1. Traffic Modelling and Related Queueing Problems Presenter: Moshe Zukerman ARC Centre for Ultra Broadband Information Networks EEE Dept., The University of Melbourne Presented at EE Dept., City University of Hong Kong, 11 April, 2002 Credit: R. Addie (USQ), T. Neame (EEE, Melbourne)

2. 1.The big picture 2.Stream traffic modelling – ground rules 3.Growth and efficiency 4.Long Range Dependence (LRD) and LRD vs. SRD + traffic models review 5.Poisson Pareto Burst Process (PPBP) 6.PPBP queueing performance 7.PPBP fitting 8.Fundamentals and ideas 9.The Resurrection of the MMPP OUTLINE

3. The Big Picture

4. Traffic Modelling Queueing Theory Performance Evaluation Simulations and Fast Simulations Numerical Solutions Formulae in Closed Form Traffic Measurements Link and Network Design and Dimensioning Traffic Prediction

5. Of Course, there are short cuts: Just Do It! and then see … Pros and Cons But let’s forget about these short cuts for now …

6. Research in Performance Evaluation 1.Exact analytical results (models) 2.Exact numerical results (models) 3.Approximations 4.Simulations (slow and fast) 5.Experiments 6.Testbeds 7.Deployment and measurements 8.Typically, 4-7 validate 1-3.

7. Black Box ParametersPerformance The black box, can have: Traffic (model or trace), and a system or a network model, and (1) Performance Formulae, or (2) Numerical solution, or (3) Fast Simulation, or (4) Simulation Performance evaluation tool

8. Black Box ParametersPerformance Very Fast (micro-millisecond) for congestion control Fast (100s milliseconds) for Connection Admission Control Slow (days) for network design and dimensioning How fast should the black box be?

9. Traffic Modelling Ground Rules: For a given TT, S finds an SP defined by a small number of parameters such that: (1)TT and SP give the same performance when fed into an SSQ for any buffer size and service rate. (2)TT and SP have the same mean and autocorr. (3)Preferably, SP SSQ is amenable to analysis. Traffic Trace (TT) Stochastic Process (SP) S

10. “Do Not” Rules We Do not take retransmissions into account. We ignore TCP dynamics. The aim is to dimension network so that retransmissions will normally not be needed. This will be efficient as traffic, capacity and number of users increase. (why??)

11. Why does efficiency increase with growth? If load and capacity increases service rate is better- Scaling (Frank Kelly). Convergence to Gaussian - Central Limit Theorem (Addie) – but the convergence is slow.

12. Reich’s Formula: Consider two scenarios (1 and 2). Let: A 1 (s,t)= A 2 (bs,bt) S 1 (t-  t,t)=b S 2 (t-  t,t)= S 2 (b(t-  t),bt) Q(t) = queue size at time t A(s,t)=Work arrive between s and t S(s,t)=Service capacity between s and t Scaling

13. Thus, queue size is the same but served much faster!

14. LRD Convergence to Gaussian

15. Why does LRD traffic converge to Gaussian slowly? I will tell you later. First, let me explain the meaning of LRD traffic and the difference between LRD and SRD.

16. Background Modelling packet traffic is difficult because of its properties Long Range Dependence (LRD) is a widespread phenomenon LRD has been found in –Ethernet traffic(Leland, et al. 94) –VBR video traffic(Garrett & Whitt 94) –Internet traffic(Paxson 95) –MAN traffic(Zukerman, et al. 95) –ATM cell traffic(Jerkins & Wang 97) –CCS Signalling Traffic(Duffy, et al. 94) Possible causes for LRD –Distribution of file sizes –Human behaviour –VBR video

17. Long Range Dependence A process is defined to be Long Range Dependent if its autocorrelation function R(  ) decays slower than exponentially. LRD need only exist in the limit –LRD implies nothing about its short term correlations which affect performance in small buffers Traditional traffic model processes are Short Range Dependent (SRD) In practice, LRD is difficult to distinguish from non-stationarity

18. 1986 Heffes and Lucantoni IEEE JSAC IEEE Best Paper Award (MMPP) 1994 Leland et al. ACM/IEEE TON IEEE Best Paper Award (LRD) 1995 Likhanov et al. INFOCOM Best Paper Award – proposed a model equivalent to the one we call Poisson Pareto Burst Process (PPBP) Traffic Modeling: Historical Highlights

19. -5.0-4.0-3.0-2.00.01.02.03.04.05.0 The Variance v=Autocovariance Sum = The variance Arrival Process Autocovariance (IID)

20. -5.0-4.0-3.0-2.00.01.02.03.04.05.0 The Variance v=Autocovariance Sum Arrival Process Autocovariance SRD

21. For SRD v = lim n  VAR [A(n)]/n That is, for large n, VAR [A(n)] grows linearly with n.

22. -5.0-4.0-3.0-2.00.01.02.03.04.05.0 The Variance V = Autocovariance Sum =  Arrival Process Autocovariance (LRD)

23. For LRD

24. LRD SRD IID BUFFER SIZE LOSS PROBABILITY Queueing Performance Comparison

25. Self Similar Traffic (Ethernet)

26. Arrivals Poisson

27. For Poisson (IID) or SRD

28. For LRD

29. LRD vs. SRD Slope=1: Non-fractal (SRD) Slope>1: Fractal (LRD) Log V(A(t)) Log (t)

30. The Hurst Parameter

31. Burstiness on ALL scales? Not Really. Still approaches zero as t grows because < 2

32. MMPP (SRD) SRD GaussianLRD Gaussian PPBP (LRD) Queueing Analysis DoneSSQ Approx. (AZ 92,93,94) SSQ Approx. (AZN 95) Useless bounds + results here Fast/Slow Simulat’n Fast + slow FastOnly Slow + results here Queueing Performance State of the art

33. The Markov Modulated Poisson Process (MMPP) ? Several traffic activity modes The period of each activity mode has exponential distribution When in mode i – Poisson i Time

34. Two State MMPP Results Analytical results for queueing performance are available Four parameters Fitted to mean, variance, v, + Numerical algorithms (Matrix Geometric) results available for n state MMPP.

35. Gaussian Queueing Results SRD: results as a function of three parameters which can be fitted to mean, variance, v LRD: results as a function of three parameters which can be fitted to mean, variance, H.

36. Exponential Pareto Poisson Pareto Burst Process (PPBP) Total work from multiple overlapping bursts Burst arrivals according to a Poisson process of rate Burst durations are Pareto distributed During a burst, bit rate is constant All bursts have constant bit rate

37. Pareto Distribution For the Pareto distribution –Is a heavy-tailed distribution –Has infinite variance, but finite mean

38. Why the PPBP? Assume each user transmits at maximum capacity, or zero capacity –Each user can be represented as an on-off process Assume users are independent of one another Assume duration of on times is heavy-tailed The PPBP is a limiting case for the sum of multiple independent heavy-tailed on-off processes – shown in Likhanov, et al. 95

39. Heavy Tailed Distribution Complementary distribution functions for exponential distribution and Pareto distribution with infinite variance. Both distributions have mean = 3

40. The Poisson Pareto Burst Process (PPBP) A n is total work arriving in a fixed size interval of length t Exponential ( ) Pareto ( ,  ) r n AnAn

41. PPBP Parameter Fitting

42. Recall: LRD Convergence to Gaussian To fit we choose the best curve!

43. Why does LRD traffic converge to Gaussian slowly? I can tell you now.

44. time period Long Bursts Short Bursts: A very good way to consider LRD traffic

45. PPBP Initial Conditions “Steady state” process has Poisson number of active bursts at time t Mean number of bursts is E(D) Remaining duration in each burst distributed according to its forward recurrence time

46. Pareto Forward Recurrence Times Even heavier tail than ordinary Pareto Has infinite mean and infinite variance

47. Forward Recurrence Times – Even Heavier Tails Complementary distribution functions for Pareto distribution with mean 3, and corresponding forward recurrence times

48. Long Bursts and Short Bursts Consider a PPBP over the interval (t, t + W) At time t, there will be B t bursts active Each of these B t bursts will last the entire interval with probability Label the bursts present at the start and the end of the interval as long bursts Rest of the bursts are the short bursts

49. t t+W Long burst process Short burst process t+W t t Long Burst – Short Burst Division

50. Properties Long bursts and short bursts processes are independent of one another. Number of long bursts, n, is Poisson distributed with mean E(D)Pr(R > W) For a given interval, long bursts process is a CBR component Short bursts process is a correlated Poisson process with mean r E(D)(1-Pr(R > W)) Short bursts process is not LRD

51. Single Server Queue In interval n: –A n is the number of cells arriving –C is the fixed service rate –Y n is the net arrival process –Q n is the amount of work in the buffer QnQn C AnAn

52. Our Queueing Performance Results Overflow probability for short burst process depends on n, number of long bursts Define S(x,n) as overflow probability when number of long bursts is n Probability of n long bursts is Poisson with mean E(D)Pr(R > W) Overall overflow probability for PPBP is:

53. Instability in a Simulation Probability of n long bursts is Poisson with mean E(d)Pr(R > W) n long bursts leaves capacity C – nr for short bursts Mean arrival rate for short bursts process is  r E(d)(1-Pr(R > W)) There is non-zero probability that the system will be unstable for the duration of the simulation Can choose simulation length, W, such that this probability is negligible Have demonstrated that overall impact of instability is limited

54. Simulation with Random Number of Long Bursts Long bursts may have significant impact on simulation results Initialise simulation with a random number of bursts and let a random number of these be long bursts Will require a large number of simulations to be sure that the state space is explored thoroughly

55. Weighted Sum to Account for Long Bursts Create a process with no long initial bursts Simulate and find losses in systems with rate C-nr Sum the losses from these systems, weighted by the probability that N = n

56. Effect of Improved Simulations IP trace fitting W = 22,000,000 x 60 simulations Simple simulation

57. Quasi-Stationary Estimate Long burst process is constant for period W Use existing techniques to estimate overflow probabilities for short burst process fed into server with capacity C - nr As with simulation, combine estimates according to probabilities of n long bursts Which W gives best estimate? –Choose the W which gives highest overflow probability.

58. Quasi-Stationary (QS) Estimates W W W W QS Estimate for W

59. Large Deviations Results Large deviations is a good approach for Gaussian queues. It has failed to provide useful results for LRD non Gaussian queues. There was an attempt to use Large deviation to obtain analytical results for the continuous time counterpart of the queue fed by PPBP (called the M/G/  process) by Tsybakov & Georganas. We will discuss it now.

60. Large Deviations Results (cont.) Large deviations results for queues fed by LRD sources have been derived by a number of authors. –Results only hold as, i.e. for large buffers Most useful results for M/G/  input are due to Tsybakov & Georganas –They give upper and lower bounds:

61. Large Deviations Results (cont.) A and B are constant with respect to the buffer size x. Note that the upper and lower bounds have the same form. We will compare our results against these bounds.

62. Comparison of Results Prob {Q >  }

63. Second Set of Results Prob {Q >  }

64. Quasi-Stationary Estimate Previously have used repeated simulation to fit final parameter Using the estimate, fitting is faster and more reliable Quasi-stationary value is still only an estimate –Most accurate when is large – for PPBP fitted to real data is usually small  Reliability is questionable

65. Modelling Measured Traffic PPBP has 4 parameters –Burst arrival rate –Rate of work per burst r –2 Pareto parameters,  and  Parameter Fitting –Can fit r, ,  to measured mean, variance, H –Can produce PPBPs with same r, ,  values which give very different queueing performance results  Fitting is vital to give a model which predicts the performance of real traffic

66. PPBP Convergence to Gaussian

67. Good News! - Fitting the PPBP to Real Traffic

68. PPBP model (from analysis) PPBP model (from simulation) Trace Autocovariance Fitting

69. How Are We Doing? Our aim was to find a stochastic process with the following properties: Defined by a small number of parameters PPBP has only 4 parameters –If the parameters of the process are fitted using measurable statistics of an actual traffic stream then 3 of 4 parameters fitted to measurable statistics Fitting the 4 th parameter can now be done systematically The first and second order statistics of the model will match those of the traffic stream

70. How Are We Doing? (cont.) If fed through an SSQ the performance results of the model will accurately predict those of the real traffic stream in an identical SSQ. This should be true for a wide range of buffer sizes and service rates. PPBP does well If performance results can be calculated analytically, so much the better Quasi-stationary estimate provides an accurate analytic estimate

71. Summary (PPBP) Described the Poisson Pareto Burst Process (PPBP) Long bursts may have a significant impact on PPBP simulation results. We can factor long bursts into simulations. Quasi-stationary analysis gives a reasonable estimate of the queueing performance of the PPBP. Using quasi-stationary estimate, reliable fitting of the PPBP to real traffic streams is possible.

72. Speculation: MMPP Resurrection Anti-thesis?? The PPBP at this stage is not amenable to state dependent queueing analysis. Such analysis is needed in many Telecommunications systems. MMPP or other Markovian models are amenable to such analyses. I will present now intuitive arguments for the resurrection of MMPP.

73. Single Server Queue Realization G/G/1

74. Unfinished Work Distribution Reich’s Formula (1958) PVTPABT w i i w ()max()     1 1  A i = work arrives during interval i B= Service capacity during interval i

75. Again, our SSQ Realization: The relevant correlation duration is from the beginning of the last busy period. Indeed, LRD  Longer busy periods. G/G/1

76. BUFFER But if we introduce a buffer … ???

77. G/G/1/2 The buffer breaks long busy periods to many short ones! And we may not need to consider LRD in our traffic modelling.

78. Queueing Performance Comparison LRD SRD IID BUFFER SIZE LOSS PROBABILITY LOG

79. LRD SRD IID BUFFER SIZE LOSS PROBABILITY The resurrection: SRD (MMPP?) as a model for LRD LOG

80. LRD vs. SRD Slope=1: Non-fractal (SRD) Slope>1: Fractal (LRD) Log V(A(t)) Log (t)

81. The resurrection: SRD (MMPP?) as a model for LRD Slope=1: Non-fractal (SRD) Slope>1: Fractal (LRD) Log V(A(t)) Log (t)

82. There are arguments for resurrection of the MMPP as A traffic model. Conclusion

83. Two approaches: (1)PPBP (State Dependent (?)) (2)MMPP (Accurate traffic model (?)) The Game goes on