wost-case. On the othe hand, n both VtualClock and self-clocked fa queueng the tmestamp calculaton nvolves only a constant numbe of computatons, esultng n a wost-case complexty of O(1). 2. Inseton n a soted poty lst: The st cell of each sesson's queue must be stoed n a soted poty lst. When a cell aves nto an empty queue, ts nseton nto the poty lst eques O(log V ) steps. 3. Selecton of the cell wth the mnmum tmestamp fo tansmsson: Snce the cells ae stoed n a soted-poty stuctue, the cell wth the hghest poty may be eteved n O(log V ) tme [20]. The last two opeatons ae dentcal fo any soted-poty achtectue. A paallel mplementaton of these opeatons wth O(1) tme complexty by usng a set of O(V ) smple pocessng elements has been shown [21, 22]. Fame-based algothms such as Weghted Round Robn and Dect Round Robn can be mplemented n O(1) tme, wthout any tmestamp calculatons. Unfotunately, these algothms yeld delay bounds that may gow lnealy wth the numbe of sessons shang the outgong lnk. Thus, n pactce, the schedulng algothm must tade o the complexty of mplementaton wth the othe desable popetes of low delay and bounded shot-tem unfaness. 3 LR-Seves In ths secton we ntoduce LR-seves and deve some key esults on the delay behavo. Ths model wll allow us to deve detemnstc bounds on end-to-end delays n an abtay topology netwok. In addton, t wll help us dene the necessay popetes of a schedulng algothm to utlze the netwok esouces ecently. We assume a packet swtch whee a set of V connectons shae a common output lnk. The tems connecton, ow, and sesson wll be used synonymously. We denote wth the ate allocated to connecton. 6

8 A ρ ρ t1 t2 t3 t4 Fgue 3.1: Intevals (t 1 ; t 2 ] and (t 3 ; t 4 ] ae two deent busy peods. We assume that the seves ae non-cut-though devces. Let A (; t) denote the avals fom sesson dung the nteval (; t] and W (; t) the amount of sevce eceved by sesson dung the same nteval. In a system based on the ud model, both A (; t) and W (; t) ae contnuous functons of t. Howeve, n the packet-by-packet model, we assume that A (; t) nceases only when the last bt of a packet s eceved by the seve; lkewse, W (; t) s nceased only when the last bt of the packet n sevce leaves the seve. Thus, the ud model may be vewed as a specal case of the packet-by-packet model wth nntesmally small packets. Denton 1: A system busy peod s a maxmal nteval of tme dung whch the seve s neve dle. Dung a system busy peod the seve s always tansmttng packets. Denton 2: A backlogged peod fo sesson s any peod of tme dung whch packets belongng to that sesson ae contnuously queued n the system. Let Q (t) epesent the amount of sesson tac queued n the seve at tme t, that s, Q (t) = A (0; t)? W (0; t): A connecton s backlogged at tme t f Q (t) > 0. Denton 3: A sesson busy peod s a maxmal nteval of tme ( 1 ; 2 ] such that fo any tme t 2 ( 1 ; 2 ]; packets of sesson ave wth ate geate than o equal to, o, A (; t) (t? 1 ): A sesson busy peod s the maxmal nteval of tme dung whch f the sesson wee sevced wth exactly the guaanteed ate, t would be eman contnuously backlogged (Fgue 3.1). Multple sesson- busy peods may appea dung a system busy peod. The sesson busy peod s dened only n tems of the aval functon and the allocated ate. It s mpotant to ealze the basc dstncton between a sesson backlogged peod and a sesson busy peod. The latte s dened wth espect to a hypothetcal system whee a backlogged connecton s sevced at a constant ate, whle the fome s based on the actual system whee 7

9 Offeed Sevce A( τ, t ) W( τ, t ) τ θ ρ Tme Fgue 3.2: An example of the behavo of an LR schedule. the nstantaneous sevce ate vaes accodng to the numbe of actve connectons and the sevce ates. Thus, a busy peod may contan ntevals dung whch the actual backlog of sesson tac n the system s zeo; ths occus when the sesson eceves an nstantaneous sevce ate of moe than dung the busy peod. Fo a gven sesson- backlogged peod, the coespondng busy peod can be longe. In addton, f (s 1 ; f 1 ] s a busy peod fo sesson, multple backlogged peods may occu n the actual system dung the nteval (s 1 ; f 1 ]. The begnnng of a busy peod of sesson always maks the begnnng of a backlogged peod, but the convese s not always tue. In addton, the begnnng of a busy peod s always caused by the aval of a packet nto the system. A busy peod cannot end befoe the backlogged peod that stated t. Note that, when the same tac dstbuton s appled to two deent schedules wth dentcal esevatons, the esultng backlogged peods can be qute deent. Ths makes t dcult to use the sesson-backlogged peod fo analyzng a boad class of schedules. The sesson busy peod, on the othe hand, depends only on the aval patten of the sesson and ts allocated ate, and can theefoe be used as an nvaant n the analyss of deent schedules. Ths s the fundamental eason why the followng denton of an LR-seve s based on the sevce eceved by a sesson ove a busy peod. Snce we ae nteested n a wost-case analyss of the system, the sesson busy peod povdes us a convenent means to bound the delay wthn the system. We can now dene the geneal class of Latency-Rate (LR) seves. A seve n ths class s chaactezed by two paametes: latency and allocated ate. Let us assume that the jth busy peod of sesson stats at tme. We denote by W;j S (; t) the total sevce povded to the packets of the sesson that aved afte tme and untl tme t by seve S. Notce that the total sevce oeed to sesson n ths nteval (; t] may actually be moe than W;j S (; t) snce some packets fom a pevous busy peod, that ae stll queued n the system, may be sevced as well. Denton 4: A seve S belongs n the class LR f and only f fo all tmes t afte tme that the jth busy peod stated and untl the packets that aved dung ths peod ae sevced, W S ;j(; t) max(0; (t?? S )): S s the mnmum non-negatve numbe that satses the above nequalty. 8

10 The denton of LR seves nvolves only the two paametes, latency and ate. The ght-hand sde of the above equaton denes an envelope to bound the mnmum sevce oeed to sesson dung a busy peod (Fgue 3.2). It s easy to obseve that the latency S epesents the wost-case delay seen by the st packet of a busy peod of sesson. Notce that the estcton mposed s that the sevce povded to a sesson fom the begnnng of ts busy peod s lowebounded. Thus, fo a schedulng algothm to belong to ths class, t s only equed that the aveage ate of sevce oeed by the schedule to a busy sesson, ove evey nteval statng at tme S fom the begnnng of the busy peod, s at least equal to ts eseved ate. Ths s much less estctve than GPS multplexng, whee the nstantaneous bandwdth oeed to a sesson s bounded. That s, the lowe bound on the sevce ate of GPS multplexng holds fo any nteval (; t] that a sesson s backlogged, wheeas n LR-seves the estcton holds only fo ntevals statng at the begnnng of the busy peod. Theefoe, GPS multplexng s only one membe of the LR class. The latency paamete depends on the schedulng algothm used as well as the allocated ate and tac paametes of the sesson beng analyzed. Fo a patcula schedulng algothm, seveal paametes such as ts tansmsson ate on the outgong lnk, numbe of sessons shang the lnk, and the allocated ates, may nuence the latency. Howeve, we wll now show that the maxmum end-to-end delay expeenced by a packet n a netwok of schedules can be calculated fom only the latences of the ndvdual schedules on the path of the sesson, and the tac paametes of the sesson that geneated the packet. Snce the maxmum delay n a schedule nceases dectly n popoton to ts latency, the model bngs out the sgncance of usng lowlatency schedules to acheve low end-to-end delays. Lkewse, uppe bounds on the queue sze and bustness of ndvdual sessons at any pont wthn the netwok can be obtaned dectly fom the latences of the schedules. In ou denton of LR seves, we made no assumptons on whethe the seve s based on a ud model o a packet-by-packet model. The only equement that we mpose, howeve, s that a packet s not consdeed as depatng the seve untl ts last bt has depated. Theefoe, packet depatues must be consdeed as mpulses. Ths assumpton s needed to bound the avals nto the next swtch n a chan of schedules. We wll emove ths assumpton late fom the last seve of a chan to povde a slghtly tghte bound on the end-to-end sesson delay n a netwok of schedules. In a ud system, we eque that all schedules opeate on a ud bass and the maxmum packet sze to be nntesmally small. We wll now pesent delay bounds fo LR schedules. We wll st consde the behavo of a sesson n a sngle node, and subsequently extend the analyss to netwoks of LR seves. In both cases, we wll assume that the nput tac of the sesson we analyze s leaky-bucket smoothed and the allocated ate s at least equal to the aveage aval ate. That s, f s the sesson unde obsevaton, ts avals at the nput of the netwok dung the nteval (; t] satsfy the nequalty A (; t) + (t? ); (3.1) whee and denote ts bustness and aveage ate, espectvely. Howeve, we make no assumptons about the nput tac of othe sessons. 9

11 3.1 Analyss of a Sngle LR Seve Assume a set of V sessons shang the same output lnk of an LR seve. Fst, we show that the packets of a busy peod n the seve complete sevce no late than S afte the busy peod ends. Lemma 1: Let (t 1 ; t 2 ] be a sesson- busy peod. All packets that aved fom sesson dung the busy peod wll be sevced by tme t 2 + S. Poof: Let us assume that the last packet of the jth busy peod completes sevce at tme t. Then at t, A (t 1 ; t 2 ) = W S ;j (t 1; t): (3.2) But we know fom the denton of the busy peod that Fom the denton of LR seves and equatons (3.2) and (3.3), o equvalently, A (t 1 ; t 2 ) = (t 2? t 1 ): (3.3) (t? t 1? S )? (t 2? t 1 ) 0 (3.4) t t 2 + S : (3.5) 2 The followng theoem bounds the queueng delays wthn the seve, as well as the bue equements, fo sesson. Theoem 1: If S s an LR-seve, then the followng bounds must hold: 1. If Q S (t) s the backlog of sesson at tme t, then 2. If D S s the delay of any packet of sesson n seve S, Q S (t) + S : (3.6) D S + S : (3.7) 3. The output tac confoms to the leaky bucket model wth paametes ( + S ; ). Poof: We wll pove each of the above statements sepaately. 1. Uppe bound on sesson- backlog: Assume that the jth busy peod coveed the tme nteval (s j ; f j ]. Let us denote wth W;j S (; t) the sevce oeed to the packets of the jth busy peod of sesson n the nteval (; t], wth s j and t f j + S. We wll denote by W S (; t) the total sevce oeed to sesson dung the same nteval of tme. Note that W S (; t) W ;j S (; t), snce, dung the nteval (; t] the seve may tansmt packets of a pevous busy peod as well. Dung a system busy peod, let us denote wth j the tme at whch the last packet of the jth busy peod completes sevce. At ths pont of tme all backlogged packets belong to busy peods that stated afte tme f j. Note also that j f j + S. We wll pove the theoem by nducton ove the tme ntevals ( j?1 ; j ]. 10

12 Base step: Assume that a sesson busy peod stated at tme s 1 = 0. Snce ths s the st busy peod fo sesson, thee ae no othe packets backlogged fom sesson at 0. By the denton of LR-seves, we know that fo evey tme t wth 0 t 1, Fom the denton of leaky bucket, W S ;1( 0 ; t) = W S ;1(s 1 ; t) max(0; (t? s 1? S )): A (s 1 ; t) + (t? s 1 ): The backlog at tme t s dened as the total amount of tac that aved snce the begnnng of the system busy peod mnus those packets that wee sevced. Theefoe, Q (t) = A (s 1 ; t)? W S ;1(s 1 ; t): (3.8) We wll consde two cases fo t: Case 1: If t < s 1 + S then, Theefoe, W S ;1(s 1 ; t) 0: Q S (t) A (s 1 ; t) + (t? s 1 ) + S : (3.9) The st nequalty follows fom the denton of the leaky bucket and the second fom the estcton t < s 1 + S. Case 2: If t s 1 + S, then fom the denton of LR-seve, Fom equatons (3.1) and (3.8), W S ;1 (s 1; t) (t? s 1? S ): Q S (t) + (t? s 1 )? (t? s 1? S ) + S : (3.10) Inductve step: We wll assume that the theoem holds untl the end of the nth busy peod. That s, fo evey tme t n, Q S (t) + S. We wll now pove that t holds also dung the nteval ( n ; n+1 ]. Afte tme n, only packets fom the (n + 1)th busy peod wll be sevced, and by tme n+1 all of the packets that belong n the (n + 1)th busy peod wll complete sevce. In addton, no packet fom the (n + 1)th busy peod has been sevced befoe tme n. Theefoe, fo evey tme t wth n < t n+1, we can wte Fom the denton of the LR-seves, W S ;n+1( n ; t) = W S ;n+1(s n+1 ; t): W S ;n+1( n ; t) max(0; (t? s n+1? S )): 11

13 The amount of packets that ae backlogged n the seve s equal to those packets that aved afte the end of the nth busy peod mnus those packets that wee sevced afte tme n ; but we know that, snce s n+1 s the begnnng of the (n + 1)th busy peod, the st packet of the busy peod aved at tme s n+1. Theefoe, Q S (t) A (s n+1 ; t)? W S ;n+1( n ; t) + (t? s n+1 )? max(0; (t? s n+1? S )) + S : 2. Uppe bound on delay: Let us assume that the maxmum delay D S was obtaned fo a packet that aved at tme t dung the jth busy peod. Ths means that the packet was sevced at tme t + D S. Hence, the amount of sevce oeed to the sesson untl tme t + D S s equal to the amount of tac that aved fom the sesson untl tme t. Snce s j s the begnnng of the jth busy peod, W;j(s S j ; t + D S ) = A (s j ; t ): (3.11) Snce the seve may povde no sevce fo tme S, DS S. Fom the denton of LR-seve, W S ;j(s j ; t + D S ) (t + D S? s j? S ): (3.12) Fom (3.11),(3.12), and the leaky bucket constant (3.1), we have o equvalently, (t + D S? s j? S ) + (t? s j ); (3.13) D S + S : (3.14) 3. Uppe bound on bustness of output tac: We wll now pove that the output tac of the schedule confoms to the leaky-bucket model as well. It s sucent to show that, dung any nteval (; t] wthn a sesson- busy peod, W (; t) + S + (t? ): Let us denote wth out the bustness of sesson at the output of the swtch. We wll ty to nd the maxmum value fo ths bustness. We wll assume that the seve s wok-consevng and that t can sevce any amount of tac fom a gven sesson wthout nteupton, f the gven sesson s the only actve sesson n the system. Let us denote wth c () the amount of tokens that eman n the leaky bucket at tme. Q S () s the amount of backlogged packets n the seve at tme. If we assume that we have nnte-capacty nput lnks, at tme + t s possble that an amount of c () packets s added to the seve queue and no packet s sevced; but we aleady poved that the maxmum backlog of the sesson s bounded by + S. Theefoe, c () + Q S () + S : (3.15) The avals A (; t) cannot exceed the amount of tokens n the leaky bucket at tme plus the amount of tokens that aved dung the nteval (; t], mnus the amount of tokens at tme t. That s, A (; t) c () + (t? )? c (t): (3.16) 12

14 We can also calculate the sevce oeed to sesson n the nteval (; t] as Then, fom (3.16) and (3.17), Theefoe, W S (; t) = Q () + A (; t)? Q (t) Q () + A (; t): (3.17) W S (; t) Q () + c () + (t? )? c (t) ( + S ) + (t? ): (3.18) out + S : (3.19) We can show that ths bound s tght. If sesson s the only actve sesson n the system, t wll be explctly sevced. Let us assume that the maxmum backlog fo ths sesson s eached at tme t. Afte t, packets ave fom the sesson wth ate. The seve wll st sevce the backlogged packets and then contnue sevcng wth ate. Theefoe, we have out + S : (3.20) Fom Equatons (3.19) and (3.20), we can conclude that out = + S Analyss of a Netwok of LR Seves In the pevous secton we consdeed a sngle LR-seve and analyzed the delay behavo of a sesson when the sesson tac s leaky-bucket contolled. We wll now poceed to pove bounds on backlog and delay ove multple nodes. The staghtfowad appoach would be to accumulate the maxmum delays ove each node. Ths appoach was used by Cuz [14]. Howeve, ths method gnoes the coelaton between avals at two seves n sees, and theefoe esults n vey loose bounds. Tghte bounds can be povded by followng the appoach used by Paekh and Gallage [16] that tes to captue the behavo of a sesson ove multple nodes at the same tme. The only estctons that we mpose n the netwok s that all the seves belong to the LR class and that the tac of sesson unde obsevaton s shaped at the souce wth a leaky bucket ( ; ). We wll also assume that the bandwdth esevaton of the sesson at evey node n ts path s at least. We st pove the followng lemma: Lemma 2: The tac pocess afte the kth node n a chan of LR seves s a leaky bucket pocess wth paametes + kx j=1 (S j ) ; and ; whee (S j ) s the latency of the jth schedule on the path of the sesson. 13

15 Poof: We aleady poved n Theoem 1, that the output tac of an LR seve confoms to the leaky bucket model wth paametes ( + S ; ). But ths means that the nput tac n the next node confoms to the same model. Theefoe, afte the (k? 1)th node, the output tac of sesson, and coespondngly the nput tac of node k, wll confom to a leaky bucket model. That s, 0 k?1 A k (; + X 1 (S j ) A + (t? ): (3.21) j=1 2 Ths s a an mpotant esult that allows us to bound the bustness of sesson- tac at each pont n the netwok. Notce that the ncease n bustness that a sesson may see n the netwok s popotonal to the sum of the latences of the seves t has tavesed. Theefoe, even a sesson wth no bustness at the nput of the netwok may accumulate a sgncant amount of bustness n the netwok because of the latency of the seves. We can now state the followng lemma that wll bound the backlog of sesson n each node of the netwok. Lemma 3: The maxmum backlog Q (S k ) (t) n the kth node of a sesson s bounded as Q (S k ) (t) + kx j=1 (S j ) : Poof: Let k denote the maxmum bustness of sesson at the output of the kth node. By Theoem 1 and Lemma 2 we have, Q (S k) k?1 + (S k ) + kx j=1 (S j ) : (3.22) 2 As a esult of the nceased bustness, the maxmum backlog and theefoe the maxmum bue equements fo each node n the netwok ae also popotonal to the latency of the seves. In Lemma 3 we bounded the maxmum backlog fo sesson n any node of the netwok. Ths bound may be seen as the maxmum numbe of bues needed n the coespondng node of the netwok to guaantee zeo packet loss fo sesson. Howeve, ths does not mean that we can bound the maxmum numbe of packets n the netwok fo sesson by addng the backlog bounds of each swtch on the path. Such a bound does not take nto account the coelaton among aval pocesses at the ndvdual nodes and theefoe can be vey loose. To deve tghte bounds fo sesson delay n a netwok of LR seves, we st show that the maxmum end-to-end delay n a netwok of two LR-seves n sees s the same as that n a sngle LR seve whose latency s the sum of the latences of the two seves t eplaces. Ths esult allows an abtay numbe of LR seves n the path of a sesson to be modeled by combnng the ndvdual latences. 14

16 Busy peod on node 1 Busy peods on node 2 1 W (t) A (t) α s 1 1 f 1 s 2 β f s 2 3 f 3 1 θ 1 Fgue 3.3: Illustaton of busy peods of a sesson n two seves n sees. The netwok busy peod fo sesson n ths case s splt nto multple busy peods n the second seve. The busy peod n the st seve s ( 1 ; 1 ]. The packets avng at the second seve fom ths busy peod fom multple busy peods (s 1 ; f 1 ]; (s 2 ; f 2 ]; (s 3 ; f 3 ] n the second seve. The lne wth slope that stats at 1 bounds all these busy peods. Analyzng two LR seves n sees ntoduces a dculty: If the st seve has non-zeo latency, the busy peod of a sesson n the second seve may not concde wth the coespondng busy peod n the st seve. That s, a packet that stated a busy peod n the st seve may not stat a busy peod n the second, but nstead mght ave whle a busy peod s n pogess at the second seve. Also, snce the actual sevce ate seen by sesson n the st seve can be moe than, a sngle busy peod n the st seve may esult n multple busy peods n the second seve. Ths s llustated n Fgue 3.3. We wll take these eects nto account n ou analyss of a netwok of LR seves. In the followng, we wll use the tem netwok busy peod to mean a busy peod of the sesson beng analyzed n the st seve on ts path. Smlaly, when we efe to sevce oeed by the netwok to sesson dung an nteval of tme, we mean the amount of sesson- tac tansmtted by the last seve on the path dung the nteval unde consdeaton. We wll use W ;j (; t) to denote the amount of sevce oeed by the netwok to sesson dung an nteval (; t] wthn ts jth netwok busy peod. Also, we wll use W 1 ;j ( 1; t 1 ) to denote the amount of sevce oeed by the st seve dung an nteval ( 1 ; t 1 ) wthn ts local busy peod, and W 2 ;j ( 2; t 2 ) the same paamete fo the second seve dung an nteval ( 2 ; t 2 ) wthn ts busy peod. We st pove the followng lemma to bound the sevce povded by the netwok to a sesson dung an nteval wthn a netwok busy peod. Lemma 4: Consde a netwok of two LR-seves S 1 and S 2 n sees, wth latences (S 1) and, espectvely. If s the ate allocated to sesson n the netwok, the sevce oeed by the (S 2) netwok to the packets of the jth netwok busy peod of that sesson dung an nteval (; t] wthn the busy peod s bounded as 15

17 W ;j (; t) max 0; (t?? ( (S 1) + (S 2) )) : Poof: We wll pove the lemma by nducton on the numbe of netwok busy peods. Let us denote wth ( j ; j ] the statng and endng tmes of the jth netwok busy peod of sesson. W;j 1 ( j; t) denotes the sevce oeed by the st seve S 1 to the packets of the jth netwok busy peod untl tme t. Note that the jth netwok busy peod s the same as the jth sesson-busy peod on the st seve. Howeve, the busy peods seen by the the second seve S 2 may be deent fom these peods. Let (s k ; f k ] denote the kth busy peod of sesson n S 2. Then W;k 2 (s k; t) s the sevce oeed by S 2 dung ts kth busy peod untl tme t. Let k denote the tme at whch packets fom the kth busy peod of sesson n S 2 stat sevce n S 2. Smlaly, let t k be the nstant at whch the last packet fom the kth busy peod n S 2 s completely sevced by S 2. Then, t s easy to obseve that k s k ; t k f k + (S 2) ; and t k k+1 : Base step: Consde any tme t dung the st netwok busy peod, that s, 1 t 1. When the st busy peod fo sesson stats thee ae no othe packets fom that sesson n the system. As obseved eale, the st netwok busy peod of sesson may esult n multple busy peods n the second seve. Let (s 1 ; f 1 ]; (s 2 ; f 2 ]; : : :; (s m ; f m ] be the successve busy peods n the second seve n whch packets fom the the st netwok busy peod wee sevced. Let t denote the last nstant of tme at whch a packet fom the st netwok busy peod was n sevce n S 2. We need to show that, fo any tme t, 1 t t, W ;1 ( 1 ; t) max 0; (t? 1? ( (S 1) + (S 2) )) : We need to consde thee sepaate cases fo t. Case 1: t 1. In ths case, no sevce has occued n the second seve, that s, W ;1 ( 1 ; t) = 0. Also, Theefoe, Hence, we can wte s (S 1) ; and 1 s 1 + (S 2) : t? 1 (S 1) + (S 2) : (3.23) W ;1 ( 1 ; t) max 0; (t? 1? ( (S 1) + (S 2) )) : (3.24) Case 2: k t t k, 1 k m (wthn the kth busy peod n S 2 ). But, W ;1 ( 1 ; t) = W ;1 ( 1 ; k ) + W ;1 ( k ; t) (3.25) W ;1 ( 1 ; k ) = W 1 ;1( 1 ; s k ); and W ;1 ( k ; t) = W 2 ;k( k ; t) = W 2 ;k(s k ; t): 16

18 Substtutng n eq. (3.25), W ;1 ( 1 ; t) = W;1( 1 1 ; s k ) + W;k(s 2 k ; t) max 0; (s k? 1? (S 1) max ) 0; (t? 1? ( (S 1) + (S 2) )) Case 3: t k t k+1 (between busy peods n S 2 ): We can wte + max 0; (t? s k? (S 2) : ) W ;1 ( 1 ; t) = W ;1 ( 1 ; k+1 )? W ;1 (t; k+1 ): (3.26) Snce no packet aves at the second seve between ts busy peods, the second tem on the ght-hand sde s zeo. Also, W ;1 ( 1 ; k+1 ) = W 1 ;1 ( 1; s k+1 ). Theefoe, eq. (3.26) educes to But, t k+1 s k+1 + (S 2), o Theefoe, we can ewte eq. (3.27) as W ;1 ( 1 ; t) = W 1 ;1 ( 1; s k+1 ) W ;1 ( 1 ; t) max max 0; (s k+1? 1? (S 1) ) : (3.27) s k+1 t? (S 2) : 0; (t? 1? ( (S 1) + (S 2) )) : If f m maks the last nstant of tme when a packet fom the st netwok busy peod was sevced by S 2, then we have accounted fo all the tac aved n the netwok dung ( 1 ; 1 ]. Howeve, t s possble that the tansmsson of packets aved dung ( 1 ; 1 ] extends nto the (m + 1)th busy peod n S 2. If so, let t be the last nstant of tme when a packet fom the st netwok busy peod was sevced by S 2. Then we need to consde the addtonal tme ntevals (t m ; m+1 ] and ( m+1 ; t ] n the poof. These can be handled exactly lke n case 3 and case 2, espectvely. Ths concludes the base step of the poof. Inductve step: We assume that fo all netwok busy peods 1; : : :; n, the lemma s tue, and thus W ;n ( n ; t) max 0; (t? n? ( (S 1) + (S 2) )) We wll now pove that, fo the (n + 1)th busy peod, W ;n+1 ( n+1 ; t) max 0; (t? n+1? ( (S 1) + (S 2) )) : In the smple case, the st sesson- packet of the (n + 1)th netwok busy peod also stats a busy peod n the second seve. Ths case can be handled exactly as n the base step. The moe dcult case occus f, when the (n + 1)th netwok busy peod stats, some packets of sesson fom eale busy peods ae stll backlogged n the second seve. That s, the begnnng of a netwok busy peod of sesson may not always concde wth the begnnng of a busy peod fo S 2, as was the case n the base step. Howeve, we wll now show that ths does not aect ou analyss. 17

19 Let us assume that the mth busy peod of S 2 s n pogess when the st packet fom the (n + 1)th netwok busy peod aves at S 2. Assume that the mth busy peod of S 2 stated at tme s m, and the st packet fom the (n + 1)th netwok busy peod was sevced by S 2 at tme. Then, W ;n+1 ( n+1 ; t) = W ;n+1 (; t) = W 2 ;m(s m ; t)? W 2 ;m(s m ; ) (3.28) Note that the mth busy peod cuently n pogess n S 2 may contan packets fom multple netwok busy peods, snce multple busy peods of S 1 may mege nto a sngle busy peod n S 2. Assume that the mth busy peod of S 2 contaned packets fom the netwok busy peods n? L; n? L + 1; : : :; n + 1. The total numbe of packets that ae seved dung the mth busy peod of the second seve untl tme s equal to the total numbe of packets seved fom the netwok busy peods n? L; n? L + 1; : : :; n mnus those packets seved fom the (n? L)-th busy peod befoe tme s m. Thus, o equvalently, nx j=n?l W 2 ;m (s m; ) = A ( j ; j ) = W 1 ;n?l( n?l ; s m ) + W 2 ;m(s m ; ); (3.29) nx nx j=n?l j=n?l A ( j ; j )? W 1 ;n?l ( n?l; s m ) ( j? j )? max(0; (s m? n?l? (S 1) )) ( n? n?l )? max(0; (s m? n?l? (S 1) )) (3.30) ( n? s m + (S 1) ) (3.31) Fom equatons (3.28) and (3.31), and the denton of LR seves, W ;n+1 ( n+1 ; t) = W 2 ;m(s m ; t)? W 2 ;m(s m ; ) max(0; (t? s m? (S 2) ))? ( n? s m + (S 1) max 0; (t? n? ( (S 1) + (S 2) )) max 0; (t? n+1? ( (S 1) + (S 2) ) : The last nequalty holds snce n n+1. Now, f the (n + 1)th netwok busy peod s late splt nto multple busy peods n S 2, we can use the same appoach used n the base step fo subsequent busy peods n S 2 to complete the poof. 2 Lemma 4 assets that the sevce oeed to a sesson n a netwok of two LR seves n sees s no less n the wost case than the sevce oeed to the sesson by a sngle LR seve whose latency s equal to the sum of the latences of the two seves t eplaces. Snce we make no assumptons on the maxmum sevce oeed by the seves to a sesson, we can mege an abtay numbe of seves connected n sees to estmate the sevce oeed afte the kth node. We can theefoe state the followng coollay. 18 )

20 Coollay 1: Let be the stat of the jth netwok busy peod of sesson n a netwok of LR seves. If s the mnmum bandwdth allocated to sesson n the netwok, the sevce oeed to packets of the jth netwok busy peod afte the kth node n the netwok s gven by W S k ;j (; t) max (t?? kx (S j ) ) j=1 whee (S j ) s the latency of the jth seve n the netwok fo sesson. Usng the above coollay we can bound the end-to-end delays of sesson f the nput tac s leaky-bucket shaped and the aveage aval ate s less than. Theoem 2: The maxmum delay D of a sesson n a netwok of LR seves, consstng of K seves n sees, s bounded as D + KX j=1 1 A ; (S j ) ; (3.32) whee (S j ) s the latency of the jth seve n the netwok fo sesson. Poof: Fom Coollay 1, we can teat the whole netwok as equvalent to a sngle LR-seve wth latency equal to the sum of the latences. By usng Theoem 1, we can dectly conclude that the maxmum delay s D + kx j=1 (S j ) : 2 Ths maxmum delay s ndependent of the topology of the netwok. The bound s also much tghte than what could be obtaned by analyzng each seve n solaton. Note that the end-to-end delay bound s a functon of only two paametes: the bustness of the sesson tac at the souce and the latences of the ndvdual seves on the path of the sesson. Snce we assumed only that each of the seves n the netwok belongs to the LR class, these esults ae moe geneal than the delay bounds due to Paekh and Gallage [16]. In the next secton, we wll show that all well-known wok-consevng schedules ae n fact LR seves. Thus, ou delay bound apples to almost any netwok of schedules. The delay bound n Eq. (3.32) shows that thee ae two ways to mnmze delays and bue equements n a netwok of LR seves: ) allocate moe bandwdth to a sesson, thus educng the tem =, o ) use LR seves wth low latences. Snce the latency s accumulated though multple nodes, the second appoach s pefeed n a lage netwok. The st appoach educes the utlzaton of the netwok, thus allowng only a smalle numbe of smultaneous sessons to be suppoted than would be possble wth mnmum-latency seves. Mnmzng the latency also mnmzes the bue equements of the sesson at the ndvdual seves n the netwok. Poposton 1: The end-to-end delay and ncease n bustness of a sesson n a netwok of LR seves s popotonal to the latency S of the seves. We can mnmze both of these paametes by desgnng seves wth mnmum latency. 19

21 Note that the latency of a seve depends, n geneal, on ts ntenal paametes and the bandwdth allocaton of the sesson unde consdeaton. In addton, the latency may also vay wth the numbe of actve sessons and the allocatons. Such a dependence of the latency of one sesson on othe sessons ndcates the poo solaton popetes of the schedule. Lkewse, n some schedules the latency may depend heavly on ts ntenal paametes, and less on the bandwdth allocaton of the sesson unde obsevaton. Such schedules do not allow us to contol the latency of a sesson by contollng ts bandwdth allocaton. On the othe hand, the latency of a PGPS seve depends heavly on the allocated bandwdth of the sesson unde consdeaton. Ths exblty s geatly desable. 3.3 Delay Bound fo Sessons wth Known Peak Rate Snce the denton of an LR seve s not based on any assumptons on the nput tac, t s easy to deve delay bounds fo tac dstbutons othe than the (; ) model. Fo example, when the peak ate of the souce s known, a moded uppe bound on the delay of an LR seve can be obtaned. Let us denote wth g the sevce ate allocated to connecton, and let and P espectvely denote the aveage and peak ate at the souce of connecton. The avals at the nput of the seve dung the nteval (; t] now satsfy the nequalty A (; t) mn ( + (t? ); P (t? )) : (3.33) We can pove the followng lemma: Lemma 5: The maxmum delay D of a sesson n an LR seve, whee the peak ate of the souce s known, s bounded as D S P? g g P? + S : (3.34) Poof: Let us assume that the maxmum delay D S was obtaned fo a packet that aved at tme t dung the jth busy peod. Ths means that the packet was sevced at tme t + D S. Hence, the amount of sevce oeed to the sesson untl tme t + D S s equal to the amount of tac that aved fom the sesson untl tme t. Snce s j s the begnnng of the jth busy peod, Fom Eq. (3.33), ths becomes W S ;j (s j; t + D S ) = A (s j ; t ): (3.35) W S ;j (s j; t + D S ) mn ( + (t? s j ); P (t? s j )) : (3.36) Snce the seve may povde no sevce fo tme S, DS S. Fom the denton of LR-seve, W;j(s S j ; t + D S ) g (t + D S? s j? S ): (3.37) Fom (3.36) and (3.37), we have g (t + D S? s j? S ) mn ( + (t? s j ); P (t? s j )) : (3.38) 20

22 Case 1: When P (t? s j ) + (t? s j ), we have and D S P? g g (t? s j ) + S ; (3.39) Substtutng fo (t? s j ) n Eq. (3.39), we get P (t? s j ) + (t? s j ); o, t? s j : P? D S P? g g Case 2: When P (t? s j ) > + (t? s j ), we get Fom Eq. (3.38), and by substtutng fo t? s j fom Eq. (3.41), P? + S : (3.40) (t? s j ) > P? : (3.41) (g? )(t? s j )? g D S + g S ; (3.42) D S P? g g P? + S : (3.43) 2 A netwok of LR seves can be modeled as a sngle LR seve wth latency equal to the sum of the latences. Thus, the followng man esult can be deved: Coollay 2: The maxmum delay D of a sesson n a netwok of LR seves, consstng of K seves n sees, whee the peak ate of the souce s known, s bounded as D P? g g P? + KX j=1 S j : (3.44) whee S j s the latency of the jth node. 4 Schedules n the Class LR In ths secton we wll show that seveal well-known wok-consevng schedules belong to the class LR and detemne the latences. Recall that ou denton of LR seves n the pevous secton s based on sesson-busy peods. In pactce, howeve, t s ease to analyze schedulng algothms based on sesson backlogged peods. The followng lemma enables the latency of an LR seve to be estmated based on ts behavo n the sesson backlogged peods. We wll use ths as a tool n ou analyss of seveal schedules n ths secton. 21