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The Impact of Long-Range-Dependent Traffic on Network Performance George Lin Ph.D. Defense Presentation Aug. 18, 2000 Outline Introduction – long-range-dependent traffic – motivation – overview and contributions Performance analysis with LRD input traffic – off-line performance analysis finite buffer queueing analysis reassembly and multiplexing queueing analysis – on-line sensitivity queueing analysis – summary Concluding remarks and future directions Aug. 18, 2000 George Lin - Defense Presentation 2 Introduction Long-Range-Dependent Traffic Network traffic exhibits long-range-dependent (LRD) property – LAN – WAN – Internet (WWW) – VBR video Implication – autocorrelation function decays slower than exponential – bursty at a wide range of time scales Aug. 18, 2000 George Lin - Defense Presentation 4 Long-Range-Dependent Traffic Ethernet trace Poisson arrivals / i.i.d. packet sizes 1.5 104 1.5 104 Bytes per sample Bytes per sample 1 104 1 104 5000 5000 0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 sample count (Sample interval = 0.01 sec.) sample count (Sample interval = 0.01 sec.) 1 105 1 105 Bytes per sample Bytes per sample 5 104 5 104 0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 sample count (Sample interval = 0.1 sec.) sample count (Sample interval = 0.1 sec.) 4 106 4 106 Bytes per sample Bytes per sample 2 106 2 106 0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 sample count (Sample interval = 10 sec.) sample count (Sample interval = 10 sec.) Aug. 18, 2000 George Lin - Defense Presentation 5 Long-Range-Dependent Traffic LRD Traffic Models – Fractional Gaussian noise (FGN) – Fractional autoregressive integrated moving average (FARIMA) – Wavelet Heavy-tailed on/off sources compelling physical evidence file sizes and packet trains are heavy tailed M/G/ – video, FTP, TELNET; source behavior – model LRD traffic with G having a heavy tailed distribution – model SRD traffic with G having an exponential distribution Aug. 18, 2000 George Lin - Defense Presentation 6 Motivation It’s important to study the impact of LRD traffic Off-line performance analysis – existing methods assume infinite buffer queues driven by LRD input traffic study the asymptotic tail behavior of the buffer overflow probability – buffer overflow probability is the probability that the buffer occupancy exceeds a given threshold value in the steady state – potential limitations of existing methods asymptotic tail behavior often capture only the most slowly decreasing term of the buffer overflow probability apply to a limited range of parameter values crucially rely on the infinite buffer assumption – derive the buffer overflow probability rather than the loss probability – loss probability is the fraction of lost work in the steady state Aug. 18, 2000 George Lin - Defense Presentation 7 Motivation On-line sensitivity queueing analysis – existing methods determine performance sensitivities with respect to network parameters – determine how performance measures vary with the changes in network parameters determine the performance sensitivities by observing a sample path – potential limitations of existing methods only determine performance sensitivities with respect to continuous parameters crucially rely on the Markov structure of the systems Aug. 18, 2000 George Lin - Defense Presentation 8 Overview and Contributions Study the impact of LRD traffic – Finite buffer qeueing analysis develop an off-line method which assumes a buffer with finite capacity and are based on non-asymptotic method determine the significance of finite buffer assumption and non-asymptotic analysis – loss probability – we show that existing analysis assuming infinite buffer significantly underestimates the network performance – Reassembly and multiplexing queueing analysis develop off-line methods which are based on non-asymptotic analysis determine practical impact of LRD traffic and non-asymptotic analysis – buffer overflow probability – frame loss probability – for reassembly queueing, we show that LRD traffic has no significant impact – for multiplexing queueing, we show that existing asymptotic analysis significantly underestimates the impact of LRD traffic when the buffer size is small Aug. 18, 2000 George Lin - Defense Presentation 9 Overview and Contributions Study the impact of LRD traffic – On-line sensitivity queueing analysis develop an on-line method which utilizes the proportional relationship determine performance sensitivity with respect to discrete parameters for a queueing system with LRD traffic – loss probability – mean queue length – mean delay – we show our method is useful for systems which are not amenable to existing on-line methods Aug. 18, 2000 George Lin - Defense Presentation 10 Performance Analysis with LRD Input Traffic Off-Line Performance analysis Finite buffer queueing analysis Reassembly and multiplexing queueing analysis On-line sensitivity queueing analysis Summary Finite Buffer Queueing Analysis Goal: Investigate the significance of the finite buffer assumption – analyze the performance of a network multiplexer with finite buffer capacity network multiplexers are fundamental building block for sharing network resources such as bandwidth and buffer space – obtain loss probability based on non-asymptotic method rather than buffer overflow probability determine the implication of using buffer overflow probability as the performance measure (or to approximate loss probability) – determine the impact of LRD traffic Aug. 18, 2000 George Lin - Defense Presentation 12 Queueing Model General fluid input process – include a large class of LRD and SRD processes M/G/ Gaussian process – the instantaneous input rate Lu Lu takes on integer values P[Lu=k], E[Lu] t total work flow in [0,t], L u du. if (0 < X u B) 0 Buffer with finite capacity B dX u L u c or ( X u , 0, L u c) Single server with constant output rate c du or ( X u B, L u c) 0 otherwise buffer occupancy Xu Aug. 18, 2000 George Lin - Defense Presentation 13 Loss Probability ~ Let X t denote the stationary buffer occupancy in the steady state, ~ X t lim X s s ~ Let L t denote the stationary input rate in the steady state, ~ L t lim L s s amount of lost work in [0,s] Loss Probability lim s amount of arriving work in [0,s] E[ work lost rate] E[work arriving rate] 1 ~ ( k c) P[ X t B, L t k ] ~ ~ E [ L t ] k c Aug. 18, 2000 George Lin - Defense Presentation 14 Loss Probability The joint probability in the loss probability is given as follows. P[ X t B, L t k ] (c n) ~ ~ ~ ~ P[Ws w, L t s n, L t k ]ds , 0 n0 w w B cs t ~ where Ws L u du. t s Proposition If the following condition holds: i) the input process in the steady state is stationary and ergodic, and ii) the average input rate is less than the constant output rate, then buffer full probability is given as follows. Buffer Full Probability = P[ X t B] (c n) ~ ~ P[Ws w, L t s n]ds , 0 n0 w w B cs t ~ where Ws L udu. t s Aug. 18, 2000 George Lin - Defense Presentation 15 Loss Probability High level proof for the Proposition: By extending Benes analysis, we show that P[ X t x ] 1 (c n) ~ ~ ~ P[Ws w, L t s n, X t s {0, B}]ds 0 n w w x cs 1 ( c n) ~ P[Ws w, L t s n]ds 0 n w w x cs ~ ~ ( c n) P[Ws w, L t s n,0 X t s B]ds 0 n w w x cs ~ ~ P[ X t B] 1 (c n) P[Ws w, L t s n]ds 0 n w w x cs ~ ~ ~ P[ X t B] 1 P[ X t B] (c n) P[Ws w, L t s n]ds 0 n w w x cs Aug. 18, 2000 George Lin - Defense Presentation 16 Finite Buffer Queueing Analysis Example ~ L t is characterized by a M / G / process M/P/ , H1=0.55, M/P/ , H3=0.9, M/M/ ~ ~ ( E[ L t ]) k E [ Lt ] ~ P[ L t k ] e k! ~ System load = E[ L t ] / c c = 1.55 Mbps Aug. 18, 2000 George Lin - Defense Presentation 17 Finite Buffer Queueing Analysis Results M/P/ , H1=0.55 This figure compares loss probability System load = 0.77, 0.58, 0.38 with buffer overflow probability – buffer overflow probability is the probability that buffer occupancy exceeds a given threshold value, where the threshold value equals to the buffer capacity of the corresponding finite buffer system – simulation results agree with our analysis. – buffer overflow probability significantly overestimate the loss probability Aug. 18, 2000 George Lin - Defense Presentation 18 Finite Buffer Queueing Analysis Results This figure shows the impact of LRD traffic when buffer size is small – LRD traffic has significant impact on network performance even when the buffer size is small Aug. 18, 2000 George Lin - Defense Presentation 19 Summary of Finite Buffer Queueing Analysis Investigate the significance of finite buffer assumption – Buffer overflow probability (existing analysis) significantly overestimates the loss probability, and designing networks using buffer overflow probability as the performance measure will cause inefficient network utilization – Existing analysis underestimate the impact of LRD traffic when the buffer capacity is small Aug. 18, 2000 George Lin - Defense Presentation 20 Performance Analysis with LRD Input Traffic Off-Line Performance analysis Finite buffer queueing analysis Reassembly and multiplexing queueing analysis On-line sensitivity queueing analysis Summary Reassembly and Multiplexing Queueing Analysis Goal: study the buffer requirements of reassembly and multiplexing operations in networks and determine practical impact of LRD traffic based on non-asymptotic mehod – intermediate network elements routers connectionless servers – interworking units network gateways application gateways (e.g., transcoders) Aug. 18, 2000 George Lin - Defense Presentation 22 IP over ATM Exit Router Packets ATM Switch Higher Higher Layer Layer IP IP IP AAL AAL AAL ATM ATM ATM ATM ATM PHY PHY PHY PHY PHY Aug. 18, 2000 George Lin - Defense Presentation 23 Queueing Model Frame ON/OFF Work Batches Reassembly Aggregated LRD Input Process Sources E[A] Queue – ON/OFF Sources R Aggreg. 1 LRD X1 Frame – Work-Batches; frames (MTU), cells Multiplexing R Input + Queue – M/G/; , E[A], 2 Process X2 – LRD, SRD MTU R + Nt XNt Frame Reassembly Queue – accumulate and reassemble – infinite buffer with a given threshold value ON/OFF Source Frame Multiplexing Queue Work-Batch (A) – re-segment and transmit ON OFF State R 0 State – infinite buffer with a given threshold value Cell T3 Frame (MTU) T2 T1 Aug. 18, 2000 George Lin - Defense Presentation 24 Differences in Queueing Models Finite buffer queueing analysis Reassembly and multiplexing – assume general fluid input queueing anaylsis process, and use M/G/ as an – assume M/G/ process with example the notion of frame (because – multiplexing queue with finite we obtain frame loss buffer capacity probability) – reassembly queue with infinite buffer – multiplexing queue with infinite buffer Aug. 18, 2000 George Lin - Defense Presentation 25 Analysis of the Both Queues Performance measures – buffer overflow probability the probability that buffer occupancy exceeds a certain threshold value in an infinite buffer system provides an upper bound to loss probability of the corresponding finite buffer system – frame loss probability the ratio between the number of lost frames and the number of total frames in the steady state a frame with cells arriving when the buffer occupancy exceeds the threshold value is lost Aug. 18, 2000 George Lin - Defense Presentation 26 Frame Reassembly Queue Results: Impact of LRD Traffic • This figure indicates that LRD traffic and Markov traffic yield similar queueing behavior Aug. 18, 2000 George Lin - Defense Presentation 27 Frame Multiplexing Queue Results: Impact of LRD Traffic • The Figure indicates that LRD traffic and Markov traffic yield similar behavior when the buffer size is small, but yield diverse behavior when the buffer size is large. Aug. 18, 2000 George Lin - Defense Presentation 28 Summary of the Reassembly and Multiplexing Queueing Analysis Frame reassembly operation – LRD does NOT have a significant impact finite MTU size reduces the negative effects of LRD – MTU size has a significant impact Frame multiplexing operation – LRD has a significant impact especially when target loss probability is small – MTU size is not a factor Aug. 18, 2000 George Lin - Defense Presentation 29 Performance Analysis with LRD Input Traffic Off-Line Performance analysis Finite buffer queueing analysis Reassembly and multiplexing queueing analysis On-line sensitivity queueing analysis Summary On-Line Sensitivity Queueing Analysis Goal: develop a new on-line performance sensitivity estimation method for systems with discrete parameters and LRD traffic – examine in real-time how performance measures would vary with the changes in system parameters example application: simulate the system with a given set of parameters, then obtain the entire performance measure vs. system parameter(s) curve with the simulation data – discrete parameters + LRD traffic raise difficulties exiting methods rely on the partial or complete knowledge of the Markov structure of the system Aug. 18, 2000 George Lin - Defense Presentation 31 Overview of Our Method Proportional Relationship Method Obtain the steady state probabilities of the nominal system from the observed sample path Obtain the performance measure of the nominal system Obtain the steady state probabilities of the perturbed system by utilizing the proportional relationship Obtain the performance measure of the perturbed system Calculate the differences between the performance measures of the two systems Aug. 18, 2000 George Lin - Defense Presentation 32 Queueing Model Controlled stream – customers arrive in batches – batch size is probabilistically determined by the queue length Uncontrolled stream – customer arrive in batches – batch size is governed by an underlying Markov chain a (j ,N ,c) P[ An1 ,c) k |Yn( N ) j ] i k (N i i infinite or finite number of states structure of the Markov chain is unknown ai(,N,k,u ) P[ An1 ,u ) k , Pn1 j| Pn i ] j i (N i Buffer and single server P[ A ( N i ,u ) n 1 k | Pn 1 j ] P[ Pn 1 j| Pn i ] – nominal system with capacity N1 – perturbed system with capacity N0 Yn(Ni ) min((Yn( Ni ) 1) An 1i ) , N i ) 1 (N Zn 1i ) ((Yn( Ni ) 1) An 1i ) N i ) (N (N Aug. 18, 2000 George Lin - Defense Presentation 33 Example 1 Markov Uncontrolled Stream Nominal buffer capacity, N1=50 Perturbed buffer capacity, N0=40 Aug. 18, 2000 George Lin - Defense Presentation 34 Example 2 LRD Uncontrolled Stream Nominal buffer capacity, N1=200 Perturbed buffer capacity, N0=150 Aug. 18, 2000 George Lin - Defense Presentation 35 Summary of On-Line Sensitivity Queueing Analysis Develop the proportional relationship for on-line sensitivity queueing analysis – apply the proportional relationship method and perform sensitivity analysis of the feedback controlled queueing system with respect to buffer capacity – we show the proportional relationship method successfully perform sensitivity analysis with respect to discrete parameters for a system with LRD traffic – we show that our method is comparable with simulation methods Aug. 18, 2000 George Lin - Defense Presentation 36 Performance Analysis with LRD Input Traffic Off-Line Performance analysis Finite buffer queueing analysis Reassembly and multiplexing queueing analysis On-line sensitivity queueing analysis Summary Summary Performance analysis with LRD traffic – Finite buffer queueing analysis loss probability we show that existing analysis assuming infinite buffer significantly underestimates the network performance Existing analysis underestimate the impact of LRD traffic when the buffer capacity is small – Reassembly and multiplexing queueing analysis buffer overflow probability and frame loss probability we show that LRD has no impact on reassembly operations we show that existing asymptotic analysis significantly underestimates the impact of LRD traffic when the buffer size is small Aug. 18, 2000 George Lin - Defense Presentation 38 Summary Performance analysis with LRD traffic – On-line sensitivity queueing analysis utilize proportional relationship determine performance sensitivity with respect to discrete parameters for a queueing system with LRD traffic – loss probability – mean queue length – mean delay – we show our method is useful for systems which are not amenable to existing on-line methods Aug. 18, 2000 George Lin - Defense Presentation 39 Concluding Remarks and Future Directions Concluding Remarks Contributions Provide new methods for performance analysis and performance optimization with LRD traffic Aug. 18, 2000 George Lin - Defense Presentation 41 Future Directions Off-line performance analysis extension – application to designing static network components or configuring quasi-static network parameters – Finite buffer multiplexing queueing analysis extension Obtain asymptotic loss probability in close form apply to admission control based on a priori characterizations On-line sensitivity analysis extension – application to configuring dynamic network parameters apply the proportional relationship method to more realistic models apply the proportional relationship method to dynamic buffer allocation apply to admission control based on measurement data – improve estimation efficiency for LRD traffic Aug. 18, 2000 George Lin - Defense Presentation 42

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posted: | 12/19/2011 |

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