1 Introduction 3. 2 Preliminaries A GPS server with variable service rate Leaky bucket... 4

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1 Abstract Future ntegrated-servces packet networks wll carry a wde range of applcatons whch could dffer sgnfcantly n ther Qualty-of-Servce (QoS) requrements. Generalzed processor sharng (GPS) s the most mportant and deal flud schedulng dscplne, whch s useful n guaranteeng QoS. Many practcal packet schedulng algorthms buld on t. However, most studes of GPS assume constant servce rate. Ths assumpton mght be nvald, especally, n heterogeneous nternetworkng envronments. In realty, the servce rate at communcaton servers could be tme-varyng owng to multple-access mechansms n subnetworks, lnk-level flow/error control, and user moblty. In ths thess, we analyze GPS networks wth varable servce rate n the determnstc and stochastc settngs and extend the results of Parekh, Zhang, et al. to the varable servce rate case. In the determnstc settng, we assume that sources are constraned by leaky buckets and the servce processes are gven. We frst show bounds of backlog, delay, and output burstness for each sesson n a sngle GPS server. We extend these results to networks wth arbtrary topology, whch belong to a broad class of GPS assgnments, the so-called Consstent Relatve Sesson Treatment (CRST) GPS assgnment. We show the stablty of CRST networks and then show bounds of backlog and delay for each sesson n CRST networks. Moreover, we relate the results for GPS to those for GPS, whch closely approxmates GPS. In the stochastc settng, on the other hand, we use exponentally bounded burstness process and exponentally bounded fluctuaton process to characterze source traffc and server fluctuaton, respectvely. We show statstcal bounds on the dstrbutons of backlog and delay for each sesson n CRST networks wth arbtrary topology. 1

2 Contents 1 Introducton 3 2 Prelmnares A GPS server wth varable servce rate Leaky bucket Analyss of a sngle GPS server Defnton and prelmnary results Fcttous worst-case regme bound Analyss of a PGPS server Analyss of GPS networks Prelmnares GPS networks under consstent relatve sesson treatment Computng delay and backlog bounds for stable networks wth known nternal burstness and servce processes Analyss of PGPS networks Statstcal analyss of GPS networks Prelmnares Sample path behavor of a sngle GPS server Statstcal analyss of a sngle GPS Server Statstcal analyss of GPS networks Concluson 25 A Proofs n secton 3 27 A.1 Proof of Lemma A.2 Proof of Lemma A.3 Proof of Theorem A.4 Proof of Theorem A.5 Proof of Theorem B Proofs n secton 4 32 B.1 Proof of Lemma B.2 Proof of Lemma B.3 Proof of Lemma B.4 Proof of Theorem B.5 Proof of Theorem B.6 Proof of Lemma B.7 Proof of Theorem B.8 Proof of Theorem C Proofs n secton 5 38 C.1 Proof of Lemma C.2 Proof of Lemma C.3 Proof of Lemma C.4 Proof of Lemma C.5 Proof of Lemma C.6 Proof of Theorem

3 1 Introducton The provson of Qualty-of-Servce (QoS) guarantees has become mportant n the desgn of hgh-speed networks. One mportant ssue n the provson of QoS guarantees s what schedulng dscplnes should be employed at network swtches. Ideally these schedulng dscplnes should, on the one hand, provde solaton between sessons, so that the msbehavor of one sesson wll not affect other sessons and, on the other hand, explot statstcal multplexng gan [8]. It s desrable that ther (determnstc or statstcal) bounds can be derved. One of the most wdely studed non-fcfs schedulng s the Generalzed Processor Sharng (GPS) schedulng dscplne (also known as Weghted Far Queueng n [8]). In [1] and [2], Parakh and Gallager examned of GPS wth leaky-bucket controlled ncomng traffc [10]. If each sesson s leaky-bucket controlled and that the total arrval rate s smaller than the servce rate, t was shown, n the case of a sngle server n solaton, that the backlog and delay of each sesson are bounded from above; and n the case of a network of GPS servers, that under a broad class of GPS assgnments known as Consstent Relatve Sesson Treatment (CRST) GPS assgnments, the network s stable. These bounds are actually attaned n the worst-case scenaro. Smulaton results n [9] show that determnstc upper bounds are usually very conservatve, and that, f these bounds are used as admsson control crtera, low utlzaton of network bandwdth wll result. In [4], Zh-L Zhang presented bounds on the ndvdual sesson backlog and delay dstrbuton under GPS schedulng dscplne n a stochastc settng. They model the source sesson traffc as an exponentally bounded burstness (E. B. B.) process whose total long-term upper rate s smaller than the servce rate. These studes assume constant servce rate. Ths assumpton of steadness, especally n a heterogeneous nternetworkng envronment, mght be nvald owng to subnetwork multple-access mechansm, lnk-level flow/error control, and user moblty [7]. In [7], Lee presented the technque exponentally fluctuated bounded (E. B. F.) process and fluctuaton constraned (F. C.) process to characterze and analyze work-conservng communcaton servers wth varyng servce rate. Ths work s motvated by, and s an extenson of, [1], [2], and [4]. We wll extend ther result to a varable servce rate case. We analyze varable servce rate GPS servers and networks n both determnstc and stochastc settngs by usng the technques smlar to [1], [2], and [4]. In the determnstc settng, we show the exstence of bounds of each sesson s delay and backlog n a sngle server and a CRST network under the followng assumptons: the arrval process of all sessons n a server or network are leaky-bucket constraned, the average servce rate s greater than the total arrval rate and the maxmum servce rate s upper-bounded. We also show that bounds can be computed and attanable n the worst-case scenaro f the output rate s upper-bounded and fluctuaton constraned process. Moreover, we relate results for GPS to those for PGPS, whch closely approxmates GPS. In the stochastc settng, we present statstcal bounds on the dstrbutons of backlog and delay of each sesson for a sngle server and a CRST network, under the assumptons that the arrval and output processes are E. B. B. and E. B. F. processes, respectvely. The rest of ths thess s organzed as follows. In secton 2, GPS and leaky bucket are defned and explaned. In secton 3 and 4, we proceed wth an analyss of a sngle GPS server n solaton and a GPS network, respectvely. In secton 5, we analyze sngle and mult-node GPS systems n the stochastc settng. Fnally, the concluson s provded n secton 6. 2 Prelmnares 2.1 A GPS server wth varable servce rate We consder a GPS server that s work conservng and can operate at rate r(t) at tme t. By work conservng, we mean that the server must operate at full rate r(t) f there are packets watng n the system at t. A GPS server that serves N sessons s characterzed by postve real numbers φ 1, φ 2,..., φ N. Let S (, t) denote the amount of sesson traffc served n an nterval (, t]. A sesson s backlogged at tme t f a postve amount of that sesson s traffc s queued at tme t. A GPS server s defned as one for whch S (, t) S (, t) φ, (1) 3

4 for any sesson that contnuously backlogged n the nterval (, t]. If both sesson and are contnuously backlogged n (, t], S (, t) = S (, t) φ. (2) Note that ths s the same defnton as n the constant rate case [1]. To smplfy notatons, we defne R(, t) = r(x)dx. R(, t) means the amount whch the server can transmt durng an nterval [, t]. In (2), summng over all sesson, S (, t) and sesson s served at greater than or equal to N φ R(, t), g (t) = N φ r(t), at tme t. Ths s one of the mportant dfferences between the constant and varable cases. When r(t) = r, sesson s guaranteed a rate of g = N φ r, regardless of t or other sessons. Ths s very attractve. For example, f the sesson s locally stable,.e., ρ s less than g, the sesson can be guaranteed a throughput and delay bound and ts bound can be obtaned easly (σ /g ). Moreover, t s well known that the delay and backlog upper-bounds of Rate Proportonal Processor Sharng networks, where all sessons locally stable, can be obtaned easly regardless the topology of the network. Ths s owng to the above pont. Thus we should use another approach to analyze RPPS networks. 2.2 Leaky bucket Token enters at rate ρ σ bts PSfrag replacements Buffer Rate C A (, t) Incomng (bursty) traffc To the network Fgure 1: Leaky bucket. We summarze some results on the leaky bucket scheme n [1]. Fg. 2.2, Fg. 2.2, and Fg. 2.2 are the same as Fgs. 3, 4, and 5 n [1], respectvely. Fg. 2.2 depcts the leaky bucket scheme that we wll use 4

5 Empty bucket PSfrag replacements slope = ρ A (0, t) K (t) l b () σ l (a) Full bucket a t Fgure 2: A (t) and l (t). slope = ρ A (0, t) K (t) PSfrag replacements σ t Fgure 3: A sesson arrval process that s greedy from tme. 5

6 to descrbe traffc that enters the network. Token s generated at constant rate, ρ, and packets can be admtted nto the network only after removng the requred amount of token from the token bucket. There s no bound on the number of packets that can be buffered, but the token bucket contans at most σ bts worth of token. In addton to securng the requred amount of token, traffc s constraned to leave the bucket at a maxmum rate of C > ρ. The constrant mposed by the leaky bucket s as follows. If A (, t) s the amount of sesson flow that leaves the leaky bucket and enters the network n an nterval (, t], then A (, t) mn{c (t ), ρ (t ) + σ }, t 0, (3) for every sesson, and we say that sesson conforms to (σ, ρ, C ), or A (σ, ρ, C ). The arrval constrant s attractve snce t restrcts the amount of traffc n terms of the average sustanable rate (ρ), peak rate (C), and burstness (σ and C). Fg. 2.2 shows how a farly bursty source mght be characterzed usng the constrants. Represent A (0, t) as n Fg Let there be l (t) bts worth of token n the sesson token bucket at tme t. We assume that the sesson starts out wth a full bucket of token. If K (t) denote the total amount of token accepted at the sesson bucket n an nterval (0, t], then Thus for all t, We may now express l (t) as From (5) and (6), we obtan the useful nequalty K (t) = mn 0t {A (0, ) + ρ (t )}. (4) K (t) K () ρ (t ). (5) l (t) = σ + K (t) A (0, t). (6) A (, t) l () + ρ (t ) l (t). (7) In ths thess we assume that C =, because ths s the easest case to vsualze (We do not have to worry about the nput lnks.), and t bounds the performance of the fnte capacty case. If A (, t) ρ (t ) + σ, t 0, (8) we say that sesson conforms (σ, ρ ) or A (σ, ρ ). Fnally, we ntroduce greedy regme. Sesson s called greedy start at tme f A (, t) = ρ (t ) + l (), t. In terms of the leaky bucket, ths means that the sesson uses as much token as possble for all tme t (Fg. 2.2). Let  denote the arrval process that all sessons greedly start at tme 0. We wll call ths arrval process all-greedy arrval process at tme 0 or all greedy regme at tme 0. Let Â(, t) denote the amount of sesson flow n an nterval, t under Â. We then have 3 Analyss of a sngle GPS server  (0, t) = ρ t + σ, t 0. (9) In ths secton, we show bounds of backlog and delay for each sesson under the assumpton that a servce process r(t) s gven. To obtan those bounds, we defne a fcttous worst-case servce process, ˆr(t) for the gven servce process, r(t). We show the bounds under ˆr(t) are greater than those under r(t) and those bounds are fcttously acheved under a fcttous worst-case regme, whch s an extenson of all-greedy regme n the constant rate case [1]. We also show a way to characterze the burstness of the output traffc for every sesson, whch wll especally useful n our analyss of GPS networks. Fnally, we relate the results of GPS to those of PGPS. We analyze a sngle GPS server that serves N leaky-bucket constraned sessons (1,..., N). Let A (, t) denote the amount of sesson flow that enters the network n an nterval (, t]. We then have A (, t) ρ (t ) + σ, t 0. (10) 6

7 A (0, t) S (0, t) A (0, ) D () PSfrag replacements Q (a) a t Fgure 4: A (0, t), S (, t), Q (t), and D (t). Let r(t) denote the servce process. Suppose r(t) satsfes and R(, + t) lm > t t N ρ, 0, (11) r(t) r, t 0. (12) (11) and (12) mean that the average servce rate s greater than total average sustan rate and the maxmum servce rate s upper-bounded by r, respectvely. Note that the constant rate case,.e., r(t) = r, satsfes (11) and (12). We assume that the server s empty at tme 0. We now defne backlog and delay for each sesson. Note that S (0, t) s contnuous and nondecreasng for all t. (see Fg. 3.) The sesson backlog at tme, Q (), s defned to be Q () = A (0, ) S (0, ). The sesson delay at tme s denoted by D (), whch s the amount of tme that t would take for the sesson backlog to clear f no sesson flow arrved after tme. Thus we have D () = nf{t : S (0, t) = A (0, )}. From Fg. 3, we see that D () s the horzontal dstance between curves A (0, t) and S (0, t) at the ordnate value of A (0, ). We are nterested n obtanng the upper bound of the backlog and delay over all tme and all arrval process, gven a servce process r(t). We defne the maxmum backlog and delay for sesson : Q = max (), A 0 (13) D = max (). A 0 (14) The problem we wll solve n the followng s to obtan the upper bound of Q and D for every sesson and any arrval process, gven weghts φ 1,... φ N and servce process r(t) for a GPS server and (σ, ρ ), = 1,..., N. 7

8 3.1 Defnton and prelmnary results In ths subsecton, we dscuss a GPS server for an arbtrary arrval process that satsfes (10). We defne σ for sesson and tme 0 as σ = Q () + l (). (15) Thus σ denotes the sum of the amount of token left n the bucket and the sesson backlog at tme. From (7) and (15), we have Note that Thus we establsh the followng lemma: Q () + A (, t) Q (t) σ σt + ρ (t ). S (, t) = Q () + A (, t) Q (t). Lemma 3.1 For every sesson and n any nterval [, t]: S (, t) σ σ t + ρ (t ). (16) Note that ths lemma s an extenson of Lemma 2 n [1]. Defne a system busy perod as the maxmal nterval B such that for any nterval [, t] B: N S (, t) = R(, t). We show an mportant proposton wth regard to a system busy perod. =1 Proposton 3.1 For any servce process r(t), the length of a system busy perod s upper-bounded. Let denote the tme when a system busy perod starts. We denote the length of the system busy perod as t (). Suppose each sesson has sent any traffc before tme and s greedy startng from. Let ˆt () denote the length of the system busy perod that starts at,.e., ˆt () = nf{ t R(, + t) > 1N ρ t + σ }. (17) Snce ˆt () > t (), we wll prove ˆt () <. From (11), for any small ɛ > 0, there exsts M < such that for all t M. Thus we have R(, + t) ρ + ɛ. t Defne t() as max{m, σ /ɛ}. For t t(), We then have R(, + t) Ths argument holds for any. We defne ˆt as 1N 1N 1N ρ t + ɛt (18) ρ t + σ. (19) ˆt () t() <. (20) ˆt = max 0 ˆt (). (21) Note that ˆt s the upper-bound of the length of the system busy perod for any arrval process,.e., t () ˆt for any and ˆt s fnte from (20). 8

9 Snce the system s work conservng, f B = [t 1, t 2 ], then N =1 Q (t 1 ) = N =1 Q (t 2 ) = 0. We now defne a sesson busy perod as the maxmal nterval B n a sngle system busy perod, such that for all, t B : S (, t) S (, t), = 1,..., N. φ In the next lemma, we show an useful nequalty. Lemma 3.2 Assume that an nterval [, t] s contaned n a sesson p busy perod. For any subset M of m sessons, 1 m N and any tme t : S p (, t) R(, t) / M {ρ (t ) + σ } M φ φ p. (22) By defnton of GPS, From (16) and σ t 0, We then have S (, t) φ p S p (, t), = 1, 2,..., N. S (, t) σ + ρ (t ), = 1, 2,..., N. S (, t) mn{σ + ρ (t ), φ p S p (, t)}, = 1, 2,..., N. (23) Snce the system s busy n [, t), the server serves exactly R(, t) unts of traffc n [, t). Thus we have R(, t) N M mn{σ + ρ (t ), φ φ p S p (, t)} (24) σ + ρ (t ) + M φ φ p S p (, t). (25) Rearragng the terms yelds the result. Note that we can derve Lemma 6 n [1] by replacng R(, t) wth (t ),.e., assumng r(t) = Fcttous worst-case regme bound We defne a fcttous worst-case servce process. Suppose r(t) satsfes (11) and (12). ˆr(t) s called a fcttous worst-case servce process for r(t) f ˆr(t) satsfes the followngs. ˆr(t) satsfes (11) and (12). For any tme 0, where ˆR(, t) = ˆr(x)dx and ˆt () s defned n (17). ˆr(t) s monotone nondecreasng for t 0. From Proposton 3.1, we establsh the followng result. ˆR(0, t) R(, + t), 0 < t < ˆt (), (26) Proposton 3.2 Suppose a servce process r(t) satsfes (11) and (12). There exsts at least one fcttous servce process ˆr(t) for r(t). We choose ˆr(t) such that ˆr(t) = { 0, f 0 t ˆt r, otherwse, P N ρ ˆt +σ r, 9

10 .e., ˆR(0, t) = [r(t ˆt ) + σ + ρ ˆt ] +, (27) 1N where [x] + stands for max{0, x} and ˆt s defned n (21). Ths functon satsfes frst and thrd condtons. Suppose there exsts t ( < x < ˆt ) whch does not satsfy nequaton (26) and each sesson has sent any traffc before tme and s greedy startng from. If the server works at maxmal rate r from tme t, the busy perod does not termnate n ˆt. Ths contradcts the defnton of ˆt. We call the case where the arrval process s  (all-greedy regme from tme 0) and the servce process s ˆr(t) as a fcttous worst-case regme from tme 0. We now defne Ŝ(, t) as the amount of servce whch sesson receves between [, t] under the fcttous worst-case regme from tme 0. In Lemma 3.3, we show that the amount of servce n (0, t] can be computed under the fcttous worst-case regme. Lemma 3.3 Let ˆB(t) denote the set of sessons that are busy at tme t under a fcttous worst-case regme. We then have Ŝ (0, t) = ˆR(0, t) / ˆB(t) {ρ t + σ } ˆB(t) φ. (28) See Appendx A.1. In the next lemma we establsh the relatonshp between S and Ŝ. Lemma 3.4 Suppose that tme t s contaned n a sesson p busy perod that begns at. We then have S (, t) Ŝ(0, t ). (29) See Appendx A.2. Note that Lemmas 3.3 and 3.4 are extensons of Lemma 6-() and 10 n [1], respectvely, when r(t) = 1. These lemmas are very mportant to analyze backlog and delay. We now defne ˆQ (t) (resp. ˆD (t)) to be backlog (resp. delay) of sesson at tme t under a fcttous worst-case regme from tme 0,.e., ˆQ = max 0t ˆD = max 0t ˆQ (t), ˆD (t). The followng theorem s the man result n ths secton. The backlog and delay bounds for a gven servce process s upper-bounded by those under ts fcttous worst-case regme Theorem 3.1 For any servce process that satsfes (11) and (12) and any sesson, Q and D bounded by ˆQ and ˆD, respectvely,.e., are upper Q ˆQ, D ˆD. See Appendx A.3. Fnally, we focus on determnng, for every sesson, the least quantty σ out S (σ out, ρ, r). such that Ths defnton of output burstness s due to Cruz [5]. By characterzng n ths manner, we can begn to analyze the networks of GPS servers. There s a convenent relatonshp between σ out and Q n the constant servce rate (Lemma 12 n [1]). Ths relatonshp also holds n the varable servce rate case. 10

11 Lemma 3.5 σ out = Q. The argument n the proof of Lemma 12 of [1] holds n the varable rate case. See [1]. From ths Lemma and Theorem 3.1, we have the followng result. Corollary 3.1 A Fluctuaton-Constraned GPS Server σ out ˆQ. (30) We now consder a fluctuaton-constraned GPS server,.e., we assume a convex fluctuaton constrant nstead of (11). Let r(t) denote the nstantaneous output transmsson capacty of a varable-rate server. The server s sad to be fluctuaton constraned [7], r (δ, µ), f: r(x)dx {µ(t ) δ} +. (31) Ths constrant s vald n consderng the followng stuaton: there are two classes served by a server. Sessons n class 1 are served n GPS dscplne and other sessons (class 2) are served n FCFS. The servce rate s r (constant). Each sesson n class 1 s gven prorty to ones n class 2,.e., a packet n the queue of class 2 does not begn to be served untl there s no packet n class 1. Suppose that the servce of any packet s not nterrupted. The nstantaneous output transmsson capacty of the server for class 1, r 1 (t) s fluctuated constraned, r 1 (L max, r), where L max s the maxmum length of a packet of class 2. We wll extend ths constrant. Let f(t) denote non-decreasng convex functon for t 0 and f(0) 0. A server s sad to be convex fluctuaton constraned, r(t) f(t), f r(x)dx f(t ) +,. (32) Note that ths constrant agrees wth (31) when f(t) = µt δ. We now consder a GPS server that serves N sessons (1,..., N). The sessons are leaky-bucket constraned, A (σ, ρ ) ( = 1,..., N) and ts servce process r(t) s convex fluctuaton constraned, r(t) f(t) and upper-bounded by r, r(t) r for all t 0. We assume that the server s empty at tme 0. Let Q max and D max denote the maxmum backlog and delay, respectvely, for sesson over all tme t, arrval process A, and servce process r(t), respectvely,.e., Q max D max = max max r(t) A = max max r(t) A max Q () (33) 0 max D (). (34) 0 Note that we can choose f(t) + as ˆR(0, t) for any r(t). So, we have the next result from Theorem 3.1. Theorem 3.2 If there exsts t such that f(t ) N ρ t + σ, for every sesson, Q max and D max are acheved (not necessarly at the same tme) when the arrval process s  and the servce process s f(t)+. Thus we can compute Q max 3.3 Analyss of a PGPS server and D max n a way smlar to [12]. A problem wth GPS s that t s an dealzed dscplne that does not transmt packets as enttes. It assumes that the server can serve multple sessons smultaneously and that traffc s nfntely dvsble. Packet-by-packet GPS (PGPS) s a smple packet-by-packet transmsson scheme that s an excellent approxmaton to GPS even when the packets are of varable length. We wll adopt the conventon that a packet arrved only after ts last bt has arrved. In ths secton we study the relatonshp between PGPS and GPS dscplne under the same arrval and servce processes n the varable rate case. Let d p denote the tme when packet p wll depart under GPS. A good approxmaton scheme s workconservng scheme that serves packets n ncreasng order of d p. PGPS server pcks the frst packet that would complete servce n the GPS smulaton f no addtonal packets arrved after tme [1]. In the varable servce rate case, ths PGPS scheme s vald and we have the same result as n [1]. 11

12 Lemma 3.6 Let p and p denote packets n a GPS system at tme, and suppose that packet p completes servce before packet p f there are no arrval after tme. Packet p wll also complete servce before packet p for any arrval process after tme. See the proof of Lemma 1 n [1]. Let d p denote the tme at whch packet p departs under PGPS. We show that Theorem 3.3 For any packet p, dp where L max represents the maxmum packet length. See Appendx A.4. d p r(t)dt L max, Let S (, t) and S (, t) denote the amount of sesson traffc (n bts) served under GPS and PGPS n thenterval [, t]. Theorem 3.4 For all tme t and sesson : S (0, t) S (0, t) L max. See Appendx A.5. Let Q (t) and Q (t) denote the sesson backlog at tme t under PGPS and GPS, respectvely. It then mmedately follows from Theorem 3.4 that Corollary 3.2 For all tmes t and sesson Q (t) Q (t) L max. Vrtual Tme Implementaton of PGPS Vrtual tme mplementaton of PGPS for a constant rate s showed n [1] and for a varable rate s n [13]. In ths secton we ntroduce defnton of vrtual tme for varable rate n [13]. We consder arrvals and departures from the GPS server as events. Let t denote the tme when the th event occurs (smultaneous event are ordered arbtrarly). We denote the tme of the frst arrval of a system busy perod by t 1 = 0. Suppose that, the set of sessons that are busy n the nterval (t 1, t ) s fxed for each = 2, 3..., and we may denote ths set as B. Vrtual tme V (t) s defned to be zero for all tmes when the server s dle. Consder any system busy perod, and assume that t begns at tme zero. V (t) then evolves as follows: V (0) = 0, V (t 1 + ) = V (t 1 ) + R(t 1, t 1 + ) B, for t t 1, ( = 2, 3,...). (35) The rate of change of V, namely δv (t 1+) δ, s equal to r(t 1+) P B and each backlogged sesson receves δv (t servce at rate φ 1+) δ. Thus V can be nterpreted as ncreasng at the margnal rate at whch backlogged sessons receve servce. Suppose that the k th sesson packet arrves at tme a k and has length L k. We denote the vrtual tmes at whch ths packet begns and completes servce by S k and F k 0, respectvely. Defnng F = 0 for all, we have S k = max{f k 1, V (a k )}. F k = S k + Lk. (36) 12

13 In the constant rate case, there are three attractve propertes of the vrtual tme nterpretaton from the standpont of mplementaton. These propertes are vald n the varable case, too. Frst, the vrtual tme fnshng tmes can be determned at the packet arrval tme. Second, packets are served n order of vrtual tme fnshng tme. Fnally, we need only update vrtual tme when there are events n the GPS system. However, the prce to be pad for these advantages s some overhead n keepng track of sets B and R(t 1, t ), whch s essental n updatng of vrtual tme. Gven ths mechansm for updatng vrtual tme, PGPS for varable rate s defned as follows: When a packet arrves, the vrtual tme s updated and the packet s stamped wth ts vrtual tme fnshng tme. The server s work conservng and serves packets n an ncreasng order of the tme-stamp. However, PGPS s not feasble for hgh speed networks [13], because t may not be possble to accurately estmate r(t) due to the unpredctable and multple tme-scale varaton n VBR vdeo bt rate and ths would make the computaton of the vrtual tme V (t) more expensve. To overcome these dffcultes, some far queueng dscplnes are formulated (e.g. [13]). 4 Analyss of GPS networks In ths secton, we analyze a GPS network that serves N sessons. We frst explan a model of the GPS network. We show the stablty of Consstent Relatve Sesson Treatment (CRST) networks. We then show bounds of backlog and delay for each sesson, when servce processes of all nodes and nternal traffc are gven. Fnally, we relate the results of GPS to those of PGPS. 4.1 Prelmnares A network s modeled as a drected graph where nodes represent swtches and arcs represent lnks. A route s a path n the graph and the path taken by sesson s defned as P (). Let P (, k) denote the k th node n P () and K denote the total number of nodes n P (). The rate of the lnk assocated wth server m at tme t s denoted by r m (t). The amount of sesson traffc that enters the network n an nterval [, t] s gven by A (, t). We denote the amount of sesson traffc served at node P (, k) n the same nterval [, t] by S (k) (, t), k = 1,..., K. Thus S (K) represents the amount of traffc that leaves the network. We characterze the servce process by pseudo leaky bucket parameters σ (k) and ρ, so that S (k) (, t) σ (k) + ρ (t ), t 0,.e., S (k) (σ (k), ρ ). Often, we wll analyze a partcular server, m. Defne I(m) as the set of sessons that are served by server m. For every sesson I(m), let A m (σ m, ρ ) and S m (σ m,out, ρ ) denote the arrval process and the departure process at that node, respectvely. The weght of sesson at node m s denoted by φ m. Suppose arrval processes of all sessons and servce processes of all nodes are gven. We defne Q (k) (t) as the sesson backlog at node P (, k) at tme t. Smlarly, let Q m (t) denote the sesson backlog at node m P (). Thus f m = P (, k), then Defne the total sesson backlog at t as Q (k) (t) = Q m (t) = A m (0, t) S m (0, t). K Q (t) = Q (k) (t). Also, let D (t) denote the tme spent n the network by sesson flow that arrves at tme t. A network s called stable f D () < or Q () < for any sesson and any tme 0. We now ntroduce Assumptons 4.1. Assumptons 4.1 We assume the followngs. 13

14 (A) Each sesson s leaky bucket constraned. A (, t) ρ (t ) + σ, t 0. (B) The system s empty at tme 0. For any sesson, (C) For each node m and all 0, where R m (, t) = rm (x)dx. Q (0) = 0. R m (, + t) lm > t t N ρ, (D) For each node m, the maxmum servce rate s upper-bounded by r m. r m (t) r m, t 0. We are nterested n obtanng the upper bound of backlog and delay over all tme and over all arrval processes, when servce processes of all nodes are gven. We defne the maxmum backlog and delay for sesson : Q = max (), A 0 (37) D = max (). A 0 (38) We now defne a system (resp. sesson ) busy perod n the network as the maxmal nterval B (resp. B ) such that for every B (resp. B ), there s at least one server that s n a system (resp. sesson ) busy perod at tme. Fcttous Worst-case Bounds for a sngle Server There are two steps to provde bounds on delay and backlog. The frst step conssts of characterzng nternal traffc, so that at each node m and for I(m) we have σ m such that A m (σ m, ρ ). In the second step, the nternal characterzaton s used to analyze the sesson route for delay and backlog. Accordng to [2], we calculate upper bounds on the mnmum value σ m,out such that S m (σ m,out, ρ ) for each node m. Suppose that for every I(m), A m (σ m, ρ ) s gven. In secton 3, t was shown that upper bounds of backlog and delay of node m for a gven servce process r m (t) are acheved under a fcttous worst-case regme. Let  m denote the resultng sesson arrval process for all I(m) and Ŝ m denote the resultng servce functon at node m. Recall Ŝm (0, t) s convex n t. From Lemma 3.5, we can fnd the smallest value ˆσ m,out such that Ŝm (ˆσ m,out, ρ ). From the dscusson above, Thus we may bound the burstness of S m ˆσ m,out by ˆσ m,out. σ m,out. (39) 4.2 GPS networks under consstent relatve sesson treatment In ths subsecton we show that a CRST network s stable f Assumptons 4.1 hold. We wll provde an algorthm for characterzng nternal traffc for every sesson n a way smlar to [2]. Sesson s sad to mpede sesson at a node m f φm φ < ρ m ρ. Note that for any two sessons, and, that contend for a node m, ether sesson mpedes sesson or vce-verse, unless φm φ m = ρ ρ. Constant Relatve Sesson Treatment (CRST) GPS assgnment s one for whch there exsts a strct orderng of sessons such that for any two sessons and, f sesson s less than sesson n the orderng, then sesson does not mpede sesson at any node of the network. All sessons of a CRST network can be parttoned nto nonempty class H 1,..., H L, such that sessons n H k are mpeded only those n H l, l < k. If two sessons, are n the = ρm ρ m, at every node, m, that s common to the same class, ther routes are ether edge dsont or φm φ m routes of sessons and. Clearly, each sesson H 1 s not mpeded by any other sessons. 14

15 Lemma 4.1 For any sesson H 1, there exsts ˆt m < such that : ˆt m = nf{ t φ I(m) ˆr(t) > ρ }. See Appendx B.1. Under Âm and ˆr(t), the guaranteed backlog cleanng rate of sesson H 1 exceeds ρ after ˆt m. Thus we have ˆσ m,out = σ m + ρ ˆt m. From (39), we can upper-bound nternal traffc of all the sessons n H 1. We now ntroduce Lemma 3 and 4 n [2], because these lemmas hold n the varable rate case, too. They wll be crucal to us n contnung the process to the sessons whch belongs to the hgher ndexed classes. Lemma 4.2 (Lemma 3 n [2]) Suppose sessons and contend for a node m and that sesson does not mpede sesson. The value of ˆσ m,out s ndependent of the value of σ m. Lemma 4.3 (Lemma 4 n [2]) Suppose sessons I(m) for node m and that for every sesson I(m) that can mpede, σ m s bounded. σ m,out must be bounded as well. See the proofs of Lemmas 3 and 4 n [2]. These lemmas can be used to sequentally characterze nternal traffc of the sessons n classes H 2, H 3,..., H L. We can use the same procedure as n [2]. Compute H 1,..., H L. k = 1 Whle k L, for each sesson H k For p = 1 to K m = P (, p) Compute ˆσ m gven: σ m = ˆσ m for all sessons that mpede at m (computed earler steps) σ m as computed earler. = 0 for all sessons that do not mpede at m σ m. Set σ (p) k = k + 1. = ˆσ m,out. From (39), we have bounds on σ m theorem. for every sesson and node m P (). Thus we establsh the followng Theorem 4.1 A CRST GPS network s stable f Assumptons 4.1 hold. 4.3 Computng delay and backlog bounds for stable networks wth known nternal burstness and servce processes We consder a stable GPS system whose nternal traffc burstness can be known. We assume that servce processes of all nodes are gven. We are nterested n computng bounds of end-to-end backlog and delay for each sesson. Note that the maxmum backlog and delay at a sngle node of the network can be upper-bounded by applyng a fcttous worst-case regme bound. We can obtan the end-to-end bounds for sesson by addng the bounds of each node m P () (called Addtve Method [6]). However, these bounds are loose. The problem s that we are gnorng strong dependences among the queues at the nodes n P () [2]. So, we treat the sesson route as a whole. We defne a fcttous unversal servce curve, whch s an extenson of a unversal servce curve n the constant rate case [2]. Moreover we show that the bounds are fcttously acheved under a staggered fcttous worst-case regme. To smplfy notatons, we focus on a partcular sesson,, that follows the route 1, 2,..., K. Fg. 5 llustrates the system to be analyzed. Ths fgure s the same as Fg. 3 n [1]. We now assume ndependent sessons relaxaton. 15

16 PSfrag replacements A (σ, ρ ) A 2 A K K Leaky buckets Fgure 5: Analyzng the sesson route as a whole, under the ndependent sessons relaxaton. Sesson traffc enters the network so that t s consstent wth (σ, ρ ), and A m = S m 1 for m = 2, 3,..., K. The ndependent sessons at node m are free to send traffc n any manner as long as A m (σ m, ρ ) for every sesson I(m) {}, m = 1,..., K. The sessons I(m) {} (for m = 1,..., K) are free to send traffc n any manner as long as A m (σ m, ρ ). Sesson traffc s constraned to flow along ts route so that A m = S m 1. The value of D and Q, whch hold under the ndependent sessons relaxaton, must be upper bounds on the true values of these quanttes [2]. Thus we wll obtan the upper bounds for D and Q that hold under the ndependent sessons relaxaton. Fcttous Unversal Servce Curve A fcttous unversal curve of sesson s easly constructed by applyng a fcttous worst-case bound at each node of sesson. Intutvely, the value of the fcttous unversal curve at tme t s a bound on the amount of flow that can traverse the network n the frst t tme unt of a network sesson busy perod. To smplfy notatons, we wll focus on a sesson such that P () = (1, 2,..., K). The functons Ŝ 1,..., ŜK can be obtaned from the nternal traffc characterzaton of secton 3. We now defne U, as a sesson fcttous unversal servce curve,.e., The curve G k s defned as G k (t) = { U (t) = mn{g K (t), Â(0, t)}. (40) Ŝ 1 (0, t), f k = 1, mn [0,t] {G k 1 () + Ŝk (0, t )}, t tb k, otherwse, where t B m represents the duraton of a sesson busy perod at m under a fcttous worst-case regme. For t k m=1 tb m, we set k k G k (t) = G k ( t B m) + Â( t B m, t). (41) m=1 We now expand the recurson n terms of 1,..., k, where m corresponds to the mnmzng value for node m. Clearly, 1 = 0 and defne k+1 = t. Note that m+1 m t B m for each m = 1,..., k. We then have k G k = mn mn... mn Ŝ m (0, m+1 m ) k [0,t] k 1 [0, k ] 2 [0, 3] = mn k t m=1 m=1 m=1 k Ŝ m (0, m+1 m ). (42) 16

17 In the next Lemma we show that G k (t) must meet Â(0, t) at some tme before k m=1 tb m. Lemma 4.4 If Assumptons 4.1 hold, we have See Appendx B.2. G k ( k t B m) Â(0, m=1 k t B m). m=1 From (41), we must have G k (t) Â(0, t), for any t k m=1 tb m. There exsts B k k that < Â (0, t), f t < B k, G k (t) = Â (0, t), f t = B k, > Â (0, t), otherwse t > B k. Thus we have U (t) = { G K (t), t B K, Â (0, t), t > B K. m=1 tb m such Havng defned U, we now relate t to the sesson departure from the network. In the followng lemma, we establsh a relatonshp between S m and G m. Lemma 4.5 Suppose the sesson arrval process, A, and servce processes of each node m, r m (t) (m = 1,... K), are gven. Consder some tme such that Q () = 0. For each m and any t >, See Appendx B.3. S m Thus we have the followng theorem. Theorem 4.2 For every sesson : See Appendx B.4. (, t) mn {A (, V ) + G m V [,t] (t V )}. (43) Q max {Â(0, ) G K ()}, (44) 0 { } D max mn{ t G K (t) = Â(0, )}. (45) 0 (K, t)-staggered Fcttous Worst-case Regme In ths subsecton we make clear the relatonshp between a staggered fcttous worst-case regme and a sesson unversal servce curve. As n the prevous sectons, we wll focus on a staggered fcttous worst-case regme wth respect to sesson and assume that P () = {1, 2,..., K}. Any staggered fcttous worst-case regmes can be characterzed by a vector, (T 1,..., T K ), T 1 T 2... T K. Ths vector means that node works at rate r n the nterval [T 1, T ), all ndependent sessons at node do not send any traffc n the same nterval, and they smultaneously start a fcttous worst-case regmes at tme T. A (K, t)-staggered fcttous worst-case regme, t B K, s the staggered fcttous worst-case regme characterzed by (0, T 2,..., T K ) such that K Ŝ k (0, T k+1 T k ) = G K (t), (46) where T 1 = 0, T K+1 = t and T k+1 T k t B k for k = 1,..., K. Note that 17

18 Snce t B K, G K (t) = U (t). For each k = 1, 2,..., K 1, the staggered fcttous worst-case regme defned by (0, T 2,..., T k ) descrbes a (k, T k+1 )-staggered fcttous worst-case regme. Comparng (46) wth (42) t s clear that (T 2,..., T K ) s a mnmzng vector n (42). Thus the unversal servce curve can be used to determne (T 2,..., T K ). Thus we have the followng theorem. Theorem 4.3 For any (K, t)-staggered fcttous worst-case regme: S K (0, t) = G K (t). See Appendx B.5. Let ˆQ () and ˆD () denote the sesson backlog and delay at under a (K, t)-staggered fcttous worst-case regme for gven servce processes. From Theorems 4.2 and 4.3, we have the followng theorem. Theorem 4.4 Suppose the ndependent sessons relaxaton. Under (K, t)-staggered fcttous worst-case regmes, max 0 max 0 ˆQ () = max {Â(0, ) G K ()}, 0 ˆD () = max 0 A Convex Fluctuaton Constraned GPS Network { mn{ t G K (t) = Â(0, )} We now consder a GPS network that satsfes Assumptons 4.1-(A), (B), and (D). Moreover, we assume that ts nternal traffc can be known and also that the GPS servers n the network are convex-fluctuatonconstraned, nstead of Assumptons (4.1)-(C). Suppose f m (t) s non-decreasng convex functon for t 0 and f(0) 0 for each node m. We assume r m (t) f m. Let Q max denote the maxmum backlog and delay for sesson over all tme t, arrval process A, and each servce process r m (t), respectvely,.e., Q max = max max r m (t) m P () A D max = max max r m (t) m P () A and D max }. max Q (), (47) 0 max D (). (48) 0 Note that for each node m, we can choose f m (t) + as a fcttous worst-case servce process, ˆRm (0, t), for any servce process, r m (t). Thus we can compute Q max and D max from Theorems 4.2 and Analyss of PGPS networks We analyze a PGPS network n a way smlar to [2]. When packet szes are not neglgble, there are two effects to consder. Frst, packets must be served non-preemptvely,.e., once a server has begun servng a packet, t must contnue to do so untl completon. Secondly, no packet s elgble for servce untl ts last bt has arrved, snce n most networks wth heterogeneous lnk speeds, packets are not transmtted untl they have completely arrved. We assume that servce s not vrtual cut-through. If m 1 and m are successve nodes on a sesson s route, we cannot assume, as we dd n the prevous secton, that S m 1 = A m. In fact, for P () = {1, 2,..., K }: S m 1 (, t) A m (, t) Sm 1 (, t) L, m = 2,..., K, < t, (49) where L σ represents the maxmum packet sze for each sesson. Ths effect s llustrated n Fg. 6, whch s the same fgure as Fg. 8 n [2]. Note that snce the GPS server does not begn servng a packet untl ts last bt has arrved, t sees the arrvals as a seres of mpulses, such that the heght of each mpulse s at most L. However, snce we are not assumng any peak rate constrant n the nput characterzatons, A m s consstent wth (σ m + L, ρ ). Accordng to [2], we frst enforce the non-vrtual cut-through effect, but allow preemptve servce, and then ncorporate the effects of non-preemptve servce. 18

19 S m 1 (0, t) (σ m, ρ ) S m (0, t) PSfrag replacements Arrval as see: A m (0, t) (σ m + L, ρ ) tme Fgure 6: A system n whch the packet szes are non-neglgble, A m (0, t) represents the cumulatve arrvals seen by server m. The length of each mpulse of A m (0, t) s bounded by L, the maxmum packet sze for sesson. Snce L σ, t can be seen from the fgure that A m (σ m + L, ρ ). Noncut-Through GPS To analyze networks of GPS servers wth non-neglgble packet szes, we follow the same steps as we dd n secton 4.3,.e., we frst characterze nternal traffc n terms of leaky bucket parameters, and then bound the worst-case delay and backlog for each sesson by analyzng ts route as a whole. To ncorporate effects of packet lengths, we stpulate that (49) holds. Consder a GPS network wth CRST assgnments. Internal traffc can be characterzed usng essentally the same procedure as n secton 4.3 to compute the fcttous worst-case bounds. To analyze the sesson route, we proceed as follows. We defne Ŝm as the sesson output at node m under a fcttous worst-case regme. Thus the sesson fcttous unversal servce curve s computed. In what follows we assume for smplcty n descrpton that P () = {1, 2,..., K}. Lemma 4.6 Consder some tme such that Q () = 0. For each m (1 m K) and each t >, S m (, t) mn V [,t] {A (, V ) + G m (0, t V )} ml. (50) See Appendx B.6. We now defne Q and D as the maxmum sesson packet backlog and delay, respectvely. From Lemma 4.6, we have the followng theorem. Theorem 4.5 For every sesson : Q max {Â(0, ) G K ()} + KL, (51) 0 { } D max mn{t G (t) = Â(0, ) + (K 1)L }. (52) 0 See Appendx B.7. Nonpreemptve Servce: PGPS Suppose a network of PGPS servers s gven, where the assgnments of the s meet the CRST requrements of secton 4.2. Consder H 1 and let P () = {1, 2,..., K }. We know from Corollary 3.2 that Q 1 () Q1 () L max, (53) 19

20 for all where Q m () and Qm () represent the sesson backlogs at node m, under PGPS and GPS, respectvely. Thus we have Q 1, Q 1, L max. (54) From (54) and Lemma 3.5, we can characterze nternal traffc at each node n P () from the same procedure as n the prevous secton. If applyng a fcttous worst-case bound to a node yelds a bound of σ out,m, then the bound on ths quantty under PGPS s ust σ out,m + L max. Thus we can characterze the nternal traffc at each node n P () under PGPS. The next step s the analyss of delay along the sesson route. We have the followng theorem: Theorem 4.6 Suppose Assumptons 4.1 hold. For each sesson : K D max mn{t : 0 GK (t) = Â(0, ) + (K 1)L } + nf{ t ˆRm (0, t) L max }. See Appendx B.8. m=1 5 Statstcal analyss of GPS networks In ths secton, we frst analyze a sngle GPS server whose ncommng traffc and servce rate are consstent wth exponentally bounded burstness (E. B. B.) process and exponentally bounded fluctuaton (E. B. F.) process, respectvely. Next, we show the stablty of CRST networks wth an arbtrary topology. 5.1 Prelmnares We consder a sngle GPS server that serves N sessons. For 1 N, we defne A and S as the arrval process for sesson and the correspondng servce process, respectvely. For any t, let A (, t) denote the amount of traffc from sesson that arrves durng the nterval [, t], and S (, t) denote the amount of servce sesson receved durng the same perod. The sesson backlog at tme t, denoted by Q (t), s gven by Q (t) = sup t {A (, t) S (, t)}. The delay experenced by sesson traffc arrvng at tme t s denoted by D (t). The nterval of a sesson busy perod s denoted by B and that of the system busy perod s denoted by B. We use E. B. B. process and E. B. F. process, whch are ntroduced n [11] and [7] to model source traffc and a fluctuated server, respectvely. We assume that sesson arrval process s a (ρ, Λ, α )-E. B. B. process,.e., for any and t, A (, t) has the followng property: for any x 0, P r(a (, t) > ρ (t ) + x) < Λ e αx. (55) We wll call ρ the long-term upper rate of the arrval process, Λ the prefactor, and α the decay rate of the exponental decay functon. Let r(t) denote the servce rate of the server at tme t. We assume that the servce process s a (r, M, β)-e. B. F. process,.e., for any and t, r(t) has the followng property: for any x 0, P r( r(x)dx < r(t ) x) < Me βx. (56) We wll call r the long-term lower rate of the servce process, M the prefacter, and β the decay rate of the exponental decay functon. As a necessary stablty condton, we requre that N =1 ρ < r. To smplfy notatons, we defne R(, t) = r(x)dx. A correspondng concept of these processes s the concept of an exponentally bounded (E. B.) process. In [11], stochastc process X(t) s called an (α, Λ)-E. B. process f for any t and any x 0, P r (X(t) x) Λe αx. (57) In [1], [2], Parakh and Gallager showed that, gven N =1 ρ < r, there exsts an orderng among the sessons such that, after relabelng the sessons, ρ < N = φ 1 (r ρ ), 1 N. (58) Such an orderng s called a feasble orderng wth respect to {ρ } 1N, { } 1N, and r. In general, there are many feasble orderngs assocated wth these factors. 20

21 5.2 Sample path behavor of a sngle GPS server We wll use Decomposton [4] to study sample path behavor of sessons. By abuse of notaton, let A and r(t) also denote a sample path of a random arrval process A and r(t) respectvely, so A (, t) denotes PSfrag the amount replacements of traffc from sesson durng the tme nterval [, t] and r(t) denotes the rate of the server at tme t on these sample paths. Smlarly, we wll use S, Q, and D to denote the correspondng sample paths of the correspondng random process S, Q, and D. Imagne that we dvde a GPS server nto a set of N fcttous servers. Let r (t) denote the servce process of each server for 1 N. See Fg. 7. Ths fgure s almost the same as Fg. 1 n [4]. We set sesson 1 Q 1(t) sesson 1 δ 1(t) r 1(t) sesson 2 Q 2(t) r(t) sesson 2 δ 2(t) r 2(t).... sesson N Decomposton Q N(t) sesson N δ N (t) r N(t) Fgure 7: Decomposton of a GPS Server. r (t) = r r r(t), where r > N =1 r. Note that r (t) s (r, M, βr r )-E. B. F. process. We now defne M = M, β = βr r. Thus r (t) s (r, M, β )-E. B. F. process. For each sesson n the decomposed system, let δ (t) denote ts backlog at tme t. We want to bound the actual sesson backlog Q (t) of the real-gps system n terms of the fcttous sesson backlog δ (t) s of the magnary decomposed system. We now should decde r. Obvously we must have N =1 r (t) r(t) and ρ < r for δ to be well-defned. We need to mpose an addtonal relaton on the r s n order to reflect the GPS schedulng dscplne, correspondng to relaton (58). We choose a set of r s such that, after some relabelng of the sessons, ρ < r N = φ 1 (r r ), 1 N. (59) As long as N =1 ρ < N =1 r < r, such a feasble orderng of the sessons always exsts [1]. Wthout loss of generalty, we assume that 1, 2,..., N s a feasble orderng of the N sessons wth respect to {r } 1N. From the study of G/G/1 queue, t s well known that δ (t) can be expressed n the form, δ (t) = sup{a (s, t) R (s, t)}. (60) st Clearly, δ (t) 0 for any t. For t, applyng (60) to δ () and δ (t), we can derve the followng useful nequaltes: A (, t) R (, t) + δ (t) δ () (61) R (, t) + δ (t). (62) For any t, let η (t) = Q (t) δ (t). As S (, t) = A (, t) + Q () Q (t), from (62) we have where the last nequalty follows from Q (t) 0. We now state an mportant fact. Lemma 5.1 For each sesson and any t, S (, t) R (, t) + η () η (t) (63) R (, t) + η () + δ (t), (64) Q (t) δ (t). (65) 21

22 See appendx C.1. Ths lemma says that on a sample path bass, the sum of the actual backlogs of the frst sessons of a feasble orderng n the real GPS system s upper-bounded by the sum of the fcttous backlogs of the correspondng sessons n the magnary decomposed system. We can also bound the actual backlog of each ndvdual sesson of the real GPS system n terms of the δ (t) s, but frst we need to establsh a lower bound on the sesson servce functon S, when t s n a sesson busy perod. Ths s stated n Lemma 5.2. The bounds on ndvdual sessons are then gven n Lemma 5.3. Lemma 5.2 For any t, let denote the begnnng of a sesson busy perod that constrants t. Thus S (, t) R (, t) N = φ 1 δ (t). (66) See Appendx C.2. Lemma 5.3 For any t, Q (t) δ (t) + N = φ 1 δ (t). (67) See Appendx C.2. For t, S (, t) A (, t) + Q (), and therefore applyng Lemma 5.3 to Q () yelds the followng. Lemma 5.4 For any t, S (, t) A (, t) + δ (t) + N = φ Note that these three lemmas correspond to the results n [4]. 5.3 Statstcal analyss of a sngle GPS Server 1 δ (t). In the prevous secton we have establshed several useful relatons on backlog on a sample path bass. In ths secton, we wll use E. B. B. process and E. B. F. process to establsh upper bounds on the tal dstrbutons of backlog and delay for each sesson. From Lemmas 5.3 and 5.5, we see that n order to bound the tal dstrbutons of Q (t) we only need to bound δ (t) for any t. We can use the result n [7], whch provdes an upper bound on δ (t). Lemma 5.5 For any x > 0, P r(δ (t) x) (Λ + M )e γρξ 1 e γɛξ e γx, (68) where 1 γ = 1 α + 1 β, ɛ = r ρ, and ξ > 0 s arbtrary dscrtzaton parameter consstent wth (Λ +M )e γ ρ ξ 1 e γ ɛ ξ > 1. See Appendx C.4. We wll choose ξ = 1 n the rest of the thess. We are nterested n boundng the moment generatng functon of δ (t),.e., E[e θδ(t) ] for some θ. By modfyng the proof of Lemma 5 n [4], we can establsh the followng result. 22