Stochastic Network Calculus

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1 Stochastic Network Calculus Assessing the Performance of the Future Internet Markus Fidler joint work with Amr Rizk Institute of Communications Technology Leibniz Universität Hannover April 22, 2010 c Markus Fidler IKT LUH 1/27

2 Outline Motivation Traffic characteristics Related results Research goal FBm sample path envelope Definition of envelopes FBm envelopes Backlog bound End-to-end analysis Leftover service curves Network service curves Scaling properties c Markus Fidler IKT LUH 2/27

3 Internet applications by volume News 11% Telnet 14% HTTP 5% 8% FTP 20% 2008/09 HTTP 74% Video 31% P2P 44% Mix of heterogeneous traffic characteristics and service requirements c Markus Fidler IKT LUH 3/27

Network performance evaluation 3 2 2 3 5 5 3 2 5 0 7 2 1

News 11% Telnet 14% application requirements quality of

4 Network performance evaluation traffic matrix links, routers, topology, routing traffic statistics network calculus evaluate, plan, optimize, design HTTP 5% 8% traffic mix FTP 20% News 11% Telnet 14% application requirements quality of service delay packet loss throughput fairness c Markus Fidler IKT LUH 4/27

5 Queuing networks Queuing theory has been used since the 60 s to understand the performance of computer networks, most prominently to prove the efficiency of packet switching over circuit switching. A th 1 2 n... A cr,1 A cr,2 A cr,n c Markus Fidler IKT LUH 5/27

6 Queuing networks Queuing theory has been used since the 60 s to understand the performance of computer networks, most prominently to prove the efficiency of packet switching over circuit switching. A th 1 2 n... A cr,1 A cr,2 A cr,n c Markus Fidler IKT LUH 5/27

7 Queuing networks Queuing theory has been used since the 60 s to understand the performance of computer networks, most prominently to prove the efficiency of packet switching over circuit switching. A th 1 2 n... A cr,1 A cr,2 A cr,n Basic assumption: Memoryless arrivals and service requirements closed form results for single queues product forms for tandem queues simple results for multiplexing and de-multiplexing of traffic c Markus Fidler IKT LUH 5/27

8 Self-similarity Zooming in/out does repeatedly reveal the same basic structures. c Markus Fidler IKT LUH 6/27

9 Memoryless versus Internet traffic s aggregation s aggregation data [kbyte] 10 5 data [kbyte] time [s] time [s] statistical self-similarity of a random process X(t): z H X(t) d = X(zt) long-range dependence: Hurst parameter H (0.5, 1) auto-covariance v(t) at lag t: t=0 v(t) = c Markus Fidler IKT LUH 7/27

10 Memoryless versus Internet traffic s aggregation s aggregation data [kbyte] 40 data [kbyte] time [s] time [s] statistical self-similarity of a random process X(t): z H X(t) d = X(zt) long-range dependence: Hurst parameter H (0.5, 1) auto-covariance v(t) at lag t: t=0 v(t) = c Markus Fidler IKT LUH 7/27

11 Memoryless versus Internet traffic s aggregation s aggregation data [kbyte] data [kbyte] time [s] time [s] statistical self-similarity of a random process X(t): z H X(t) d = X(zt) long-range dependence: Hurst parameter H (0.5, 1) auto-covariance v(t) at lag t: t=0 v(t) = c Markus Fidler IKT LUH 7/27

12 Memoryless versus Internet traffic s aggregation s aggregation data [kbyte] data [kbyte] time [s] time [s] statistical self-similarity of a random process X(t): z H X(t) d = X(zt) long-range dependence: Hurst parameter H (0.5, 1) auto-covariance v(t) at lag t: t=0 v(t) = c Markus Fidler IKT LUH 7/27

13 Fractional Brownian motion Fractional Brownian motion (fbm) is a model for self-similar traffic with long memory. FBm arrivals A(t) are composed of a mean rate λ and a correlated Gaussian increment process Z(t) with zero mean A(t) = λt + Z(t). The increment process is fully characterized by two parameters variance σ 2 Hurst parameter H c Markus Fidler IKT LUH 8/27

14 Fractional Brownian motion Fractional Brownian motion (fbm) is a model for self-similar traffic with long memory. FBm arrivals A(t) are composed of a mean rate λ and a correlated Gaussian increment process Z(t) with zero mean A(t) = λt + Z(t). The increment process is fully characterized by two parameters variance σ 2 Hurst parameter H FBm traffic has effective bandwidth α(θ, t) = 1 θt log E[eθA(t) ] = λ + θσ2 2 t2h 1 α lies between the mean rate and peak rate and characterizes the resource requirements of a flow. Note the continuous growth with t. c Markus Fidler IKT LUH 8/27

15 Lindley s recursion Model of a buffered work-conserving link with capacity C A(t), D(t) cumulative arrivals, resp., departures in [0, t) B(t) = A(t) D(t) backlog, i.e. unfinished work, at time t A(t) C D(t) Lindley equation B(t) B(t + 1) = max{0, B(t) + A(t, t + 1) C} By induction with B(0) = 0 B(t) = max {A(τ, t) C(t τ)} τ [0,t] c Markus Fidler IKT LUH 9/27

16 Largest term approximation Lindley s recursion for the backlog B(t) at a system with capacity C B(t) = sup {A(τ, t) C(t τ)}. τ [0,t] permits a stochastic formulation of a backlog bound b [ ] P[B > b] = P sup {A(τ, t) C(t τ)} > b τ [0,t]. c Markus Fidler IKT LUH 10/27

17 Largest term approximation Lindley s recursion for the backlog B(t) at a system with capacity C B(t) = sup {A(τ, t) C(t τ)}. τ [0,t] permits a stochastic formulation of a backlog bound b [ ] P[B > b] = P sup {A(τ, t) C(t τ)} > b τ [0,t]. For fbm arrivals known backlog bounds are derived from an approximation by the largest term { } P[B > b] sup P [A(τ, t) C(t τ) > b] τ [0,t] that strictly provides, however, only a lower bound. c Markus Fidler IKT LUH 10/27

18 Performance bounds for fbm traffic The fundamental finding is the asymptotic resp. approximate result ( lim P[B > b] = exp 1 ( ) C λ 2H ( ) ) b 2 2H b 2σ 2 = ε a H 1 H that has been derived from the Gaussian increments [Norros, 94] large deviations theory [Duffield, O Connell, 95] statistical envelopes [Fonseca, Mayor, Neto, 00]. c Markus Fidler IKT LUH 11/27

19 Performance bounds for fbm traffic The fundamental finding is the asymptotic resp. approximate result ( lim P[B > b] = exp 1 ( ) C λ 2H ( ) ) b 2 2H b 2σ 2 = ε a H 1 H that has been derived from the Gaussian increments [Norros, 94] large deviations theory [Duffield, O Connell, 95] statistical envelopes [Fonseca, Mayor, Neto, 00]. It provides clear rules for dimensioning, e.g. regarding packet loss is it better to double the buffer size b or is it better to double the unused capacity C λ? The result is, however, limited to through traffic at a single system. c Markus Fidler IKT LUH 11/27

20 Networks under fbm cross-traffic The goal of this work is to provide end-to-end performance bounds for a through flow in a network under fbm cross-traffic. A th 1 2 n... A cr,1 A cr,2 A cr,n We seek to answer: Given the delay of a through flow at single system is w. What is the end-to-end delay for n tandem systems? c Markus Fidler IKT LUH 12/27

21 Networks under fbm cross-traffic The goal of this work is to provide end-to-end performance bounds for a through flow in a network under fbm cross-traffic. A th 1 2 n... A cr,1 A cr,2 A cr,n We seek to answer: Given the delay of a through flow at single system is w. What is the end-to-end delay for n tandem systems? To this end, we use the framework of stochastic network calculus. Essential intermediate steps are an envelope for fbm sample paths a leftover service curve for systems under fbm cross-traffic a convolution form for service curves of tandem systems c Markus Fidler IKT LUH 12/27

22 Table of contents Motivation Traffic characteristics Related results Research goal FBm sample path envelope Definition of envelopes FBm envelopes Backlog bound End-to-end analysis Leftover service curves Network service curves Scaling properties c Markus Fidler IKT LUH 13/27

23 Point-wise versus sample path envelopes data envelopes sample paths time [from Liebeherr, statcalc talk, 05] the point-wise envelope is violated at most with probability 1/4 at any point in time c Markus Fidler IKT LUH 14/27

24 Point-wise versus sample path envelopes data envelopes sample paths time [from Liebeherr, statcalc talk, 05] the point-wise envelope is violated at most with probability 1/4 at any point in time the sample path envelope is violated at most by a share of 1/4 of the sample paths c Markus Fidler IKT LUH 14/27

25 Construction of sample path envelopes A point-wise definition of stochastic envelope E(t) is P[A(τ, t) E(t τ) > 0] ε p for any τ [0, t] as opposed to the sample path definition P[ τ [0, t] : A(τ, t) E(t τ) > 0] ε s c Markus Fidler IKT LUH 15/27

26 Construction of sample path envelopes A point-wise definition of stochastic envelope E(t) is P[A(τ, t) E(t τ) > 0] ε p for any τ [0, t] as opposed to the sample path definition P[ τ [0, t] : A(τ, t) E(t τ) > 0] ε s that is equivalent to the backlog-like form (E(t) = b + Ct) [ ] P sup {A(τ, t) E(t τ)} > 0 τ [0,t] ε s c Markus Fidler IKT LUH 15/27

27 Construction of sample path envelopes A point-wise definition of stochastic envelope E(t) is P[A(τ, t) E(t τ) > 0] ε p for any τ [0, t] as opposed to the sample path definition P[ τ [0, t] : A(τ, t) E(t τ) > 0] ε s that is equivalent to the backlog-like form (E(t) = b + Ct) [ ] P sup {A(τ, t) E(t τ)} > 0 τ [0,t] ε s For a set of supporting points the union bound yields [ t ] t P {A(τ, t) E(t τ) > 0} ε p. τ=0 For t the difficulty is to select ε p such that its sum is bounded. τ=0 c Markus Fidler IKT LUH 15/27

28 Construction of fbm envelopes With Chernoff s bound an envelope with point-wise violation probability ε p (t) follows from the effective bandwidth of fbm traffic. The optimal solution is E(t) = λt + 2 log ε p (t)σt H. c Markus Fidler IKT LUH 16/27

29 Construction of fbm envelopes With Chernoff s bound an envelope with point-wise violation probability ε p (t) follows from the effective bandwidth of fbm traffic. The optimal solution is E(t) = λt + 2 log ε p (t)σt H. To derive a sample path envelope we have to select ε p (t) such that it has a finite sum. We choose resulting in ε p (t) = η t2β E(t) = λt + 2 log ησt H+β. Selecting β (0, 1 H) ensures a linear growth of E(t) in t. c Markus Fidler IKT LUH 16/27

30 Example fbm envelopes β = 0.12 the parameter β relaxes the envelope E(t) [Mb] β = t [ms] ε p (t) β = 0 β = 0.04 β = 0.08 β = t [ms] the point-wise violation probability decays with t the decay is, however, slower than exponential c Markus Fidler IKT LUH 17/27

31 Sample path violation probability Summing the point-wise violation probability ε p (t) = η t2β for all t [0, ) the envelope is violated by sample paths at most with ε s = Γ( 1 2β ) 2β( log η) 1 2β. c Markus Fidler IKT LUH 18/27

32 Sample path violation probability Summing the point-wise violation probability ε p (t) = η t2β for all t [0, ) the envelope is violated by sample paths at most with ε s = Γ( 1 2β ) 2β( log η) 1 2β We optimize the parameters η and β to find a closed form backlog bound from the envelope: the optimal η follows by fitting E(t) below b + Ct when deriving the backlog bound b at a server with capacity C a near optimal β is derived using Stirling s approximation of the Gamma function Γ(x) 2π/x (x/e) x for x 1 After some algebra we express the sample path violation probability ε s as a multiple of the known approximation ε a. This supports earlier conclusions from ε a by a rigorous sample path argument.. c Markus Fidler IKT LUH 18/27

33 Tail decay of backlog bounds ε fbm H=0.8 H= T=0.2 EBB T=0.4 T=0.6 H= b [kb] the EBB model includes all traffic types that have exponentially bounded burstiness, e.g. Markov on-off traffic the decay of tail probabilities for fbm traffic log ε b 2 2H is much slower Weibull-type making buffering less effective ms c Markus Fidler IKT LUH 19/27

34 Table of contents Motivation Traffic characteristics Related results Research goal FBm sample path envelope Definition of envelopes FBm envelopes Backlog bound End-to-end analysis Leftover service curves Network service curves Scaling properties c Markus Fidler IKT LUH 20/27

35 Leftover stochastic service curve Stochastic service curves make service guarantees subject to a violation probability ε [Cruz, 96] [ P D(t) < inf {A(τ) + S(t τ)} τ [0,t] ] ε. inf τ {A(τ) + S(t τ)} =: A S(t) is known as min-plus convolution. c Markus Fidler IKT LUH 21/27

36 Leftover stochastic service curve Stochastic service curves make service guarantees subject to a violation probability ε [Cruz, 96] [ P D(t) < inf {A(τ) + S(t τ)} τ [0,t] ] ε. inf τ {A(τ) + S(t τ)} =: A S(t) is known as min-plus convolution. A th C D th A cr D cr Given cross-traffic with sample path envelope E cr (t) at a server with capacity C. A leftover service curve for the through traffic is S th (t) = max{0, Ct E cr (t)}. c Markus Fidler IKT LUH 21/27

37 End-to-end convolution of service curves A th 1 2 n... A cr,1 A cr,2 A cr,n A network service curve can be derived by recursive insertion of the definition of service curve. In case of deterministic service curves D n = A n S n = D n 1 S n = A n 1 S n 1 S n = = A 1 n i=1s i c Markus Fidler IKT LUH 22/27

38 End-to-end convolution of service curves A th 1 2 n... A cr,1 A cr,2 A cr,n A network service curve can be derived by recursive insertion of the definition of service curve. In case of deterministic service curves D n = A n S n = D n 1 S n = A n 1 S n 1 S n = = A 1 n i=1s i A simple recursive insertion of stochastic service curves [ ] P D(t) < inf {A(τ) + S(t τ) b} ε(b). τ [0,t] is, however, not possible since the definition requires sample path arguments for the arrivals A(t) makes only point-wise statements for the departures D(t) c Markus Fidler IKT LUH 22/27

39 Construction of sample path service curves The construction of stochastic service curves that provide sample path guarantees for the departures has been open for a while. The method [Ciucu, Burchard, Liebeherr, 05] resorts to the union bound introduces a slack rate and thus achieves integrable violation probabilities We provide a solution for end-to-end leftover service curves under fbm cross-traffic and optimize its parameters as before. c Markus Fidler IKT LUH 23/27

40 Scaling of end-to-end delays w/n [ms] EBB T cr =0.6 ms T cr =0.4 ms T cr =0.2 ms fbm H cr =0.8 H cr =0.75 H cr = n Performance bounds for networks under fbm cross-traffic grow as O ( 1 ) n(log n) 2 2H for n tandem systems. For H = 0.5 the EBB result O(n log n) [Ciucu, Burchard, Liebeherr, 05] is recovered. c Markus Fidler IKT LUH 24/27

41 Impact of spare capacity on end-to-end delays w [ms] H cr = 0.8 H cr = 0.75 H cr = 0.7 H cr = mean spare capacity [Mb/s] Under LRD spare capacity is essential for network performance, e.g. for H cr = 0.75 (0.5) halving the spare capacity increases the delay bound tenfold (twofold). c Markus Fidler IKT LUH 25/27

42 Conclusions We contributed a sample path envelope for fbm traffic that complements and agrees with previous approximate results a sample path leftover service curve for systems under fbm cross-traffic to derive performance bounds for networks under fbm cross-traffic. We showed that end-to-end bounds under fbm cross-traffic grow as O ( 1 ) n(log n) 2 2H. Our result complements O(n) for deterministic traffic [LeBoudec, Thiran, 01] Θ(n log n) for EBB traffic [Ciucu, Burchard, Liebeherr, 05] O(n) for statistically independent EBB traffic [Fidler, 06] c Markus Fidler IKT LUH 26/27

43 References Amr Rizk, Markus Fidler: End-to-end Performance Bounds for Networks under Long-memory fbm Cross-traffic, September Amr Rizk, Markus Fidler: Sample Path Bounds for Long-memory fbm Traffic, Proc. of IEEE INFOCOM MC, March Amr Rizk, Markus Fidler: End-to-end Performance Bounds for Networks under Long-memory fbm Cross-traffic, Proc. of IEEE IWQoS, June c Markus Fidler IKT LUH 27/27

Delay Bounds in Communication Networks with Heavy-Tailed and Self-Similar Traffic

Delay Bounds in Communication Networks with Heavy-Tailed and Self-Similar Traffic Jörg Liebeherr, Almut Burchard, Florin Ciucu 1 Abstract Traffic with self-similar and heavy-tailed characteristics has