Effective Capacity-Based Quality of Service Measures for Wireless Networks

Transcription

1 Effective Capacity-Baed Quality of Service Meaure for Wirele Network Dapeng Wu Rohit Negi Abtract An important objective of next-generation wirele network i to provide quality of ervice (QoS) guarantee. Thi require a imple and efficient wirele channel model that can eaily tranlate into connection-level QoS meaure uch a data rate, delay and delay-violation probability. To achieve thi, in [8], we developed a link-layer channel model termed effective capacity, for the etting of a ingle hop, contant-bit-rate arrival, fluid traffic, and wirele channel with negligible propagation delay. In thi paper, we apply the effective capacity technique to deriving QoS meaure for more general ituation, namely, ) network with multiple wirele link, 2) variable-bit-rate ource, 3) packetized traffic, and 4) wirele channel with non-negligible propagation delay. Key Word: Wirele channel model, QoS, delay, effective capacity, large deviation theory. Pleae direct all correpondence to Dapeng Wu, Univerity of Florida, Dept. of Electrical & Computer Engineering, P.O.Box 630, Gaineville, FL 326, USA. Tel. (352) , Fax (352) , wu@ece.ufl.edu. URL: Carnegie Mellon Univerity, Dept. of Electrical & Computer Engineering, 5000 Forbe Avenue, Pittburgh, PA 523, USA. Tel. (42) , Fax (42) , negi@ece.cmu.edu. URL:

2 Tranmitter Data ource Link-layer channel Receiver Data ink Intantanteou channel capacity log(+snr) Network acce device Network acce device Channel encoder Channel decoder Modulator Demodulator Phyical-layer channel Wirele channel Received SNR Introduction Figure : A wirele communication ytem. Providing QoS guarantee i crucial in the development of next-generation packet-baed wirele communication network [4]. To upport QoS guarantee, QoS proviioning mechanim are required. A major problem in deigning QoS proviioning mechanim i the high complexity in characterizing the relation between the control parameter of QoS proviioning mechanim, and the calculated QoS meaure, baed on exiting channel model, i.e., phyical-layer channel model (ee Fig. ). Thi i becaue the phyical-layer channel model (e.g., Rayleigh fading model with a pecified Doppler pectrum) do not explicitly characterize a wirele channel in term of the link-level QoS metric pecified by uer, uch a data rate, delay and delay-violation probability. To ue the phyical-layer channel model for QoS upport, we firt need to etimate the parameter for the channel model, and then extract the link-level QoS metric from the model. Thi two-tep approach i obviouly complex, and may lead to inaccuracie due to poible approximation in extracting QoS metric from the model. Recognizing that the limitation of phyical-layer channel model in QoS upport, i the

3 difficulty in analyzing queue uing them, in [8], we propoed moving the channel model up the protocol tack, from the phyical-layer to the link-layer. We call the reulting model an effective capacity (EC) channel model [8], becaue it capture a generalized link-level capacity notion of the fading channel. Figure illutrate the difference between the conventional phyical-layer channel and the link-layer channel. In [8], we preented the EC channel model under the etting of a ingle hop, contant-bit-rate arrival, fluid traffic, and wirele channel with negligible propagation delay; in thi paper, we ue the effective capacity technique to derive QoS meaure for more general ituation, namely, ) network with multiple wirele link, 2) variable-bit-rate ource, 3) packetized traffic, and 4) wirele channel with nonnegligible propagation delay. The remainder of thi paper i organized a follow. In Section 2, we preent preliminary reult to familiarize the reader with the effective capacity technique. Section 3 to 6 preent effective capacity-baed QoS meaure for network with multiple wirele link, variable-bitrate ource, packetized traffic, and wirele channel with non-negligible propagation delay, repectively. Section 7 conclude the paper. 2 Preliminarie We firt formally define tatitical QoS, which characterize the requirement of a uer. Firt, conider a ingle-hop ytem, where the uer i allotted a ingle time varying channel. Aume that the uer ource ha a fixed rate r and a pecified delay bound D max, and require that the delay-bound violation probability i not greater than a certain value ε, thati, PrD( ) >D max } ε, () where D( ) i the teady-tate delay experienced by a flow, and PrD( ) >D max } i the probability of D( ) exceeding a delay bound D max. Then, we ay that the uer i pecified by the (tatitical) QoS triplet r,d max,ε}. Even for thi imple cae, it i not immediately obviou a to which QoS triplet are feaible, for the given channel, ince a rather complex queueing ytem (with an arbitrary channel capacity proce) will need to be analyzed. The key contribution of [8] wa to introduce a concept of tatitical delay-contrained capacity 2

4 Date Rate µ Intantaneou Channel Capacity Dmax Buffer r(t) Data ource D(t) Data ink Figure 2: A queueing ytem model. termed effective capacity, which allow u to obtain a imple and efficient tet, to check the feaibility of QoS triplet for a ingle time-varying channel. That paper did not deal with general ituation, e.g., network with multiple wirele link and multi-hop, variable-bitrate ource, packetized traffic, and wirele channel with non-negligible propagation delay, which we conider in thi paper. Next, we briefly explain the concept of effective capacity, and refer the reader to [8] for detail. Let r(t) be the intantaneou channel capacity at time t. Aume that, the aymptotic log-moment generation function of r(t) t Λ( u) = lim log E[e u 0 r(τ)dτ ] (2) t t exit for all u 0. Then, the effective capacity function of r(t) i defined a α(u) = Λ( u) u, u>0. (3) That i, t α(u) = lim log E[e u 0 r(τ)dτ ], u>0. (4) t ut Conider a queue of infinite buffer ize upplied by a data ource of contant data rate µ (ee Fig. 2). It can be hown [8] that if α(u) indeed exit (e.g., for ergodic, tationary, Markovian r(t)), then the probability of D( ) exceeding a delay bound D max atifie PrD( ) >D max } e θ(µ)dmax, (5) 3

5 where the function θ(µ) of ource rate µ depend only on the channel capacity proce r(t). θ(µ) can be conidered a a channel model that model the channel at the link layer (in contrat to phyical layer model pecified by Markov procee, or Doppler pectra). The approximation (5) i accurate for large D max. In term of the effective capacity function (4) defined earlier, the QoS exponent function θ(µ) can be written a [8] θ(µ) =µα (µ) (6) where α ( ) i the invere function of α(u). Once θ(µ) ha been meaured for a given channel, it can be ued to check the feaibility of QoS triplet. Specifically, a QoS triplet r,d max,ε} i feaible if θ(r ) ρ, whereρ =. log ε/d max. Thu, we can ue the effective capacity model α(u) (or equivalently, the function θ(µ) via (6)) to relate the channel capacity proce r(t) to tatitical QoS. Since our effective capacity method predict an exponential dependence (5) between ε and D max, we can henceforth conider the QoS pair r,ρ} to be equivalent to the QoS triplet r,d max,ε}, with the undertanding that ρ = log ε/d max. In the following ection, we extend the effective capacity technique to more general ituation. The following property i needed in the propoition in the ret of thi paper. Property (i) The aymptotic log-moment generation function Λ(u) definedin(2)ifinite for all u R. (ii) Λ(u) i differentiable for all u R. 3 QoS Meaure for Wirele Network In thi ection, we conider two baic network tructure for wirele network: one with only tandem wirele link (ee Figure 3) and the other with only parallel wirele link (ee Figure 4). In the following, Propoition and 2 give QoS meaure for thee two network tructure, repectively. Denote r k (t) (k =,,K) the intantaneou capacity of channel k at time t. For a 4

6 Node Node 2 Data ource Rate = µ Q Wirele channel Q 2 Wirele channel 2 Data ink Figure 3: A network with tandem wirele link. network with K tandem link, define the ervice S(t 0,t), for t 0andanyt 0 [0,t], by S(t 0,t)= inf t 0 t t K t K =t K k= and the aymptotic log-moment generating function } r k (τ)dτ, (7) t k tk Λ tandem ( u) = lim t t log E[e u S(0,t) ] (8) where S(0,t) i defined by (7); alo define the effective capacity of channel k by t α k (u) = lim log E[e u 0 rk(τ)dτ ], u>0. (9) t ut Propoition Aume that the log-moment generating function Λ tandem (u) defined by (8) atifie Property. Given the effective capacity function α k (u),k =,,K} of K tandem link and an external arrival proce with contant rate µ, the end-to-end delay D( ) experienced by the traffic travering the K tandem link atifie lim up log PrD( ) >D max } θ, if α(θ/µ) >µ, (0) D max D max and lim log PrD( ) >D max } = θ, where α(θ /µ) =µ, () D max D max where α(u) = Λ tandem ( u)/u. Moreover, the effective capacity α(u) atifie α(u) min k α k (u). (2) 5

7 Node Wirele channel Data ource Rate = µ Q Data ink Wirele channel K Figure 4: A network with parallel wirele link. For a proof of Propoition, ee the Appendix. Note that the capacity procee of the tandem channel are not required to be independent in Propoition. Propoition 2 Aume that the log-moment generating function Λ k (u) of each channel k in the network atifie Property. Given the effective capacity function α k (u),k =,,K} of K independent parallel link and an external arrival proce with contant rate µ, the endto-end delay D( ) experienced by the traffic travering the K parallel link atifie lim up log PrD( ) >D max } θ, if α(θ/µ) >µ, (3) D max D max and lim log PrD( ) >D max } = θ, where α(θ /µ) =µ, (4) D max D max where α(u) = K k= α k(u). For a proof of Propoition 2, ee the Appendix. Propoition and 2 ugget the following approximation PrD( ) >D max } e θ D max, (5) for large D max. In addition, α(u) pecified in Propoition and 2 can be regarded a the effective capacity of the equivalent channel of the network, which conit of tandem link 6

8 only or independent parallel link only. In Section 4 to 6, we will ue α(u) to characterize the equivalent channel of the network; and we will ue (4) only ince (4) i tighter than (3). 4 QoS Meaure for Variable-Bit-Rate Source In thi ection, we develop QoS meaure for the cae where the ource generate traffic at variable bit-rate (VBR). We conider two clae of VBR ource: leaky-bucket contrained arrival [2][9, page 5] and exponential proce with it effective bandwidth function known [][9, page 6]. Propoition 3 and 4 provide QoS meaure for thee two clae of VBR ource, repectively. Propoition 3 Aume that a wirele network conit of tandem link only or independent parallel link only; the effective capacity function of the equivalent channel of the wirele network i characterized by α(u); and the log-moment generating function Λ k (u) of each channel k in the network atifie Property. Given an external arrival proce contrained by a leaky bucket with bucket ize σ () and token generating rate λ (), the end-to-end delay D( ) experienced by the traffic travering the network atifie lim D max D max σ () /λ () log PrD( ) >D max } = θ, where α(θ /λ () )=λ (). (6) For a proof of Propoition 3, ee the Appendix. Eq. (6) ugget the following approximation PrD( ) >D max } e θ (D max σ () /λ () ), (7) for large D max. For convenience, we replicate the definition of the effective bandwidth [] here. Conider an arrival proce A(t), t 0} where A(t) repreent the amount of ource data (in bit) over the time interval [0, t). Aume that the aymptotic log-moment generating function 7

9 of a tationary proce A(t), defined a Λ(u) = lim t t log E[euA(t) ], (8) exit for all u 0. Then, the effective bandwidth function of A(t) i defined a α () (u) = Λ(u) u, u>0. (9) Propoition 4 Aume that a wirele network conit of tandem link only or independent parallel link only; the effective capacity function of the equivalent channel of the wirele network i characterized by α(u); an external arrival proce i characterized by it effective bandwidth function α () (u); and the log-moment generating function Λ k (u) of each channel k in the network and the log-moment generating function Λ () (u) of the external arrival proce atify Property. Denote u the unique olution of the following equation α () (u) =α(u). (20) The end-to-end delay D( ) experienced by the traffic travering the network atifie lim log PrD( ) >D max } = θ, where θ = u α () (u ). (2) D max D max For a proof of Propoition 4, ee the Appendix. Note that a ingle-link network i a pecial cae in Propoition 3 and 4. 5 QoS Meaure for Packetized Traffic In previou ection, we aumed fluid traffic. In thi ection, we extend the QoS meaure obtained previouly for the fluid model to the cae with packetized traffic. Thi i important ince in practical ituation, the packet ize i not negligible (not infiniteimal a in fluid model). 8

10 We aume the propagation delay of a wirele link i negligible, and the ervice at a network node i non-cut-through, i.e., no packet i eligible for ervice until it lat bit ha arrived. We alo aume a wirele network conit of tandem link only or parallel link only. For a network with tandem link only, the number of hop in the network i determined by the number of tandem link in the network; for a network with parallel link only, the number of hop in the network i one. We conider two cae: ) a contant-bit-rate ource with contant packet ize, and 2) a variable-bit-rate ource with variable packet ize. Propoition 5 and 6 give QoS meaure for thee two cae, repectively. Propoition 5 Aume that a wirele network conit of tandem link only or independent parallel link only; the effective capacity function of the equivalent channel of the wirele network i characterized by α(u); the log-moment generating function Λ k (u) of each channel k in the network atifie Property ; and the network conit of N hop. Given an external arrival proce with contant bit rate µ and contant packet ize L c, the end-to-end delay D( ) experienced by the traffic travering the network atifie lim D max D max N L c /µ log PrD( ) >D max} = θ, where α(θ /µ) =µ. (22) For a proof of Propoition 5, ee the Appendix. Eq. (22) ugget the following approximation for large D max. PrD( ) >D max } e θ (D max N L c/µ), (23) Propoition 6 Aume that a wirele network conit of tandem link only or independent parallel link only; the effective capacity function of the equivalent channel of the wirele network i characterized by α(u); the log-moment generating function Λ k (u) of each channel k in the network atifie Property ; and the network conit of N hop. Given a traffic flow having maximum packet ize L max and contrained by a leaky bucket with bucket ize σ () and token generating rate λ (), the end-to-end delay D( ) experienced by the traffic travering the network atifie lim D max D max N L max /λ () σ () /λ () 9 log PrD( ) >D max } = θ, (24)

11 where α(θ /λ () )=λ (). For a proof of Propoition 6, ee the Appendix. Eq. (22) ugget the following approximation PrD( ) >D max } e θ (D max N L max/λ () σ () /λ () ), (25) for large D max. Note that a ingle-link network i a pecial cae in Propoition 5 and 6. 6 QoS Meaure for Wirele Channel with Non-negligible Propagation Delay In previou ection, we aumed the propagation delay of a wirele link i negligible. In thi ection, we extend the QoS meaure obtained previouly to the ituation where the propagation delay of a wirele link i not negligible. We conider two cae: ) a fluid ource with a contant rate, and 2) a variable-bit-rate ource with variable packet ize. Propoition 7 and 8 give QoS meaure for thee two cae, repectively. Propoition 7 Aume that a wirele network conit of tandem link only or independent parallel link only; the effective capacity function of the equivalent channel of the wirele network i characterized by α(u); the log-moment generating function Λ k (u) of each channel k in the network atifie Property ; the network conit of N hop; and the i-th hop (i =,,N) incur a contant propagation delay d i. Givenafluidtrafficflowwithcontant rate µ, the end-to-end delay D( ) experienced by the traffic travering the network atifie lim D max D max N i= d i log PrD( ) >D max } = θ, where α(θ /µ) =µ. (26) For a proof of Propoition 7, ee the Appendix. Eq. (26) ugget the following approximation PrD( ) >D max } e θ (D max N i= d i), (27) for large D max. 0

12 Propoition 8 Aume that a wirele network conit of tandem link only or independent parallel link only; the effective capacity function of the equivalent channel of the wirele network i characterized by α(u); the log-moment generating function Λ k (u) of each channel k in the network atifie Property ; the network conit of N hop; and the i-th hop (i =,,N) incur a contant propagation delay d i. Given a traffic flow having maximum packet ize L max and contrained by a leaky bucket with bucket ize σ () and token generating rate λ (), the end-to-end delay D( ) experienced by the traffic travering the network atifie lim D max D max N L max /λ () where α(θ /λ () )=λ (). σ () /λ () N i= d log PrD( ) >D max } = θ, (28) i For a proof of Propoition 8, ee the Appendix. Eq. (28) ugget the following approximation for large D max. PrD( ) >D max } e θ (D max N L max/λ () σ () /λ () N i= di), (29) 7 Concluding Remark The deign of QoS proviioning mechanim in wirele network call for a imple and effective wirele channel model. In [8], we propoed and developed uch a imple and effective channel model, called effective capacity, for the etting of a ingle hop, contantbit-rate arrival, fluid traffic, and wirele channel with negligible propagation delay. In thi paper, we employed the effective capacity technique to derive QoS meaure for more general ituation, i.e., network with multiple wirele link, variable-bit-rate ource, packetized traffic, and wirele channel with non-negligible propagation delay. In our future work, the QoS meaure developed in thi paper will be ued to deign efficient mechanim to provide end-to-end QoS guarantee in a multihop wirele network. Thi will involve developing algorithm for QoS routing, reource reervation, admiion control and cheduling.

13 Acknowledgment Thi work wa upported by the National Science Foundation under the grant ANI-088. Appendix ProofofPropoition Denote Q k (t) the queue length at time t at node k (k =,,K), Q(t) the end-to-end queue length at time t, Q( ) the teady tate of the end-to-end queue length, A(t 0,t) the amount of arrival to node (ee Figure 3) over the time interval [t 0,t]. Define S k (t 0,t)= t t 0 r k (τ)dτ, which i the ervice provided by channel k overthetimeinterval[t 0,t]. We firt prove an upper bound. It can be proved [0, page 8] that Q(t) = K k= Q k (t) = up A(t 0,t) S(t } 0,t) 0 t 0 t (30) where S(t 0,t) i defined by (7). Without lo of generality, we conider the dicrete time cae only, i.e., t N, wheren i the et of natural number. From (30) and Loyne Theorem [6], we obtain Q( ) =up A(0,t) S(0,t) } =up µt S(0,t) } (3) 2

14 Then, we have PrQ( ) >q} = Pr (a) Pr up µt S(0,t) } } >q (32) µt S(0,t) >q} } (33) (b) Pr µt S(0,t) } >q (34) (c) e uq E[e u(µt S(0,t)) ] (35) where (a) ince the event up µt S(0,t) } } >q µt S(0,t) } >q, (b) i due to the union bound, and (c) from the Chernoff bound. Since α(u) = Λ tandem ( u)/u, we have α(u) = lim t ut log E[e u S(0,t) ], u>0, (36) Hence, for any ɛ>0, there exit a number t >0 uch that for t t, wehave If µ + ɛ<α(u), we have E[e u S(0,t) ] e u( α(u)+ɛ)t, u >0. (37) e uq E[e u(µt S(0,t)) ] (a) t t (b) e uq t e uq e u(µ α(u)+ɛ)t + e uq E[e u(µt S(0,t)) ] (38) t= eu(µ α(u)+ɛ) t t e + E[e u(µt S(0,t)) ] (39) u(µ α(u)+ɛ) t= where (a) from (37), and (b) from geometric um. From (35) and (39), we have PrQ( ) >q} γ e uq, if µ + ɛ<α(u), (40) 3

15 where γ i a contant independent of q. Hence, we obtain lim up q Letting ɛ 0, we have lim up q Since Q(t) =µ D(t), t 0, (42) reult in log PrQ( ) >q} u, if µ + ɛ<α(u). (4) q log PrQ( ) >q} u, if µ<α(u). (42) q lim up log PrD( ) >D max } µ u, if µ<α(u). (43) D max D max Let θ = µ u. Then (43) become (0). Next we prove a lower bound. Let q = βt where β>0. Then, we have lim inf q log PrQ( ) >q} = lim inf q t = lim inf t lim inf t = lim inf t log PrQ( ) >βt} (44) βt log Pr βt up µt S(0,t) } } >βt βt log Pr µt S(0,t) >βt log Pr βt } S(0,t) } >β µ t (45) (46) (47) (a) β inf x>β µ Λ (x) (48) where (a) from Gärtner-Elli Theorem [] ince Λ tandem (u) atifie Property, and the Legendre-Fenchel tranform Λ (x) ofλ tandem ( u) i defined by Λ (x) =upu x Λ tandem ( u)}. (49) u R 4

16 Since (48) hold for any β>0, we have lim inf q q log PrQ( ) >q} up β>0 β inf x>β µ Λ (x) (50) Λ (y) = inf y> µ y + µ (5) It can be proved [3] that Λ (y) inf y> µ y + µ = u, where Λ tandem ( u )= µ u. (52) From the definition of effective capacity α(u) in (36), Λ tandem ( u ) = µ u α(u )=µ. Then, applying (52) to (5) lead to implie lim inf q Due to the continuity of α(u), we have q log PrQ( ) >q} u, where α(u )=µ. (53) Hence, letting u u, (42) and (53) reult in u lim inf q Hence, we have lim α(u) =µ (54) u u log PrQ( ) >q} lim up q q q log PrQ( ) >q} u (55) lim q q log PrQ( ) >q} = u, where α(u )=µ. (56) Since Q(t) =µ D(t), t 0, (56) reult in lim log PrD( ) >D max } = µ u, where α(u )=µ. (57) D max D max Let θ = µ u. Then (57) become (). 5

17 From (7), it i obviou that S(t 0,t) min k Sk (t 0,t). (58) Then we have for u>0, α(u) = lim t ut log E[e u S(0,t) ] (59) (a) lim t ut log E[e u min k S k (0,t) ] (60) min lim k t ut log E[e u S k (0,t) ] (6) = min k α k (u) (62) where (a) from (58). Thi complete the proof. ProofofPropoition2 Denote r k (t) (k =,,K) channel capacity of link k at time t. From Figure 4, it i clear that the network ha only one queue and multiple erver, each of which correpond to a wirele link. Since the total intantaneou channel capacity r(t) = K k= r k(t), the effective capacity function for the aggregate parallel link i α(u) (a) t = lim log E[e u 0 r(τ)dτ ] t ut t K = lim log E[e u 0 k= rk(τ)dτ ] t ut (b) = lim t ut K k= log E[e u t 0 r k(τ)dτ ] (c) = K α k (u) (63) k= where (a) from (4), (b) ince r k (t),k =,,K} are independent, and (c) from (9). 6

18 Given the effective capacity α(u), we can prove (3) and (4) with the ame technique ued in proving (0) and (). Thi complete the proof. ProofofPropoition3 For the traffic of contant rate λ (),denote Q( ) the teady tate of the end-to-end queue length and D( ) the end-to-end delay. Uing the reult in [5, page 30]), we can how D( ) D( ) σ () /λ (), (64) Note that D( ) i the end-to-end delay experienced by the traffic contrained by a leaky bucket with bucket ize σ () and token generating rate λ (). Hence, we have D( ) D( )+σ () /λ (), (65) From (40) and Q( ) =λ () D( ), we have } Pr D( ) >Dmax γ e u λ() D max, if α(u) >λ (), (66) where γ i a contant independent of D max. Then, we have PrD( ) >D max } (a) = Pr } D( ) >Dmax σ () /λ () (b) γ e u λ() (D max σ () /λ () ) (67) where (a) from (65), and (b) from (66). Hence, we have lim up D max D max σ () /λ () log PrD( ) >D max } θ, if α(θ/λ () ) >λ (). (68) Similar to the proof of Propoition, we can obtain a lower bound lim inf D max D max σ () /λ () log PrD( ) >D max } θ, where α(θ /λ () )=λ (). (69) Combining (68) and (69), we obtain (6). Thi complete the proof. 7

19 ProofofPropoition4 The proof i imilar to that of Propoition. Denote Q(t) the end-to-end queue length at time t, Q( ) the teady tate of the endto-end queue length, A(t 0,t) the amount of external arrival to the network over the time interval [t 0,t]. From (30), we know } Q(t) = up 0 t 0 t A(t 0,t) S(t 0,t) (70) where S(t 0,t) i defined by (7) for the tandem link, and i defined by S(t 0,t)= t 0 K k= r k(τ)dτ for independent parallel link. We firt prove an upper bound. Without lo of generality, we conider the dicrete time cae only, i.e., t N, wheren i the et of natural number. From (70) and Loyne Theorem [6], we obtain } Q( ) =up A(0,t) S(0,t) (7) Then, we have PrQ( ) >q} = Pr Pr up Pr A(0,t) S(0,t) } } >q (72) A(0,t) S(0,t) >q} } (73) A(0,t) S(0,t) } >q (74) e uq E[e u(a(0,t) S(0,t)) ] (75) From the definition of effective capacity in (36), for any ɛ/2 > 0, there exit a number t >0 uch that for t t, wehave E[e u S(0,t) ] e u( α(u)+ɛ/2)t, u >0. (76) 8

20 Similarly, from the definition of effective bandwidth in (9), for any ɛ/2 > 0, there exit a number t >0 uch that for t t, wehave E[e ua(0,t) ] e u(α() (u)+ɛ/2)t, u >0. (77) Without lo of generality, here we chooe the ame t for both (76) and (77), ince we can alway chooe the maximum of the two to make (76) and (77) hold. Then, if α () (u)+ɛ< α(u), we have e uq E[e u(a(0,t) S(0,t)) ] (a) t t t e uq e u(α() (u) α(u)+ɛ)t + e uq E[e u(a(0,t) S(0,t)) ] (78) t= eu(α() e uq (u) α(u)+ɛ) t e + t E[e u(a(0,t) S(0,t)) ] (79) u(α() (u) α(u)+ɛ) t= where (a) from (76) and (77). From (75) and (79), we have lim up q Letting ɛ 0, we have lim up q q log PrQ( ) >q} u, if α() (u)+ɛ<α(u). (80) q log PrQ( ) >q} u, if α() (u) <α(u). (8) Similar to the proof of Propoition, we can obtain a lower bound lim inf q Combining (8) and (82), we have q log PrQ( ) >q} u, where α () (u )=α(u ). (82) lim q q log PrQ( ) >q} = u, where α () (u )=α(u ). (83) Since Q( ) =α () (u ) D( ), (83) reult in (2). Thi complete the proof. 9

21 ProofofPropoition5 For the packetized traffic, denote Q k (t) the queue length at time t at node k (k =,,N), Q(t) the end-to-end queue length at time t, andq( ) the teady tate of the end-to-end queue length. Correpondingly, for the fluid traffic of contant arrival rate µ, denote Q k (t) the queue length at time t at node k, Q(t) the end-to-end queue length at time t, Q( ) the teady tate of the end-to-end queue length, and D( ) the end-to-end delay. For each node k, we have the ample path relation a below [7] Q k (t) Q k (t) L c, t 0. (84) Summing up over k, weobtain N [Q k (t) Q k (t)] = Q(t) Q(t) N L c, t 0. (85) k= Hence, for the teady tate, we have Q( ) Q( ) N L c. (86) Since Q( ) =µ D( ) and Q( ) =µ D( ), we have D( ) D( ) N L c /µ. (87) Note that D( ) i the end-to-end delay experienced by the packetized traffic with contant bit rate µ and contant packet ize L c. Then, we can prove (22) in the ame way a we prove (6) in Propoition 3. ProofofPropoition6 Denote D( ) the end-to-end delay experienced by the fluid traffic with contant arrival rate λ (). Uing the ample path relation in [5, page 35]), we obtain D( ) D( ) N L max /λ () 20 + σ () /λ (), (88)

22 Note that D( ) i the end-to-end delay experienced by the packetized traffic having maximum packet ize L max and contrained by a leaky bucket with bucket ize σ () and token generating rate λ (). Then, we can prove (24) in the ame way a we prove (6) in Propoition 3. ProofofPropoition7 Denote D( ) the end-to-end delay experienced by the fluid traffic with contant arrival rate µ and without propagation delay. Uing the ample path relation between the two cae (with/without propagation delay), it i eay to how D( ) D( ) N d i, (89) i= Then, we can prove (26) in the ame way a we prove (6) in Propoition 3. ProofofPropoition8 Denote D( ) the end-to-end delay experienced by the fluid traffic with contant arrival rate λ () obtain and without propagation delay. Uing the ample path relation in [5, page 35]), we D( ) D( ) N L max /λ () + σ () /λ () + N d i, (90) i= Then, we can prove (28) in the ame way a we prove (6) in Propoition 3. Reference [] C.-S. Chang, Performance guarantee in communication network, Springer, [2] R. L. Cruz, A calculu for network delay, Part I: network element in iolation, IEEE Tran. on Information Theory, vol. 37, no., pp. 4 3, Jan

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