bounds compared to SB and SBB bounds as the former two have an index parameter, while the latter two

Transcription

1 1 Queung Procee n GPS and PGPS wth LRD Traffc Input Xang Yu, Ian L-Jn Thng, Yumng Jang and Chunmng Qao Department of Computer Scence and Engneerng State Unverty of New York at Buffalo Department of Electrcal and Computer Engneerng Natonal Unverty of Sngapore Inttute for Infocomm Reearch, Sngapore Abtract Long range dependent (LRD) traffc whoe ngle erver queue proce Webull Bounded (WB) frt analyzed. Two upper bound on the ndvdual eon queue length of LRD traffc under the generalzed proceor harng (GPS) chedulng dcplne are then contrbuted. It hown that the ndex parameter n the upper bound of one LRD flow, (n addton to the decay rate and the aymptotc contant), may be affected by other LRD flow. A new concept, called LRD olaton, ubequently contrbuted and accompanyng t, a new technque contrbuted to check whether a flow, wth a gven GPS weght agnment, can be guaranteed to be LRD olated. Th technque alo amenable for ue n an onlne call admon control (CAC) cenaro. When extng flow have already been agned contract weght that cannot be changed, our technque can be ued to determne mnmum contract weght to be agned to a new flow n order to guaranteed the flow to be LRD olated. The reult are alo extended to a PGPS (Packet-baed GPS) cheduler and relevant numercal reult are provded to how the uefulne of our bound and LRD olaton technque. I. Introducton Schedulng dcplne ued by wtche or router are mportant to provde QoS (qualty of ervce) upport for ntegrated applcaton comprng of voce, vdeo and data traffc. An deal chedulng dcplne hould atfy two requrement: (a) Provde olaton between eon (where olaton mean that the queung proce behave no wore than t Sngle Server Queue (SSQ) proce wth a comparable ervce rate). Th guarantee that the chedulng dcplne able to protect an ndvdual flow 1 agant mbehavor from other flow. (b) Realze tattcal multplexng gan. Th ugget a flow can utlze exce ervce rate allocated to other. Perhap the mot wdely tuded non-fcfs chedulng dcplne the Generalzed Proceor Sharng (GPS) dcplne. GPS ha two attractve charactertc: (1) each backlogged eon guaranteed a mnmum ervce rate. Th enure that the mbehavor of other flow ha a lmted effect on an ndvdual eon, and provde the foundaton of olaton between eon. Achevng olaton further enable GPS to guarantee dfferentated QoS for n- Work upported by MOE Reearch Project R and Sngaren Reearch Project R A prelmnary veron of th paper appeared n IEEE Globecom [1]. 1 whch wll be ued nterchangeably wth the term eon. dvdual eon. (2) t work-conervng, and any exce ervce rate can be redtrbuted amongt backlogged flow. The econd charactertc enable GPS to obtan tattcal multplexng gan between nput flow. Becaue of thee two charactertc, GPS deemed an deal chedulng dcplne that can atfy the two requrement (a) and (b) mentoned above, and thu ha attracted a lot of reearch nteret. When GPS extended to packet wtched network, t uually referred to a Weghted Far Queung (WFQ) or Packet-baed GPS (PGPS) [2]. Long range dependent (LRD) traffc an ncreangly mportant cla of traffc n modern day network becaue long range dependency exhbted n Ethernet traffc [3], WWW traffc [4], compreed vdeo traffc [5,6], TCP traffc [7] and o on. Snce LRD traffc ha burtne extendng over varou tme cale, a Webull bound rather than a conventonal exponental bound uually aocated wth LRD traffc SSQ proce [8]. Achevng dfferentated QoS n a GPS queue reman a challengng tak, epecally for LRD traffc. Pror analy of GPS are often baed on nput traffc tream whch atfy ome well known burtne contrant that are ether determntc [9] or tochatc [10, 11] n nature. In partcular, the conventonal tochatc traffc, whoe aocated queue length dtrbuton ha an exponental form, ha been demontrated to have an Exponental Bounded Burtne (EBB) arrval proce [12]. Baed on uch contrant on the arrval proce, an upper bound ha been developed for the ndvdual eon queue length n a GPS ytem [10, 11]. However, few analy of GPS baed on LRD arrval procee have been carred out. Th paper contrbute new nght nto the behavor of the GPS erver when chedulng everal LRD flow wth arrval proce exhbtng what we call Webull Bounded Burtne (WBB) contrant. The relatonhp between LRD traffc and the Webull bound expreon tem much from emprcal tude demontratng that the queung proce of a work conervatve SSQ, whoe nput traffc exhbt LRD behavor, can be commonly characterzed by a Webull Bounded (WB) expreon [13]. The work preented here applcable to LRD traffc that are WB n t SSQ proce (although not all LRD traffc can be WB n t SSQ proce). Note that the term WB refer to the queung proce whle the term

2 2 WBB we ntroduced earler refer to the arrval proce. The relatonhp between a WBB arrval proce and a WB queung proce wll be etablhed n later ecton. In order to acheve QoS n a GPS ytem whoe nput are LRD flow, ue related to olaton between LRD flow n GPS ytem need to be carefully characterzed. Recent reearch contrbuton by Bort [14 16] have hown that under certan condton, a LRD (or heavy taled) traffc flow can be well olated from other nput flow n a GPS ytem [14,16] whle n ome other cenaro, t may nhert burtne from other flow [15]. However, good theoretcal bound for an ndvdual eon queue length n the GPS ytem reman to be found. More pecfcally, whle Bort work valuable n provdng a method to judge whether an ndvdual eon can be olated from other nput eon under the GPS chedulng dcplne, the problem of how to guarantee the olaton of an ndvdual eon by tunng GPS parameter, or how to determne the QoS performance of an ndvdual eon that not olated from other eon reman unolved. In th paper, we derve two bound for ndvdual queung procee n a GPS ytem wth LRD traffc nput, each wth t own advantage and dadvantage (to be pelt out n detal later). Thee bound provde valuable nght nto the olaton between multple GPS eon. More pecfcally, t hown that the ndex parameter n the upper bound of one LRD flow, (n addton to the decay rate and the aymptotc contant), may be affected by other LRD flow. In addton, we ntroduce a ueful noton called LRD olaton whch dffer from the uual noton of flow olaton n that a flow guaranteed to be LRD olated from other flow a long a t ndex parameter not affected by other flow. We alo develop a neceary and uffcent condton for a flow beng guaranteed to be LRD olated from other flow. Baed on th condton, a technque that can be ued to quckly check f flow can be guaranteed to be LRD olated from other flow wth a gven (GPS ervce) weght agnment propoed. When ome flow have already been agned contract weght (accordng to ome Servce Level Agreement or SLA) that cannot be changed, the propoed technque can alo be ued to determne the mnmum contract weght to be agned to flow n order for t to be guaranteed to be LRD olated from other flow, and thu epecally ueful for on-lne call admon control (CAC). The reult are alo extended to a PGPS (Packet-baed GPS) cheduler o that numercal mulaton can be preented to llutrate the uefulne of the bound and the propoed technque. Our contrbuton are dfferent from thoe preented n [17, 18] whch condered hort range dependent model. The work n [19] tuded Stochatcally Bounded (SB) SSQ and Stochatcally Bounded Burtne (SBB) arrval procee. One of the dfference that whle SB and SBB are dcrete tme procee, WB and WBB procee tuded n th work are contnuou tme procee. In addton, the WB and WBB bound developed n th paper are pecalzed bound compared to SB and SBB bound a the former two have an ndex parameter, whle the latter two do not. Accordngly, not only are the reult contrbuted n [19] not drectly applcable, t alo nontrval to derve the WB and WBB bound (a well a other reult n th paper). Reference [20] dd provde tattcal bound for queung delay n GPS ytem for the LRD traffc cenaro. However, t focu wa on the admon regon of GPS wth tattcal reource harng rather than the olaton between ndvdual queung procee. The bound provded were not derved from the queue length dtrbuton but were obtaned va certan nput/output traffc envelope approxmaton and a Gauan dtrbuton for each cla of traffc. Another gnfcant dvergence from conventonal LRD traffc analy whch uually begn wth aocated queung proce that we begn our dervaton from the arrval proce of LRD traffc. Th brng more valdty n all ubequent queue analy, nce only when the arrval proce bounded, the tablty of the queung proce can be enured. The paper organzed a follow. In Secton II, we revew fundamental knowledge related to conventonal LRD traffc analy and the GPS dcplne. In Secton III, we preent the relatonhp between a WB SSQ and a WBB arrval proce, and then derve everal upper bound on the aggregate WB SSQ procee. In Secton IV, the concept of feable orderng revewed, and the noton of LRD olaton ntroduced. Two upper bound related to the ndvdual eon queue length n a GPS ytem are alo derved. In Secton V, a neceary and uffcent condton to guarantee a flow to be LRD olated from other flow contrbuted, and a technque for determnng a mnmum contract weght to be agned to a flow to enure t LRD olaton alo propoed and llutrated va numercal example. Reult n GPS are further extended to PGPS ytem n Secton VI, and n Secton VII, we demontrate, va numercal mulaton of a two-nput PGPS ytem, the uefulne of the two bound. Secton VIII conclude the paper. II. Prelmnare In th ecton, we brefly revew the fundamental knowledge on the queue length dtrbuton of LRD traffc and the GPS chedulng dcplne. A. LRD Traffc and WB SSQ LRD traffc often characterzed by heavy traffc burt that extend over a wde range of tme cale [21, 22]. The LRD traffc backlog, buffered wthn a SSQ, often poee a tal dtrbuton that decay lower than that of tradtonal (e.g., Poon) traffc. More pecfcally, the queue length dtrbuton of tradtonal traffc obey a certan exponental form. For the cae of LRD traffc, the Webull dtrbuton often ued to characterze the lower decayng SSQ dtrbuton [5, 23, 24]. Th relatonhp between a WB SSQ dtrbuton and LRD traffc uually baed on emprcal obervaton. The queue length dtrbuton

3 3 whch Webull Bounded (WB), ha been defned a follow [25]: Defnton 2.1. A tochatc SSQ proce, denoted by W SSQ,γ (t), where γ the ervce rate of the queue, WB(C, η, υ) wth parameter C > 0 (whch denote the aymptotc contant), η > 0 (whch denote the decay rate), and 0 < υ 1 (whch denote the ndex parameter), f t atfe P r{w SSQ,γ (t) > w} < Ce ηwυ (1) for all w 0 and all t 0. Remark: The quantty P r{w SSQ,γ (t) > w} repreent the probablty that the backlog of the SSQ wth ervce rate γ wll exceed a certan queue ze w. In other word, P r{w SSQ,γ (t) > w} repreent the queue length dtrbuton of the SSQ. In addton, the decay rate η ncreae wth γ becaue when the ervce rate ncreae, the lkelhood that the queue length exceedng w wll decreae. Alo, the ndex parameter υ can be further expreed n term of the Hurt parameter H (whch commonly ued to characterze the degree of long range dependence [5, 23, 24]), and more pecfcally, υ = 2(1 H), where 0.5 H < 1. A traffc proce wth H = 0.5 correpond to conventonal traffc wth a queue length dtrbuton that decay exponentally. A larger H, or a maller υ, correpond to heaver taled LRD traffc. B. GPS Fundamental Generalzed Proceor Sharng (GPS) a chedulng dcplne defned under the aumpton that ource are decrbed by flud model [11]. Conder a GPS erver wth rate γ ervng N eon. Let each eon be agned a weght parameter, whch a fxed real-valued potve number φ. The et {φ 1, φ 2,..., φ N } thu repreent the GPS agnment. The N eon hare the erver n the followng way [26]: 1. It work conervng,.e., a long a there are packet backlogged n any of the GPS queue, the erver never dle. 2. The exce ervce rate, f any, redtrbuted among the backlogged eon n proporton to ther weght parameter. 3. Let S k (, t) denote the amount of traffc erved n the tme nterval [, t] for eon k. If eon backlogged n the ytem durng the entre nterval [, t],.e., there alway traffc queued for eon, then S (, t) S j (, t) φ φ j, j = 1, 2,..., N. (2) From (2), t clear that when eon backlogged, t guaranteed a backlog clearng rate (or equvalently, a guaranteed ervce rate) of at leat γ = φ j=1 φ j γ. (3) III. Analy of LRD Traffc Whle analy on LRD traffc uually tart wth t SSQ proce, t tll mportant to formally characterze the arrval proce becaue, a mentoned prevouly, the tablty of the SSQ proce depend on whether the arrval proce bounded. Stochatcally Bounded (SB) SSQ and Stochatcally Bounded A. Sngle WBB Arrval Proce and Sngle WB SSQ In th ubecton, we etablh the relatonhp between a WB SSQ and a Webull Bounded Burtne (WBB) arrval proce. We begn wth our defnton of the burtne contrant qualfer to decrbe the arrval proce of LRD traffc a follow: Defnton 3.1. A traffc arrval proce A(t) WBB( ρ, C, µ, υ) wth parameter ρ, C, µ and υ, f t atfe P r{ A(u)du > ρ(t ) + w} < Ce µwυ (4) for all w 0 and all 0 t. Smlar to the notaton n Defnton 2.1, C denote the aymptotc contant, υ the ndex parameter. Here, µ the decay rate 2 and ρ the long term upper rate of the arrval proce, whch wll be further elaborated n Lemma 3.1. Remark: A(u)du the amount of arrval traffc accumulated n tme nterval [, t]. In addton, the decay rate µ wll ncreae wth ρ, jut a η wll ncreae wth γ n a WB SSQ proce (ee Defnton 2.1). However, there a ubtle dfference between ρ and the parameter γ. In the former cae, ρ appled contnuouly from to t whle n the later cae, γ only appled when the queue not dle n the nterval [, t]. Several lemma and theorem, ueful to the objectve of th paper, are now preented. Lemma 3.1. An arrval proce A(t) that WBB(ρ, C, µ, υ) poee the property that t parameter ρ alway larger than or equal to t long term average rate E[ ρ lm A(u)du]. (5) t t The proof of Lemma 3.1, whch ntutvely explan why ρ called the upper rate, n Appendx A. Remark: The long term upper rate ρ ueful for the purpoe of boundng the entre enemble of ample tme obervaton that conttute the tochatc arrval proce A(t). In partcular, let A n (t) be the nth ample obervaton of A(t) n [, t], and let λ n = lm A(n,u)du t t t be the correpondng average arrval rate for th ample. If we were to repeat the obervaton of A(t) nfntely many tme ung dfferent tart tme, o that n approache nfnty, then we would have a correpondng lt of average arrval rate λ 1, λ 2,..., λ n. Th long term upper rate ρ range between the lower lmt E[λ n ] and the hgher lmt ρ max = max[λ 1, λ 2,..., λ n ]. For a conervatve (looe) 2 Not to be confued wth ymbol η, whch denote the decay rate of a WB SSQ proce ntead.

4 4 WBB bound on A(t), one may et ρ to the hgher lmt ρ max. However, notce that the long term upper rate, defned n (4), appled contnuouly even f the arrval proce nactve. Therefore, a lower value of ρ where ρ max ρ E[λ n ] may uffce to produce a tghter WBB bound on A(t). In ummary, the ue of the long term upper rate ρ n (4) eental for a general tochatc proce whch may not be tatonary (.e., λ 1 λ 2... λ n ). However, n practcal arrval procee, tatonary an mplct property for a flow that ha ome fxed arrval rate λ. Th mean that f th flow preented to the queue at dfferent tart tme, the ame average rate λ apple. Hence, for the cae of practcal flow, λ 1 = λ 2 =... = λ n = λ and therefore ρ = λ. Although many of our later dervaton followng th defnton are tll baed on ρ, reader hould be aware that for practcal conderaton, ρ ought to be replaced by λ nce practcal arrval procee are by default mplctly tatonary n property. In fact, n the conderaton of the GPS and PGPS dcplne n Secton IV, VI and VII, we conder λ ntead of ρ. Fnally, t alo noted that bede ρ, the WBB expreon n (4) alo contan other parameter lke the decay rate µ, the ndex υ and the aymptotc contant C. Thee parameter can mlarly be modfed to obtan ether looe or tght WBB bound. In the followng theorem, the relatonhp between a WBB arrval proce that we have defned, and a WB SSQ proce etablhed. Theorem 3.1. Conder a work conervng SSQ that tranmt at rate γ. Suppoe the queue fed wth a ngle arrval proce A(t), and let W SSQ,γ (t) be the amount of workload tored n the queue at tme t. Then: ()If W SSQ,γ (t) WB, then A(t) WBB, wth long term upper rate ρ = γ. ()If A(t) WBB wth long term upper rate ρ = γ ε for ome ε > 0, then W SSQ,γ (t) WB. Proof: For (), nce the workload W SSQ,γ (t) can be expreed a follow: W SSQ,γ (t) = max ( A(u)du γ(t )) (6) <t where 0 t, we have: W SSQ,γ (t) for all 0 t, therefore, P r{ A(u)du γ(t ) A(u)du > γ(t ) + w} P r{w SSQ,γ (t) > w} < Ce ηwυ Hence, A(t) a WBB(γ, C, η, υ) proce. To prove () we begn wth the condton that A(t) WBB wth long term upper rate ρ,.e. P r{ A(u)du > ρ(t ) + w} < Ce ηwυ (7) we prove by demontratng that the SSQ proce of any gven ample obervaton of A(t) WB. Now, from bac tochatc theory, we can apply the bound n (1) to a partcular ample obervaton of A(t), ay A n (t), t [0, t n ], where 0 the tart tme of the obervaton and t n the end tme of obervaton. Note that the ubcrpt n not only ued to repreent a partcular tme ample but alo the tart tme and end tme of that ample obervaton. Thu, P r{ P r{ A(u)du > ρ(t ) + w} < Ce ηwυ, 0 t n x A n (u)du > ρ(t n x) + w} < Ce ηwυ, 0 x t n (8) The relaton n (8) only ha a one way mplcaton. However, f A(t) an ergodc proce, then a double mplcaton can be ued n (8). It alo noted that the obervaton tme of the ample proce A n (t) range from 0 to t n. Th a fnte tme range nce by defnton, an obervaton requre a fnte tart tme and a fnte end tme. Let Wn SSQ,γ be the maxmum queue ze value arng out of the A n (t) arrval ample, uch that n Wn SSQ,γ = max x [0,t n] x A n (u)du γ(t n x)) (9) Snce A n (t) a contnuou tme proce between fnte tme lmt 0 and t n, there mut ext a value x n that maxmze (9), o that we can proceed on to: n Wn SSQ,γ = A n (u)du γ(t n x n) where 0 x n t n x n (10) Therefore, P r{wn SSQ,γ (t) > w} = P r{ = P r{ n x n n x n < Ce η[w+ε(tn x n )]υ A n (u)du > γ(t n x n) + w} A n (u)du > ρ(t n x n) + ε(t n x n) + w} < Ce 2 1 ηw υ e 2 1 ηε υ [t n x n ]υ (See (47) n Appendx) (11) It noted that the above bound on P r{w SSQ,γ n (t) > w} depend on the oberved t n 0 duraton. If the duraton were to change, ay for example, n the end tme t n, th wll affect the value x n and hence the bound wll change accordngly. However, a generc (but looer) bound that ndependent of the oberved duraton can be obtaned a follow: P r{w SSQ,γ n (t) > w} < Ce 2 1 ηw υ e 2 1 ηε υ [t n x n ]υ < Ce 2 1 ηw υ (12)

5 5 A cloer look at the Webull bounded expreon Ce 2 1 ηw υ n (12) alo how an abence of dependency on the ample ndex n. Th mean, that gven any ample obervaton A n (t) of A(t) of any duraton, the reultng SSQ proce WB(C, η 2, υ), or n other word, the SSQ proce of A(t) WB(C, η 2, υ). Remark: Although conventonally, LRD traffc uually decrbed n term of ome WB SSQ proce, t tll nuffcent to proceed on to GPS analy nce n GPS, we are concerned wth multple arrval procee rather than a ngle arrval proce. If there no burtne contrant on a ngle arrval proce, there not much that can be deduced on the tablty of a GPS erver that ervng a number of thee arrval procee. Wth the ntroducton of Theorem 3.1, we can now proceed further, nce t now known that any LRD arrval proce reultng n a WB SSQ proce mut atfy the WBB contrant wth ome long term upper rate ρ. Th mean that for a GPS erver ervng a number of LRD ource, a long a the um of the long term upper rate of thee LRD ource doe not exceed the ervce capacty of the GPS erver, the GPS queue wll be table and further analy can proceed. B. Bound on Aggregate WB SSQ In th ubecton, everal bound on the aggregate WB SSQ proce are derved. Thee bound wll be ued frequently n Secton IV, VI and VII where GPS and PGPS analy are preented. Lemma 3.2. Let W 1 (t) be WB(C 1, η 1, υ 1 ) and W 2 (t) be WB(C 2, η 2, υ 2 ). The two procee can ether be dependent or ndependent. Then, W 1 (t)+w 2 (t) WB(C 1 + C 2 + C ), η, υ) atfyng P r{w 1 (t) + W 2 (t) > w} < (C 1 + C 2 + C )e ηwυ (13) where η = η1η2 η 1+η 2, υ = mn(υ 1, υ 2 ) and C a contant obtaned ung the method n Appendx C. The proof of Lemma 3.2 n Appendx C. Lemma 3.2 can be ealy extended to multple WB procee a follow. Theorem 3.2. Let W (t), 1 N be N WB procee wth parameter (C, η, υ ) repectvely. Thee procee can ether be dependent or ndependent. Then, W 1 (t)+w 2 (t)+...+w N (t) WB(( =1 C +C ), η, υ) atfyng N N P r{ W (t) > w} < ( C + C )e ηwυ (14) =1 1 where η = =1 =1, υ = mn(υ 1 1, υ 2,..., υ N ) and C can η be obtaned ung the method Appendx C by applyng Lemma 3.2 tep by tep. Remark: Gven Lemma 3.2, the proof of Theorem 3.2 traght forward and hence omtted. For N WB queung procee wth the ame LRD degree (.e. the ame υ), Appendx C how that υ = mn(υ 1, υ 2,..., υ N ) the tghtet lower bound on the ndex parameter. But for N WB queung procee wth dfferent LRD degree, t a looe bound becaue the ndex parameter of multplexed LRD flow n general heaver taled than the ndvdual flow due to multplexng gan. The followng Lemma 3.3 and Theorem 3.3 preent alternate bound to thoe bound obtaned n Lemma 3.2 and Theorem 3.2 repectvely. The alternate bound are ueful nce n certan cae they are tghter (ee Fg. 4(b)). However, thee alternate bound are for large queue tuaton. For th, we now preent Lemma 3.3 and Theorem 3.3 a follow. Lemma 3.3. Let W 1 (t) and W 2 (t) be two ndependent WB(C 1, η 1, υ 1 ) and WB(C 2, η 2, υ 2 ) repectvely. If η 2 η 1 and υ 2 υ 1, then for w > 2, W 1 (t) + W 2 (t) ha an upper bound of the form P r{w 1 (t) + W 2 (t) > w} < C W B 2 (w)e ηwυ (15) where η = mn{η 1, η 2 } = η 2, υ = mn{υ 1, υ 2 } = υ 2 and C W B 2 (w) = C 2 h(c 1 ) + C 1 where h(c 1 ) = 1 + C 1 υη(e η 1) + C 1 w υ η The proof of Lemma 3.3. n Appendx D. Remark: From Appendx C, we know that η n Lemma η 3.2 1η 2 η 1+η 2, whch alway le than or equal to mn(η 1, η 2 ), whch the η n (15). Hence Lemma 3.3 yeld a larger (thu better) decay rate. In fact, f η 2 η 1, then η n Lemma 3.3 almot twce the value of η n Lemma 3.2. However, the aymptotc contant n Lemma 3.3 ncreae wth w, whch a trade off. For a heavy taled arrval proce where υ 0 o that for practcal and fnte value of w, w υ 1 and thu C2 W B (w) approache a contant, the penalty for ung Lemma 3.3 ngnfcant. Converely, f η 2 dffer gnfcantly from η 1 (.e., η 2 η 1 ), then η mn(η 1, η 2 ), makng Lemma 3.2 more attractve. Table 1 ummarze the preference (n term of whch Lemma to ue to obtan the bound) aumng that n all the cenaro, the queue ze of nteret larger than or equal to 2. Scenaro Preference η 2 η 1 & υ 2 mall Lemma 3.3 η 2 η 1 & υ 2 large Lemma 3.2 All other cae Ether ok TABLE I Preference for Lemma 3.3 or Lemma 3.2 n dfferent cenaro Smlarly to the way n whch Lemma 3.2 extended to Theorem 3.2, we now extend Lemma 3.3 to the followng Theorem 3.3. Theorem 3.3. Let W (t), 1 N, be N ndependent WB procee wth parameter (C, η, υ ) repectvely. If the queung procee can be rearranged uch that the Nth queung proce ha the property that η N η j and υ N υ j for 1 j N 1, then for w > 2,

6 6 W 1 (t)+w 2 (t)+...+w N (t) ha an upper bound of the form taled flow a to be explaned n ubecton IV.C and IV.D n more detal. N P r{ W (t) > w} < CN W B Note that the ndex parameter what dfferentate a (w)e ηwυ (16) LRD flow from a SRD flow. Although the decay rate and =1 contant parameter alo defne the queung proce, however, thee parameter form the exponental bound param- where η = mn{η 1, η 2,..., η N } = η N, υ = mn{υ 1, υ 2,..., υ N } = υ N and eter commonly aocated wth an SRD flow. Hence ther N j 1 preence, by defnton, for the purpoe of decrbng the CN W B (w) = [C j h(c l )] (17) SRD property of the flow. The ndex parameter, found j=1 where h(c l ) = 1+C l υη(e η 1)+C l w υ η, and by conventon, j 1 l=1 h(c l) = 1 when j = 1. Proof: ee Appendx D, partcularly t lat paragraph. l=1 IV. Sample Path Behavor of LRD Traffc n a GPS Sytem Recall from Theorem 3.1 that any LRD traffc nput whoe queue length dtrbuton characterzed by a WB dtrbuton ha an arrval proce that atfe the WBB contrant wth ome long term upper rate ρ. Hereafter, we conder N tatonary flow that mantan the ame long term average rate λ, = 1, 2,..., N rrepectve of the tart tme of the flow. A mentoned earler (n the remark for Lemma 3.1), the long term upper rate ρ reduce to the more famlar λ. A. Feable Orderng and LRD Iolaton of Flow Let the arrval proce for a tatonary WBB LRD eon be A (t) wth long term average rate λ, uch that =1 λ < γ. In order to characterze the effect of backlog from a et of eon, we ue the followng defnton of the o-called feable orderng of the eon that wll be frequently referred to hereafter, accordng to ther arrval rate and (GPS ervce) weght parameter. Defnton 4.1. For a gven et of nput traffc flow n a GPS erver, whoe long term average rate λ, an orderng called a feable orderng among the eon wth repect to {λ 1, λ 2,..., λ N } and (GPS ervce) weght parameter {φ 1, φ 2,..., φ N } [2] f and only f: 1 λ < ϕ (γ λ j ), 1 N (18) j=1 φ where ϕ = N, a contant aocated wth weght j= φj parameter. And by conventon, 1 j=1 λ j = 0 when = 1. We have the followng reult for feable orderng, whch ha been proved n [11]: Lemma 4.1. For a gven et of nput traffc flow n a GPS erver wth =1 λ < γ, there alway ext one or more than one feable orderng that atfe (18) after beng relabelled. Remark: The rght-hand de of (18) can be condered a the ervce rate avalable to flow. It clear, by defnton, that thoe flow ordered earler than flow wll affect the ervce rate avalable to flow. However, they wll not affect the ndex parameter of the queung proce of a heaver n the Webull bound formula, wa ntroduced to bound flow exhbtng LRD behavor whch cannot be utably bounded by jut the contant parameter and the decay rate. Hence, the LRD property of a flow, by defnton, prmarly due to t ndex parameter. Accordngly, we ntroduce the followng noton of LRD olaton. Defnton 4.2. We ay that a flow, when multplexed wth other flow n a queue ytem, LRD olated (from other flow) n that queue ytem f and only f t reultng queue proce ha the ame or larger ndex parameter (.e. le heavy taled) a the ndex parameter aocated wth t SSQ proce wth equvalent ervce rate a that guaranteed n the queue ytem. Th noton of LRD olaton dfferent from the conventonal undertandng of flow olaton. In flow olaton, the major concern the flow ervce rate, and a flow ad to be olated from other flow f th flow not adverely affected by thee flow [27]. Baed on th, a LRD flow flow olated f and only f t queue proce not adverely affected after t multplexed and erved wth other flow n the GPS erver. It can be hown that flow olaton guaranteed for a flow n a GPS erver f the flow can be ordered frt n a feable orderng. The reaon under th cae, the flow alway guaranteed a ervce rate greater than t long term average rate baed on (18), whch not affected by other flow. In addton, Lemma 4.2, whch wll be preented later, alo how that the flow queue proce n the GPS ytem not adverely affected (wth repect to t SSQ proce) by other flow. However, f the flow cannot be ordered frt n any of the feable orderng, the guaranteed ervce rate to the flow may depend on the arrval rate of ome other flow. In other word, t may vary over tme and hence the queue proce of the flow n the GPS ytem could be affected adverely. 3 A a reult, f a flow cannot be ordered frt n any of the feable orderng, the flow may or may not be guaranteed to be flow olated from other flow. However, a flow can tll be guaranteed to be LRD olated (from heaver taled flow) even f ome lghter taled flow have to be ordered before th flow n all feable orderng, a to be dcued later n th ecton. Clearly, flow olaton mple LRD olaton but not vce vera. Snce the ndex parameter the mot mportant meaure of the LRD property (heavne or lghtne of the tal) of a flow, the noton of LRD olaton a defned above ueful when tudyng LRD flow. 3 Note that, when λ = φ γ, flow cannot be ordered frt accordng to (18) although t flow olated.

7 7 B. GPS Decompoton Now let A denote a ample path (or a ngle realzaton) of the random arrval proce A (t), and Q denote the correpondng ample path of the GPS queue backlog due to the ample arrval proce A. To obtan relevant bound on Q, we ue a method mlar to that n [11] to decompoe the GPS ytem nto N fcttou WB ngle erver queue (SSQ), denoted by δ SSQ,γ (t), wth ndvdual rate γ 1, γ 2,..., γ N, where γ > λ, =1 γ γ, and γ ϕ (γ 1 j=1 γ j). Now, the reaon for conderng the N fcttou WB SSQ that ther bound are eaer to obtan and would urely bound Q a well. Th becaue the N fcttou WB SSQ do not conder multplexng gan whle the Q queue proce doe. A 1 (t) A 2 (t) Q1 (t) A N (t) Q2 (t) Q (t) N Fg. 1. γ A 1 (t) A 2 (t) A N (t) δ SSQ,γ 1 1 (t) δ SSQ,γ 2 2 (t) δ SSQ,γ N N γ 1 γ 2 (t) γ N Decompoe a GPS ytem nto N fcttou SSQ Wthout lo of generalty, let 1,2,...,N be a feable orderng of the fcttou procee wth repect to γ. From Lemma 3 of [11], the followng Lemma 4.2 can be derved: Lemma 4.2. For any t, Q 1 (t) ϕ j=1 δ SSQ,γj j (t) + δ SSQ,γ (t) (19) where each δ SSQ,γ SSQ proce ndependent. Remark: Lemma 4.2 provde an upper bound on the queue length Q (t) of an ndvdual eon n the GPS ytem n term of the queue length δ SSQ,γ (t) n the fcttou ytem. It clear from Lemma 4.2 that to bound the dtrbuton of Q (t), t uffce to bound the followng aggregate of fcttou queue length procee: ϕ δ SSQ,γ1 1 (t)+ϕ δ SSQ,γ2 2 (t)+...+ϕ δ SSQ,γ 1 1 (t)+δ SSQ,γ (t) (20) In what follow, we wll provde two bound on (20),.e., the rght hand de of (19). C. A General Bound on Indvdual Seon Queue Length For N ndvdual LRD flow harng a GPS erver on the condton of queue tablty,.e., =1 λ < γ, and under the aumpton that 1,2,...N a feable orderng wth repect to φ and λ, λ < γ for = 1, 2,..., N, we preent a GPS bound n the followng Theorem 4.1 that baed on Theorem 3.2. Theorem 4.1. Each ndvdual queue length dtrbuton n the GPS ytem ha an upper bound a follow: where P r{q (t) > q} < C GP S υ GP S e ηgp S q υgp S (21) = mn 1 j {υ j}, (22) η GP S 1 =, (23) j=1 1 η j C GP S = ( Note that, n the above, η j = { j=1 η j ϕ υ j η C j + C )e ηgp S (24) 1 j < j = and C can be obtaned mlarly a n Theorem 3.2. Proof: Frt, becaue all the nput flow are LRD flow, we have P r{δ SSQ,γ j (t) > q} < C j e ηjqυ j Secondly, let we have P r{δ SSQ,γj j,eqv, j = 1, 2,..., N. (25) δ SSQ,γj j,eqv (t) = ϕ δ SSQ,γj j (t), j < (26) (t) > q} = P r{δ SSQ,γj j (t) > q } < C j e ηjqυ j ϕ for 1 j <, where η j = ηj ϕ υ j Fnally, nce (20) can now be wrtten a δ SSQ,γ1 1,eqv (t)+δ SSQ,γ2 2,eqv. (27) (t)+...+δ SSQ,γ 1 1,eqv (t)+δ SSQ,γ (t) (28) by combnng (25), (27) and Lemma 4.2, one can ealy verfy the reult n (21) baed on Theorem 3.2. Remark: Theorem 4.1 gve a general upper bound on queue length dtrbuton n a GPS ytem. It mportant to note that the GPS upper bound on flow not affected by the flow that are ordered after flow (becaue they do not factor n the upper bound expreon for flow ). It affected only by flow 1 to 1, but the mpact on the bound neglgble a long a flow heaver taled (.e., ha a maller ndex parameter υ ) than any of the flow 1 through 1. In fact, the ndex parameter n the bound for flow not affected at all a long a flow 1 to 1 are lghter taled than flow.

8 8 D. An Alternate Upper Bound In the followng Theorem 4.2, an alternate upper bound on ndvdual eon queue length n GPS wth LRD traffc provded baed on Theorem 3.3. Such a bound may be better than the bound prevouly gven n Theorem 4.1 but can only be appled under the condton that for any gven, there ext a 1 k uch that both η k and υ k are mnmal, n addton to the condton tated for Theorem 4.1. Theorem 4.2. If there ext a k, 1 k, uch that η k = mn 1 j { η j } and υ k = mn 1 j {υ j }, then for q > 2, the upper bound for ndvdual eon queue length : where P r{q where C l w υgp S (t) > q} < C GP S υ GP S η GP S C GP S (q) = C k h GP S η GP S (q)e ηgp S q υgp S (29) = mn 1 j (υ j) = υ k, (30) = mn 1 j ( η j) = η k, (31) l=1,l k j=1,j k h GP S (C l ) + j 1 [C j l=1,l k η GP S h GP S (C l )] (32) (e ηgp S 1) + (C l ) = 1 (C l ) = 1 + C l υ GP S, and by conventon, j 1 l=1,l k hgp S when j = 1. Proof: Wthout lo of generalty, aume that k <. The aggregate proce n (28) can be re-wrtten uch that the kth proce wth the mnmum decay rate a well a wth the mnmum ndex parameter appear lat n the equence a follow: δ SSQ,γ1 1,eqv +δ SSQ,γ k+1 k+1,eqv (t) + δ SSQ,γ2 2,eqv (t) δ SSQ,γ k 1 k 1,eqv (t) (t) δ SSQ,γ (t) + δ SSQ,γ k k,eqv (t) (33) Hence, by applyng Theorem 3.3, Theorem 4.2 can be ealy verfed. Remark: Theorem 4.2 provde an upper bound on an actual eon backlog Q (t) n the GPS ytem when there ext a very heavy taled LRD flow wth the mallet ndex parameter (a well a the mallet decay rate). One mplcaton of Theorem 4.2, mlar to Theorem 4.1, that t derable to order the flow that are heaver taled a cloe to the end of a feable orderng a poble, agan nce the ndex parameter n the upper bound for the ndvdual queue length of flow wll not be affected f and only f the flow 1 through 1 are all lghter-taled than flow. V. A Technque to Check and Enure LRD Iolaton Recall from Lemma 4.1 that n a table GPS ytem where j=1 λ j < γ, there ext at leat one feable orderng for a gven weght agnment. Before we dcu LRD olaton, t ueful to revt the concept of flow olaton wth the followng Lemma 5.1. Lemma 5.1: In a table GPS ytem, f flow atfe the followng condton: φ λ < γ j=1 φ j (34) then the flow flow olated. Proof: The proof to Lemma 5.1 traght-forward nce the rght hand de of (34) the mnmum guaranteed rate, however, we tll provde the requred proof for Lemma 5.1 nce everal ntermedate reult of th proof wll be ued later to prove newer reult pertanng to LRD olaton. Relabel flow a flow 1, and all other N 1 flow to be flow 2 to N. Note that flow 1 now atfe (18), and hence all we need to how that the remanng N 1 flow can be feably ordered after flow 1. To th end, conder a new GPS ytem wth ervce rate γ = γ λ 1. Snce γ > j=2 λ j, the new GPS ytem alo table, and hence, there alway ext a feable orderng uch that (after relabellng the flow 2 to N, we have for any flow 2 N: λ < = φ j= φ j φ j= φ j 1 (γ λ j ) j=2 1 (γ λ j ). (35) j=1 Note that the above become the ame a (18), whch mean that f flow 1 ordered frt, the remanng N 1 flow can alo be ordered to yeld a feable orderng. Therefore, flow 1 flow olated. Remark: It hould be noted that (34) only a uffcent condton for flow olaton, not a neceary condton. In fact, t a uffcent condton to guarantee a flow to be flow olated. However, a mentoned earler, a flow can tll be olated even f t cannot be guaranteed to be flow olated, or even f t doe not atfy (34). Baed on Lemma 5.1, an obvou method to guarantee the flow olaton of every flow to agn weght of every flow accordng to (34) uch that every flow can be ordered n the frt place n a feable orderng. A mentoned earler, beng able to order a flow frt n any feable orderng the mot applcable condton to guarantee a flow to be flow olated for CAC purpoe. That, however, not neceary to guarantee jut LRD olaton of a flow, whch le trct than flow olaton, a to be dcued n a later ubecton. A. Lmtaton of Extng Method In th ubecton, we dcu the hortcomng of the extng method for agnng weght to acheve flow o-

9 9 laton and tetng whether a flow can be flow olated for a gven weght agnment. A GPS ytem may upport the followng three type (clae) of flow (ervce): A Type 1 flow requre a hgher QoS than that provded by flow olaton, o t requre a contract weght that much larger than λ γ j=1 φ j; A Type 2 flow requre flow olaton, and thu need a contract weght that a lttle larger than λ γ j=1 φ j; A Type 3 flow only requre LRD olaton (but not flow olaton), and thu can have a contract weght le than λ N γ j=1 φ j. Note that the contract weght cannot be changed a long a the ervce level agreement (SLA) n effect. On the other hand, a (lghtly loaded) GPS ytem may agn a flow an extra weght (f avalable) to provde the flow wth better ervce, and uch extra weght can be adjuted (e.g., tranferred to other flow) by the GPS ytem. The method of agnng weght baed on (34) ha a lmted applcablty n upportng both Type 1 and 2 but not applcable to Type 3 flow. From uer or applcaton vewpont, havng Type 3 flow ueful becaue certan applcaton may requre le trct performance guarantee than that gven by flow olaton, and uch flow can be admtted nto a GPS ytem and wth le cot to the uer or aplcaton. In addton, from the GPS ytem vewpont, upportng Type 3 flow allow t to admt more flow than otherwe poble, thu ncreang t utlzaton and potental revenue. For example, (hereafter referred to a Example 1), conder a GPS ytem wth γ = 16 and fve flow numbered 1 through 5 n the decendng order of ther ndex parameter, whoe λ = where 1 5. Aume that the total weght 5 j=1 φ j = 16, and n addton flow 1 and 2 have been agned contract weght of φ 1 = 1.1 and φ 2 = 4, repectvely. Snce the remanng weght for flow 3, 4 and but the um of ther arrval rate 12, t clear that the (34) cannot be ued to agn the weght to all the three remanng flow to guarantee ther flow olaton. In general, due to the extence of Type 1 flow (e.g., flow 2 n Example 1), flow olaton may not alway be achevable by every flow, and accordngly, the extng approach baed on (34) may not be ueful. Note that, even f flow doe not atfy (34), t may tll be LRD olated. In the above Example 1, one can agn 2.6 to flow 3 to enure t LRD olaton (whch can be verfed ung the technque to be propoed later), even though uch a weght volate (34). A another example (hereafter called Example 2) howng the defcency of the extng approache, aume that the weght agnment for the ame fve flow a n Example 1 now {1.1, 2.1, 1, 4, 7.8}. It clear that (34) cannot be ued to tet f flow 3 and 4 (both volate (34)) are LRD olated or not. In addton, (18) n Defnton 4.1 not effectve ether. More pecfcally, n order to ue t to tet whether flow 4 can be guaranteed to be LRD olated or not, a nave approach wll tet f the orderng of 1,2,3,4,5 feable, and becaue t not, t wll have to examne the orderng of 1,3,2,4,5 and then the orderng of 2,3,1,4,5 and o on. In the wort cae, to tet f flow can be guaranteed to be LRD olated or not, all poble orderng nvolvng j lghter taled flow, where 0 j ( 1), have to be teted. Thu, the (wore cae) tme complexty of the tetng proce O(!). When the number of flow large, uch an approach clearly nfeable. B. A Neceary and Suffcent Condton for the Guarantee of LRD Iolaton We now determne, for a gven flow, not only the et of lghter-taled flow, denoted by f, that can be ordered before flow n a feable orderng, but alo the mnmum contract weght to enure the LRD olaton of flow. To th end, we frt ntalze f to be empty. Then, f there ext a flow k, where 1 k <, whch atfe λ k γ j f < λ j φ N k j=1 φ j (36) j f φ j we add flow k to f, and update the rght-hand de of (36) whch wll be denoted by R(f ). We repeat the above proce untl no uch flow k ext, and denote the reultng et by F, and accordngly, the fnal value of R(f ) by R(F ). Note that th proce of obtanng F ha the wort-cae tme complexty of O( 2 ). One can ealy verfy that when a flow k that atfe (36) added to f, the reultng R(f ) ncreae..e. R(f ) < R(f k) R(F ) f f F. Converely, f we were to add a flow k that doe not atfy (36) to f, then R(f k ) R(f ). In other word, R(F ) the maxmum value that flow can obtan from all flow that are lghtertaled than flow. Th obervaton mportant for provng the followng theorem whch provde both a neceary and uffcent condton for the LRD olaton guarantee of flow. Theorem 5.1: Suppoe there are N flow n a GPS ytem whch are numbered n the decendng order of ther ndex parameter a 1, 2,..., N, and ther contract weght are φ 1, φ 2,..., φ N, repectvely. Then flow guaranteed to be LRD olated from other flow f and only f: λ γ j F < λ j φ N j=1 φ j = R(F ) (37) j F φ j Proof: () To how that (37) a uffcent condton, we note that flow alo atfe (36), jut a any flow k < n F doe. Accordngly, f we let F = F {}, and note that γ when f empty, R(f ) =, we have the followng j=1 φj (baed on the obervaton drawn precedng the Theorem): γ j F λ j j=1 φ j > j F φ j j F γ j F λ j j=1 φ j j F φ j > Accordngly, we can ealy conclude that j F λ j γ < φ N j j=1 φ j γ j=1 φ j The above mean that f we treat the flow n F a one bg flow wth arrval rate j F λ j and weght j F φ,

10 10 t atfe (34). Hence, accordng to Lemma 5.1, there ext a feable orderng wth th bg flow ordered frt. In other word, flow can be feably ordered before any heaver taled flow. Note that the exact orderng of the flow wthn F wll not affect the LRD olaton of flow. In fact, the flow n F can be feably ordered accordng to the order they are added to F n (36) wth flow beng ordered rght after them. () we now prove that (37) neceary by contradcton. Suppoe (37) doe not hold for flow, but there tll ext a feable orderng wth flow ordered before any heaver taled flow. Denote the et of all the (lghter taled) flow that are feably ordered before flow by F (whch may be empty). Accordng to (18), we hould have: λ φ < γ j F λ j j=1 φ j j F φ j = R(F ) However, nce F contan zero or more flow n F, and zero or more flow not n F, we have R(F ) R(F ) baed on the dcuon precedng the Theorem. Or n other word, λ φ < R(F ) R(F ) whch contradct wth the aumpton that (37) doe not hold for flow. Remark: If a flow atfe (34), t wll atfy (36) but not vce vera. Wth (37), whether a flow guaranteed to be LRD olated or not depend only on the weght agned to the flow n F, and flow telf. In prevou Example 1, one can ealy verfy that F 3 = {1, 2}, and R(F 3 ) = (16 3)/(16 5.1) = Hence, f φ 3 = 2.6, flow 3 atfe (37) and thu guaranteed to be LRD olated. On the other hand, n prevou Example 2 (where the weght agnment for fve flow {1.1, 2.1, 1, 4, 7.8}), one can ealy verfy that nce F 3 = F 4 = {1, 2}, and R(F 3 ) = R(F 4 ) = 13/12.8, (37) cannot be atfed by flow 3, and thu flow 3 not guaranteed to be LRD olated. On the other hand, flow 4 atfe (37), and thu guaranteed to be LRD olated. C. Weght Adjutment and Agnment to Enure LRD Iolaton Theorem 5.1 alo ueful for weght agnment and adjutment to guarantee flow LRD olaton. More pecfcally, the obervaton drawn precedng the theorem,.e., R(F ) maxmum wth repect to flow, erve a the bae for determnng a mnmal φ to guarantee the LRD olaton of flow. For ntance, conder agan prevou Example 2 but now aume that only the weght agned to flow 1, 2 and 4 are contract weght (.e., non-adjutable). If we want to enure LRD olaton of flow 3, we mut ncreae φ 3 to above 13/12.8. Such an ncreae can be accomplhed f φ 5 ha an extra weght of 2 that can be tranferred to flow 3 (and a a reult, φ 5 reduced to 5.8 from 7.8). The above technque to adjut the weght of a ngle flow to enure t LRD olaton can certanly be extended to enure LRD olaton of more than one flow provded that there are extra weght n the GPS ytem that can be adjuted/tranferred. A a lghtly dfferent example from thoe above (Example 3), conder fve flow numbered n the decendng order of ther ndex parameter whoe arrval rate are more or le randomly dtrbuted a {2, 4, 5, 1, 3}. Suppoe that γ = 17 (whch uffcent to make the ytem table) and the total weght a contant 17. In addton, uppoe that flow 2 (whch a Type 1 flow) ha been agned a contract weght of 7 (and thu the method baed on (34) cannot be ued for weght agnment to guarantee flow olaton of all the other flow a dcued earler). If all other four flow are Type 3 flow that only requre LRD olaton, we can ue Theorem 5.1 to agn contract weght to them to guarantee ther LRD olaton a follow (note that one can ealy verfy that flow 2 can be ordered frt n any feable orderng o t already flow olated). For the frt flow, from Theorem 5.1, we need to have φ 1 > λ 1 = 2, o we et φ 1 = 2.1 (theoretcally peakng, we can et φ 1 = 2 + ɛ where ɛ > 0 can have a very mall value). For flow 3, we frt obtan F 3 = {1, 2}, and then from (37), we have 5 1 φ 3 > λ φ φ 1 φ 2 3 γ λ 1 λ = = 3.59 Accordngly, we et φ 3 = 3.6 to flow 3. Smlarly, we et φ 4 = 0.72 and φ 5 = 0 4. The extra weght avalable n the ytem = 1.58, whch may be dtrbuted among the fve flow n an arbtrary manner. To further llutrate the uefulne of the propoed technque, let u conder the followng Corollary of Theorem 5.1 whch applcable to the cae of on-lne CAC. Corollary 5.1: If a flow provded a contract weght φ whch guarantee t to be ether flow olated or jut LRD olated, t wll be guaranteed to be flow olated or jut LRD olated after a new flow j admtted a long a the ytem reman table (note that flow j cannot take away any extng contract weght already agned to the other flow o t contract weght can only come from the extra weght avalable n the ytem before t admtted). Proof: If flow wa guaranteed to be flow olated before flow j admtted, flow mut atfy (34). Hence, flow guaranteed to be flow olated after flow j admtted. Now aume flow wa only guaranteed to be LRD olated but not guaranteed to be flow olated before flow j admtted. Under th aumpton, there exted a feable orderng n whch the et of flow ordered n front of flow, F, are all lghter-taled than flow. Now treat F {} a one bg flow F. Jut a hown n the Part () of the proof for Theorem 5.1, th bg flow F atfe (34) and thu can be ordered n the frt place n a feable orderng. Thu, flow tll guaranteed to be LRD olated. 4 Note that wth φ 5 = 0, flow 5 get bet effort ervce

11 11 R Regulator Fg. 2. PGPS Server PGPS Server GPS Let u contnue the above Example 3 by aumng the onlne CAC receve a requet for a new flow (flow 6). Suppoe that t arrval rate λ 6 =1, and t ndex parameter n between thoe of flow 2 and 3. To enure t LRD olaton, we frt obtan F 6 = {1, 2} and then conclude we need a contract φ 6 > Snce we have an extra weght of 1.58, we can agn φ 6 = 0.72 and re-dtrbute the remanng extra weght = 0.86 among all x flow. Note that from Corollary 5.1, admttng flow 6 a done n the above cae wll not affect ether the flow olaton or LRD olaton of any flow, or n other word, ther guaranteed (contracted) performance. There are, however, cae where a heavy taled flow ha been agned a weght more than t arrval rate, and hence, the remanng weght not enough to enure the LRD olaton of the newly arrved flow. An example that for the ame et of 5 flow decrbed n Example 3, but th tme flow 4, ntead of flow 2, a Type 1 flow that requre a contract weght of 5. To enure every other four flow are LRD olated, we need the followng weght agnment: {2.1, 4.1, 5.1, 5, 0}, whch leave an extra weght of only 0.7. Hence, when flow 6 arrve, t need φ 6 > 1 to enure t LRD olaton. In uch a cae, the ytem may decde not to admt flow 6 or admt t wthout enurng t LRD olaton. VI. Sample Path Behavor of LRD Traffc n a PGPS Sytem The reult obtaned o for the GPS ytem are now extended to the PGPS ytem. Whle the GPS dcplne aume that the nput traffc behave lke a flud uch that multple eon can be erved bt by bt, the Packet baed GPS (PGPS) a more practcal dcplne n that only one packet at a tme may be erved. In other word, a PGPS erver conder the arrval of a packet only after t lat bt ha been receved. To manage th dfference, the PGPS erver often taken to cont of two part, a regulator and a PGPS core whch a GPS cheduler (ee Chapter 4 n [28]), a llutrated n Fgure 2. Partallycomplete (or partally arrved) packet are queued n the regulator whch pae only complete (or arrved) packet to the PGPS core. The output of th regulator, whch the nput to the PGPS core, a ere of mpule, whoe heght repreent the ze of the packet. Let A be the eon nput traffc to the PGPS erver, whch alo the nput to the regulator, A,reg be the output traffc from the regulator, whch the nput traffc to the PGPS core, and fnally A(, t) be the total amount of traffc arrved n tme nterval [, t]. From Corollary 1 n [2], the queung proce of A,reg (, t) alo bounded by the queung proce of A (, t) wth an extra length L,.e., Q P GP S (, t) Q GP S (, t) + L, where L the maxmum length of all arrved packet.. From the queung proce Q of A, whch WB(C, η, υ), we obtan the queung proce of A,reg a follow: Q P GP S P r{q P GP S (, t) > q} P r{q GP S (, t) + L > q} = P r{q GP S (, t) > q L} < C e η(q L)υ 5 C e ηlυ e ηqυ (38) whch WB(Ce ηlυ, η, υ). In other word, the two GPS upper bound derved n Theorem 4.1 and 4.2 n the prevou ecton can be extended to the PGPS doman va a mple tranformaton of the aymptotc contant C C e ηlυ provded that the queue length or backlog large enough to exceed the maxmum packet length L,.e., q > L. Note that th aumpton (q > L) reaonable becaue n practce, the buffer ze B much larger than L,.e. B L, and n addton, nce the man concern whether the backlog about to exceed B, the value of q that are of nteret hould be cloe to B and thu larger than L. For completene, we now preent Theorem 6.1 and Theorem 6.2 whch are derved from Theorem 4.1 and Theorem 4.2 repectvely va the ue of the mple tranformaton C C e ηlυ a follow: P Theorem 6.1: Let Q, 1 N repreent the th queung proce of the PGPS ytem wth N LRD arrval procee, then at any tme t, for any queue length q > L where L the maxmum packet length of all the N eon, we have: P P r{q where υ GP S, η GP S (23), and C P GP S = ( > q} < C P GP S e ηgp S q υgp S (39) have already been defned n (22) and j=1 C j e ηjlυ j + C ) where C can be obtaned mlarly a n Theorem 3.2. An alternate bound gven below, whch, lke the bound n Theorem 4.1, apple only when there ext a 1 k for any gven, uch that both η k and υ k are mnmal. Theorem 6.2: Under the ame aumpton ued for Theorem 4.2 except that the erver now a PGPS erver, at any tme t, for any q > L > 2 where L the maxmum packet length of all the N eon: P P r{q where υ GP S, η GP S 5 ee (48) n the Appendx. > q} < C P GP S (q)e ηgp S q υgp S (40) have already been defned n (30) and

12 12 (31), and where C P GP S (q) = C e η kl υ k k + j=1,j k [C j e ηjlυ j j 1 l=1,l k l=1,l k h GP S (C l e η ll υ l ) = (1 + C e η ll υ l l +C e η ll υ l l j 1 l=1,l k hgp S h GP S (C l e η ll υ l ) h GP S (C l e η ll υ l )] υ GP S q υgp S η GP S ) η GP S (e ηgp S 1) and by conventon, (C l e η ll υ l ) = 1 when j = 1. Remark: Note that the above bound hed lght on the LRD olaton among LRD ource harng a PGPS erver. To llutrate th, conder a mple cae of two ndependent LRD ource wth a feable orderng of 1, 2. From Theorem 4.1, the ource whch appear frt n the feable orderng alway guaranteed to be LRD olated. Therefore, the queung proce that of nteret the lat queung P proce n the feable orderng,.e., Q2. By applyng Theorem 6.1 and 6.2, three poble et of bound can be obtaned a follow: ()If η 1 η 2 and υ 1 υ 2 then from Theorem 6.2, we have: where P P r{q2 > q} < C2 P GP S 1 (q)e η1qυ C P GP S 2 (q) = C 1e η1lυ 1 h(c 2 e η2lυ 2 ) + C 2 e η2lυ 2 ()Ele f η 2 η 1 and υ 2 υ 1 then from Theorem 6.2: where P P r{q2 > q} < C2 P GP S (q)e η2qυ 2 C P GP S 2 (q) = C 2 e η2lυ 2 h(c 1 e η1lυ 1 ) + C 1 e η1lυ 1 (41) (42) ()In general, regardle of the relatonhp between η 1 and η 2 and that between υ 1 and υ 2, from Theorem 6.1, we have P P r{q2 > q} < (C 1 e η1lυ 1 + C 2 e η2lυ 2 ) e η(qυmax 0 q υ 0 ) e ηqυ (43) where η = η 1η 2 and υ = mn {υ 1, υ 2 } η 1 + η 2 The ndex parameter (a well a the decay rate parameter) of the three bound hown n (41)-(43) ndcate the nfluence of ource 1 on ource 2. In the frt cae, the bound P on Q2 decay lower, and n fact, t adopt the ame ndex parameter a that n the bound on the more heaver taled queung proce δ SSQ,γ1 1. Th mean that ource 2 not guaranteed to be LRD olated from ource 1. In the econd cae, ource 2 not much affected by ource 1 nce P the bound on Q2 adopt the ame ndex parameter a the bound on δ SSQ,γ2 2. Fnally n the thrd cae, whch ueful when nether of the frt two cae applcable, P the bound on Q2 decay lower than both the bound on δ SSQ,γ1 1 and the bound on δ SSQ,γ2 2. VII. Expermental Study of a Two Queue PGPS Sytem In th ecton, we preent numercal reult on a PGPS ytem wth two LRD flow to demontrate the uefulne of the bound gven by Theorem 6.1 and Theorem 6.2. Two cenaro are condered, each nvolvng a HT (heaver taled) flow and a LT (lghter taled) flow. The HT flow a burty porton of a Star War MPEG trace (ftp://thumper.bellcore.com/pub/vbr.vdeo.trace), whle the LT flow an artfcally generated LRD flow ung aggregated Pareto On/Off procee [13]. In the frt cenaro, called Non-LRD-Iolated, the HT flow appear frt n the feable orderng whle the LT (lghter taled) flow appear next (lat). In the econd cenaro, called LRD-Iolated, the order of the feable orderng revered. The am of the graphcal preentaton to llutrate the LRD olaton properte of the two flow n the PGPS erver and to demontrate that, gven the WB parameter of the ndvdual flow, the derved PGPS bound wll alway bound the actual PGPS P r{q > B} dtrbuton. A. Non-LRD-Iolated Scenaro The Non-LRD-Iolated cenaro decrbed a follow, where PGPS capacty γ = 1100kbp and all other parameter are hown n the table below. Note that to mplfy calculaton, we ntroduce the normalzed long term average rate λ baed on λ ). Alo, the ndex parameter are obtaned va leat-quare matchng of WB model to the numercally generated SSQ proce. Th method of obtanng the Hurt parameter not recommended for real-tme cenaro nce the method requre the luxury of mulatng the ample LRD ource countle tme n order to obtan a reaonably table SSQ tal probablty curve. Th method nonethele uffcent for u nce the focu of th paper on the multplexng of LRD ource and not concerned wth the ntty-grtty of obtanng the ndex parameter of ndvdual LRD ource. For a more profeonal treatment n regard to a real-tme etmaton of the ndex parameter of an ndvdual LRD ource, the method uggeted n [29] can alway be ncorporated. HT Flow LT Flow λ 655kbp 345kbp λ υ η C e ηlυ φ γ Gven the above weght agnment of φ HT = 0.7 and φ LT = 0.3, the feable orderng {HT, LT }, whch n fact, accordng to (18), the only feable orderng. A a reult, the bound for the LT flow of nteret. Snce