FAKULTÄT FÜR INFORMATIK

Transcription

1 b b b b b b b b b b b b b b b b b b b b FAKULTÄT FÜR INFORMATIK der Technischen Universität München Lehrstuhl VIII Rechnerstruktur/-architektur Prof. Dr. E. Jessen Modeling of Packet Arrivals Using Markov Modulated Poisson Processes with Power-Tail Bursts Diplomarbeit Hans-Peter Schwefel

2

3 b b b b b b b b b b b b b b b b b b b b FAKULTÄT FÜR INFORMATIK der Technischen Universität München Lehrstuhl VIII Rechnerstruktur/-architektur Prof. Dr. E. Jessen Modeling of Packet Arrivals Using Markov Modulated Poisson Processes with Power-Tail Bursts Diplomarbeit Hans-Peter Schwefel Aufgabensteller: Betreuer: Prof. Dr.-Ing. Eike Jessen Prof. Lester Lipsky, Ph.D., Dr. Michael Greiner Abgabedatum: Department of Computer Science, University of Connecticut

4

5 Contents iii Contents Introduction 2 Background 3 2. Denitions Matrix Exponential Distributions LAQT Queueing Models Power-Tail Distributions Order Statistics Self Similar Behavior and Long-Range Correlation Arrival Processes with Long-Range Correlation 8 3. Merging of Power-Tail and Poisson Process A Single Source On/O-Model: -Burst N-Burst Process (Multiple Sources) Description of the Process The LAQT-Matrices Calculation of the Mean The Steady-State Process at Departure Times Reducing the State Space Entrance Vector p Autocorrelation Coecient Lag-k Estimate of a-parameter for Real Network Parameter Inuence for 2-Burst SM/M/-Queue with Arrivals from the N-Burst Process Mean Queue-Length Buer Overow Probabilities Comparison to the Poisson-Zeta Process of [Fan & Georganas, 996]. 47

6 iv Contents 5 Simulations Generation of Random Variables Pareto Distributed Random Variable TPT-T Distributed Random Variable Generating Interarrival Times of the SM-Processes Simulated Autocorrelation Curves Calculation of the Autocorrelation Coecient MERGE-Process Varying Results and Natural Truncation N-Burst Process Simulation of the Counting Process Power-Tail Renewal Process Merged Process N-Burst Process GI/M/-Queue Simulations Comparison with Real Data 7 6. Ethernet Data TCP Data Summary 74 Bibliography 76

7 Chapter Introduction Within the last decade, the volume of trac on various computer networks has grown immensely. Especially for areas like network design, development of taring schemes, and application design, a statistical model for the network trac is essential. While more and more research was going on in that eld, it turned out that classical (Poisson) models are not able to yield the eects of trac on large scale networks: [Leland et al., 994] observed large uctuations in the number of packets on a local Ethernet. In classical models these uctuations are expected to occur when measuring over small time intervals, but they should smooth out quickly with increasing interval size. However, the measurement showed these uctuations over a wide range of time scales. That eect is called high variability or self-similar behavior (also, fractal behavior). Another eect, called long-range autocorrelation, is connected with the self-similarity and will be described in detail later on. The high variability could be taken as an indication of inhomogeneity. However, we will come up with a homogeneous model that captures that eect by using special distributions, so called Power-Tail distributions. There is increasing evidence that these kinds of distributions are involved in related elds: In 986, [Leland & Ott, 986] observed that the CPU times for jobs at BELLCORE had a power-tail distribution over 5 orders of magnitude. Furthermore, [Garg et al., 992] showed that the distribution of le sizes at the local UNIX system at the University of Connecticut also had a power-tail. Most recently, [Crovella et al., 996] measured the sizes of les sent to Boston University over the Internet and found power-tail behavior as well. Motivated by these ndings, we have come up with a mathematically tractable model, called N-Burst, that uses power-tail distributions to get the aforementioned characteristics. Similar models were discussed by [Willinger et al., 995] and [Fan & Georganas, 996]. However, they do not give exact analytical solutions to the same extent as we can give for our model. The analytical tools are supplied by the Linear Algebraic Queueing Theory (LAQT) brought up by [Neuts, 98] and [Lipsky, 992]. These methods were further developed by [Fiorini & Lipsky, 996], especially for treatment of the Semi-Markov Processes. The analytical section of this work uses MATLAB for the calculations and either MATLAB or GNUPLOT to produce the graphs. The simulations are done by C-programs running on a Sun Sparc workstation with SUN OS Notational conventions: Throughout this work matrices will be symbolized by bold faced capital letters (e.g. A). Bold faced, lower case letters (e.g. p) stand for row vectors. The symbol ' expresses the transposed of a matrix or a vector. I is the unit matrix, and

8 2 Introduction is the vector with all components being. The dimensions of both of these are determined by the context. Abbreviations: CDF cumulative distribution function CLT central limit theorem IA-times inter-arrival times iid independent, identical distributed LAQT linear algebraic queueing theory pdf probability density function PT Power-Tail SM Semi-Markov TPT truncated power-tail

9 3 Chapter 2 Background 2. Denitions The center of our interest will be xed sized packet trac (such as it appears in ATMnetworks). Therefore the trac on the line is suciently described by the inter-arrival times (IA-times) T ; :::; T n between the single packets. In our models these T i will be random variables. The process ft i g is called the Inter-arrival Process. If the T i are iid, then the process is a renewal process. However, as we will see later, real network trac has characteristics that do not allow independence. Another way of looking at the trac is by counting the number of packet arrivals in some interval of length. Let N`() be the random variable denoting the number of arrivals in the `-th interval. Then the sequence fn`()g is the Counting Process associated with the Inter-arrival Process ft i g: N`() = fk : ` < kx i= T i (` + )g : The following will give some well-known denitions about random variables, since we will make ample use of them. In all cases the time variables are in the range x <. Let X be a random variable representing the time for a process to complete. Then the Cumulative Probability Distribution Function (CDF) is one-sided, and dened by: 3 F (x) := P r(x x) with F ( ) =, and lim x! F (x) =. The Reliability Function is R(x) := P r(x > x) = F (x); and the probability density function (pdf) for the process, if it exists, is: f(x) := d F (x) dx = d R(x) dx The `th moment of the distribution (or the expectation of X`), if it exists, is dened by: 3`P r(u)' means `Probability that U is true'. Z x` := E(X`) := x` f(x) dx: ()

10 4 2 Background The rst moment (mean) can also be calculated by using the reliability function: E(X) = Z The variance of the distribution (if it exists) is dened by R(x) dx: (2) Var(X) := 2 := E([X E(X)] 2 ) = E(X 2 ) [E(X)] 2 = x 2 x 2 ; and the dimensionless quantity, the coecient of variation, is dened by the ratio of the variance and the square of the mean (rst moment): C 2 := 2 x 2 Therefore multiplying a random variable by some constant a does not change the coef- cient of variation, because both, numerator and denominator, then yield an additional factor a 2. The denitions above only involve a single random variable. Since we deal with a series of those, we are also interested in the relationships between them. The Covariance of X and Y gives us an idea of that relationship: Cov(X; Y ) := E[(X E(X))(Y E(Y ))]: The Correlation Coecient of X and Y is dened as (X; Y ) := Cov(XY ) E[(X E(X))(Y E(Y ))] = (3) X Y X Y From its denition, it can be shown that (X; Y ). If (X; Y ) = then X and Y are called uncorrelated. If X and Y are independent, then it follows that they are uncorrelated. On the other hand, even if two variables are uncorrelated, they still may not be independent. The autocorrelation coecient lag-k for an inter-arrival process ft i g, or its associated counting process is dened as r i (k) := (T i ; T i+k ): For stationary processes in steady state the value is the same for all indices i, so we just use r(k). A positive r(k) means, that if the value t i is bigger than its mean, t i+k k steps later is more likely to be bigger than the mean as well. When having a realization fx i g i=;:::;n of a process (i.e. a series of n samples), the autocorrelation coecients lag-k can be estimated by r(k) n k P n k `= (x` ) (x`+k 2 ) p (4) where X = n k x`; 2 = n n k n k `= X `=k x`

11 2.2 Matrix Exponential Distributions 5 X 2 = n k (x` ) 2 ; 2 2 = n k n k `= n X `=k (x` 2 ) 2 : To distinguish between the arrival and the counting processes, we will use the notation, r(k), for the inter-arrival process, and r(kj) for the counting process. Though the interarrival times for a renewal process are uncorrelated (since they are independent by denition), the counting process in general has correlation - unless the process is memoryless, i.e. Poisson. Finally, though already mentioned, the most important distribution for all work about Markovian processes, is the negative exponential (with rate ). It has as its reliability function and density function: If follows for the mean and variance: R(x) = e x ; f(x) = e x : (5) E(X) = ; Var(X) = 2 ) C = : The associated counting process has a Poisson Distribution: P r[n j () = i] = ()i e 8j: (6) i! Strictly speaking, when talking about a Poisson process, it would be the counting process, that is referred to. However, this expression is also going to be used for the arrival process; it should be clear from the context, which one is meant. For negative exponential distributed state times, T, one neat property is the memorylessness. It does not matter at what time in the past the last event happened, the residual time is always negative exponential with the same rate: P r(t > x + hjt > x) = P r(t > x + h) P r(t > x) = e (x+h) e x = e h for h : If there are two negative exponentially distributed variables X and X 2 with rates and 2, then the probability of server nishing rst is P r(x < X 2 ) = Z R X2 (x) f X (x) dx = Z 2.2 Matrix Exponential Distributions F X (x) f X2 (x) dx = + 2 (7) The fundamental approach to all the analytical treatment of our trac models is a method called Linear Algebraic Queuing Theory (LAQT). By using a network of states (called phases) with exponentially distributed inter-event times (almost) any distribution can be approximated arbitrarily closely. This approach was rst brought up in [Neuts, 98] and is discussed in detail by [Lipsky, 992]. The following paragraph gives a brief introduction and mentions the needed formulas in reference to the latter.

12 6 2 Background P j µ µ j q j p j p P ij P jm q m p m µ m p i µ i Figure 2.: A typical subsystem with m phases. A customer enters the system and goes with probability p k to phase k. The time in each phase is exponentially distributed with rate k. Then he moves around according to the transition probabilities P kl and eventually leaves the system with probability q i for phase i. The distribution of the time in the subsystem then is an approximation of the desired distribution. The distribution of the time between a single customer entering a subsystem such as in Figure 2. and leaving it again can be neatly determined by using matrix algebra. Let (P) ij = P ij be the matrix of the transition probabilities within the subsystem. The exit vector q has as its components the probabilities of leaving the subsystem when being at the according phase. Since there are only two choices, going to another phase within the subsystem or leaving the system, the probabilities must add up to : P + q = : Next we dene the completion rate matrix M, which is a diagonal matrix of the single state leaving rates k. Further, let be a vector whose components i are the mean times to leave the system, given that the customer started at phase i. Then the following equation holds: = M + P : The rst summand gives the mean time at the current phase, while the second term stands for the mean time in the system after the next transition. Solving for gives: = [M(I P)] =: V : Therefore, V := [M(I P)] is called the service time matrix. Its elements V ij are the overall mean times spent at phase j until the customer leaves the system, if it started at phase i. The inverse of V, B := M(I P), has the name service rate matrix. The balance equations (ow into a state equals ow out of that state) for the system lead to the following dierential equation: dr(t) dt = R(t)B =) R(t) = exp( tb); where R ij (t) is the probability that the customer is in phase j at time t, given that it started in i.

13 2.2 Matrix Exponential Distributions 7 Whenever a customer leaves the system, the next one comes in immediately. The components p i of the entrance vector p are the probabilities that a new customer upon entering the system will directly go to phase i. That matrix formulation gives rise to an elegant formula for the reliability function of the time that the customer spends in the system: Dierentiation yields the pdf: R(x) = p exp( xb) : (8) f(x) = dr(x) dx = pb exp( xb) : Also, the nth moments of this distributions come out neatly using (): E(X n ) = n!pv n : (9) This way of handling distributions is only suitable for renewal processes, since the next customer makes its way through the subsystem independently of the previous one. However, investigations of real trac data showed correlation in the series of inter-arrival times. Thus we need to extend this matrix algebraic treatment to be able to capture so called Semi-Markov Processes (also known as Markov Renewal Processes). The idea is, that the next customer enters the subsystem in a phase that depends on the leaving phase of the previous customer (see [Fiorini et al., 995]). Therefore another matrix L is introduced. The components L ij give the departure rate from phase i, whereafter the next customer starts in phase j, i.e. for small time intervals, L ij is the probability that a customer at phase i leaves within and the next one starts in phase j. L and B must be consistent, i.e. L = B, because both matrices describe the departure rate of the customers. However, L does not capture what is going on before the departure, while B is not aecting the next customer. In the steady state of these semi-markov processes the entrance vector p is not relevant any more, since the entering customers are described by the L-matrix. However, for transient behavior that vector is still important. For notational convenience the matrix Y := VL is introduced. It can be shown (see [Fiorini et al., 995]), that the left eigenvector, p, of Y with eigenvalue (i.e. p = py) is the steady state equivalent of the entry vector of the renewal processes. The components of p give the probabilities that a newly arriving customer goes to the according phase, when the inuence of the startup of the process has deceased (i.e. steady state). Obviously, this p-vector now has to replace p in the formula for the moments of the distribution. From p = pvl it follows, that pv = pl. In our Markov modulated models, L is a diagonal matrix, so easily invertible; on the other hand V usually is much harder to write down. So it is easier to use for the mean: E(X) = p L : () [Fiorini et al., 995] also derive a formula for the covariance of the process in steady state: Cov(X; X +k ) := lim n! Cov(X n ; X n+k ) = pv(y k p)v for k = ; 2; : : : : ()

14 8 2 Background Putting these formulas together, we get for the autocorrelation coecient lag-k: r(k) = pv(yk p)v 2pV 2 (pv ; k = ; 2; : : : : (2) ) 2 For large k, Y k is calculated by using the method of spectral decomposition: Y k = X i k i v i u i = p + X i 6= k i v i u i : i are the eigenvalues of Y; u i ; vi are the corresponding left and right eigenvectors such that u i vi =. Since Y = VL = VB =, is the right eigenvector with eigenvalue. By its denition, p is the corresponding left eigenvector with the same eigenvalue. That leads to the following expression for r(k): r(k) = pv Pi 6= ki v i u i V 2.3 LAQT Queueing Models 2pV 2 (pv ) 2 ; k = ; 2; : : : : (3) At the end of the day, the performance of the models that we described in the last section needs to be evaluated. In our context that means, the process is fed into a queue. To make things not too complicated we assume single server queues with exponentially distributed service times, i.e. GI/M/-queues for the renewal process or SM/M/-queues for the correlated IA-process. The GI/M/-queue is analyzed in [Lipsky, 992]. Let the IA-process be sub-system with mean X, and the server be sub-system 2 with mean X 2. The utilization = X 2 = X has to be smaller than to have stable behavior (i.e. for a steady state to exist). The state space of the GI/M/-queue is the product of the state space of the arrival process and the set of possible queue-lengths. The balance equations are matrix equations that use the following matrices: A := I + X 2 B p; U := A : (4) For the open system (possibly innite number of customers) the limit U N for N! is necessary. That limit is dominated by the largest eigenvalue of U, i.e. the smallest eigenvalue of A, which is called the geometric parameter s. The balance equations nally yield for the steady state queue-length probabilities (the customer being served is also assumed to belong to the queue): () := P r(`queue length is ) = ; The mean queue-length then turns out to be: (k) = ( s)s k for k = ; 2; 3; ::: : (5) q = X k= k (k) = s (6)

15 2.3 LAQT Queueing Models 9 The mean system time (waiting time + service time) follows from Little's Law: T = X q = X 2 s Since the arrival process is non-exponential, it makes a dierence whether we look at the queue-length probabilities at randomly chosen points in time or at arrival times. The latter is important for determining buer overow probabilities for non-loss systems. If there is either a second backup buer (assumed to be innitely large, e.g. a harddisk) or assuming a feedback mechanism that signals the full primary buer to the sender and thus delays the new packet (i.e. the backup buer is spread out among the transmission sources), the probability that a newly arriving packet goes to the backup-buer is: P r(overow) = X k=b S + a(k); where B S is the size of the primary buer and a(k) is the probability that the queue-length at arrival times is k. It can be shown that (see [Lipsky, 992] p. 29) a(k) = ( s)s k ; ) q a = s (7) s ) P r(overow) = ( s) X k=b S + s k = s B S+ : Where q a is the mean queue-length at arrival times. For an exponential arrival process (M/M/-queue), s =. Another problem of importance in studying buers occurs when there is no secondary buer. As long as the primary buer is full, every arriving packet is thrown away. This would be equivalent to a GI/M//B S -queue. However, this approach is not going to be investigated here. The SM/M/-queue is a little bit more complicated, since there is no closed formula for the queue-length probabilities. It is treated in [Neuts, 98] as a so-called Quasi- Birth-Death Process. That is a Markovian-like process where the transition rate matrix is block tri-diagonal. The matrix notation of [Neuts, 98] is slightly dierent but will be changed here to make it consistent with the previous formulas. The state space of the SM/M/-queue is also the product of the state space of the arrival process and the set of possible queue-lengths. Using the previously introduced matrices B; M; L the blocktridiagonal transition rate matrix Q for the SM/M/-queue is the innite matrix: Q = B L X 2 I (B + X 2 I) L X 2 I (B + X 2 I) L X 2 I (B + X 2 I) The leftmost column of matrices stands for the states with queue-length, the next for queue-length and so on. The matrices on the main diagonal describe state transitions :

16 2 Background within the arrival process without change of the queue-length. The upper diagonal (L) describes the arrivals while the other diagonal (I= X 2 ) represents the servicing (without changing the state of the arrival process. According to [Neuts, 98], the balance equations have a matrix geometric solution for the steady state queue-length probabilities. Therefore the quadratic matrix equation, A + RA + R 2 A 2 = ; where A = L, A = (B + = X 2 I), A 2 = = X 2 I, has to be solved for a minimal R. An easy iterative procedure is used therefore: It can be shown that A has an inverse, so the equation can be rewritten as: R n+ = (A + R n2 A 2 )A : (8) Starting with R = this formula is repeatedly applied until convergence is reached. [Neuts, 98] showed that this procedure terminates eventually and delivers a correct solution. The R-matrix is the equivalent of the U-matrix of the GI/M/-queue. The steady state probabilities for the queue-length are then obtained by (k) = (I R)R k ; k = ; ; : : : : (9) is the residual vector obtained from p by multiplication with V and normalization, i.e. the components are the probabilities that the arrival process is in the corresponding state: = pv pv (2) The geometric series in the mean queue-length formula can then be simplied: q = X k= k (k) = R(I R) : (2) For the GI/M/-queues, the s-parameter is a necessity to get the queue-length probabilities and the mean queue-length. It is inherent in the sense, that it is an eigenvalue of the A-matrix. This is not the case for the SM/M/-queue. For comparison purposes we dene the equivalent s-parameter for the SM/M/-queue from (6) to be: s := q (22) Using the equivalent s-parameter has the advantage that it does not grow unboundedly with!. s < always holds, s = is equivalent to q = (smallest possible value for q). s! corresponds to q!. The vector part of (9), (I R)R k, has as its components the probabilities that the queue-length is k and the arrival process is in the corresponding state. If we look at the queue at arrival times, the states of the arrival process with a high departure rate 4 contribute more observation points. Therefore, the probability vector has to be scaled by

17 2.4 Power-Tail Distributions the L-matrix and then normalized to give the probabilities at arrival times. Adding up all the components gives the probability That leads to a mean queue-length at arrival times of q a = a(k) = (I R)Rk L L (23) X k= k a(k) = R(I R) L L Finally, the overow probability for a buer of size B S comes out as: P r(overow) = X k=b S Power-Tail Distributions a(k) = RB S L L (24) It turns out that the widely used exponential distributions are not sucient for modeling recent trac. Trac measurements show a high variability for widely ranging time-scales. In this work, this variability is achieved by using Power-tail (PT) Distributions. All nite extensions of exponential distributions (hyper-exponential, Erlangian,...) have a reliability function R(x) which drops o exponentially at some point, i.e. the likelihood of getting very large values for X vanishes very quickly. In contrast to that, the reliability function of a PT-distribution drops o with some power of x, much slower than an exponential distribution: R(x)! c x for large x: (25) c > and > are constants for a given distribution. Straight-forward dierentiation yields the asymptotic form for the pdf, f(x)! c (26) x+ Then from () and elementary calculus, it follows that all moments for ` are innite! Thus, if 2 then F () has innite variance, and if then F () has innite mean! We assume to be between and 2, so it has a nite mean, but innite variance. Since this parameter will appear in various measured trac characteristics (such as autocorrelation curves), and some real data ([Leland et al., 994]) suggested a value of = :4, that value will be used from now on. The nite mean of the distribution also implies, that the occasional huge values have to be made up by a larger number of values smaller than the mean. That means that most of the time, the observer is going to see something smaller than average. But occasionally, single huge samples occur. 4 The process itself is an arrival process when fed into a queue. However, we talk of departures from the SM-process. Thereby we look at the phases of the sub-system that build up the arrival process. A departure from one of the phases (described by the elements of L) then corresponds to an arrival at the queue.

18 2 2 Background One type of PT-distributions, which we will mainly use for simulations, is a so called Pareto Distribution: R(x) = (x + ) for x ; (27) with E(X) = for > : Though this function has a simple analytical formula and also is easy to simulate, the disadvantage is that the LAQT-methods of the last section cannot be applied to it: It can be shown that the Laplace transformation of all matrix exponential distributions is a rational function. This does not hold for the Pareto function, thus it does not have an exact matrix exponential representation. [Greiner et al., 995] have introduced one method to overcome that problem. They introduced a family of distributions, so called Truncated Power-tails (TPT), which asymptotically have PT-characteristics. For our purposes, the easiest case is using a series of hyper-exponentials. However, their paper is more general in that respect. It is straightforward to handle these hyper-exponentials by LAQT methods then, since they are a very simple case of phase distributions (with P =, so B is just diagonal). X ξ ξ 2 ξ T X 2 X T Figure 2.2: Phase diagram for the TPT distribution with truncation T. The probability i = i ( ) =( T ) of going to phase i is geometrically distributed, i.e. it gets smaller by a factor < from state i to i +. Furthermore, the state-time grows geometrically by the factor = = =. Figure 2.2 shows the state diagram for a hyper-exponential TPT distribution with T states. Let the entrance probabilities i and the state leaving rates r i be: i = i ( ) T ; r i = i where < < ; > : (28) Unless mentioned dierently, is always set to be :5. Choosing close to or often resulted in unstable simulations (see [Klinger, 997]), while the choice of outside that `critical' area did not have much inuence on any analytical results (see

19 2.5 Order Statistics 3 [Greiner et al., 995]). So :5 is a reasonable compromise. The parameter inuences the mean of the distribution. To make E(Y T ) = m, has to be = () T T m (29) Then [Greiner et al., 995] show that in the limit for T! the distribution has a PT with = log()= log(). Since we always x (mostly to be :4), will be determined from and. The reliability function is in the limit: R Y (x) = ( ) X i= i e r ix : [Klinger, 997] determined the tail-constant of (25) to be c = :2383 [E(X)] for this function. In practice it is usually not possible to work with innite series. So the calculations must be done using truncated PTs with increasing truncations T until the result converges in the sense, that increasing T does not change the results more than a negligible amount. The reliability function of the truncated reliability function is a nite sum with a dierent normalization constant: TX R YT (x) = i e rix : T In the following, the distribution with reliability function R YT will be called a TPT-T distribution. TPT- or equivalently just TPT is the name for the untruncated limit distribution with reliability R Y. Figure 2.3 shows the reliability functions for varying T keeping all the other parameters (,, E(Y T ) but not ) constant. The graphs are plotted on log-log scale, because the power-tail =x then results in a straight line with slope. Therefore, the dierence between an exponential and a PT drop-o can easily be seen. These kinds of graphs will show up very frequently in this work. Finally, though already mentioned, we give the LAQT-matrices for the TPT-T renewal process. The index is used throughout the rest of this work to distinguish them from other processes: p = [ ; 2 ; ; T ] ; P = ; B = M = 2.5 Order Statistics i= =... = T : (3) When doing simulations with PT-distributions later on, some interesting eects will arise. The explanation of those will need an introduction to order statistics for PTs, which tells

20 4 2 Background TPT T: Reliability function 2 3 R(x) T= x Figure 2.3: The reliability-function R YT (x) for T = ; 3; 5; 7; 2; 3 plotted on log-log scale keeping the mean E(Y T ) = constant. = :4 and = :5 are the TPT parameters. For T = 2 linear behavior (i.e. PT-behavior) can be observed over 3 orders of magnitude for x (i.e. until x 3 ). Smaller T 's will be used later on in the N-Burst Process, since calculations for N > 2 with large T will require extremely large matrices. something about the largest sample in a nite sample space, taken from independent distributions. Let fx i j i Ng be a set of iid random variables with distribution F (), and mean E(X). Further, let fx [] X [2] X [N] g be the same set, but ordered in size place. Then F X[N] (x) = [F (x)] N : That is, if N samples are taken from a distribution F () then the distribution of the largest of them is given by [F (x)] N (see Order Statistics in [Feller, 97] or [Trivedi, 982]). For PT distributions, the expectation value of this largest member was shown by [Lipsky et al., 996] to have the asymptotic behavior (for large N) In contrast, the negative exponential distribution yields E(X [N] ) = E(X) NX j= j E(X [N] )! E(X)N = : (3)! E(X) ln N (law of diminishing returns); The faster growth of the largest member of the order statistic of PT distributed variables again clearly shows the tail inuence; the probability of huge values occurring is too large to be negligible. Even worse, not only does the expected value E(X [N] ) grow by the factor N = with increasing N, but the distribution of X [N] also has a power-tail with the same, since: x R X[N] (x)! x ( c=x ) N = xn (x c) N N! c = Nc x (N )

21 2.5 Order Statistics 5 for x! (easily seen by using the binomial expansion of (x c) N ). As already mentioned, the main characteristic of a PT distribution is the non-negligible probability of large values. So E(X [N] ) is already large, but in its realizations, even much bigger values occur from time to time..4 Distribution of Maximum of N PTs each with Mean. N=. Distribution of Maximum of N PTs each with Mean. N= f(x[n]) f(x[n]).6..4 e x[n] e-6 x[n] Figure 2.4: Simulated Distribution (pdf) of the maximum of i.i.d. PT-samples, each with mean E(X) = :. Since = :4, E(X [N] ) = 38:95. The curve on the right is drawn on log-log scale and shows a linear behavior for large x [N], i.e. a power-tail =x +. The curve is [N] very asymmetric, its theoretical maximum is around x M = (c + N)= = 3:34, which is a factor 4:4 smaller than the mean. The empirical cumulative probabilities F (x M ) = :266 and F (E(X [N] )) = :89 give further evidence of the highly unsymmetric distribution: Most of the time the simulated maximum is going to be too small, therefore sometimes it is going to be much bigger than expected. Figure 2.4 shows the empirical pdf of X [], obtained by generating random numbers from a TPT-distribution (see Chapter 5..2) with mean, determining their maximum and repeating that process. The empirical mean of the simulated distributions of the largest order statistic was 48.4 for N = 3 and 4 runs of the simulation. For all other N = ; :::; 6 the mean was within % of the calculated E(X [N] ) E(X)N =. Also, the graph clearly shows the power-tail in the log-log plot. Moreover, the asymmetry of that distribution is obvious: The maximum of the pdf, the most likely value for X [N], can be calculated: F X[N] = (F X ) N =) f X[N] = N (F X ) N f X =) df X [N] dx = N(N )(F X) N 2 fx 2 + N (F X ) df N X dx : = N(F X ) N 2 (N )f 2 X + F X df X dx To get the maximum, the derivative of f X[N] is set to. Since for large x, F X (x) = c x ; f X(x) = c x ; and f c( + ) X(x) = : + x +2 f X [N] = =) c2 2 x (N ) = c c( + ) 2+2 x x +2

22 6 2 Background Solving that equation for x gives: r x M = c( + (N )) : + Using the tail of the distribution for that formula makes sense for large N, since then the largest order statistics is very likely to be large, so it will end up in the tail of the distribution. For large N, the equation can be further simplied: x M rc + N : Redoing that whole derivation for our Pareto function (c = ), leads exactly to the same approximation for x M. This really shows that the tail alone determines the largest order statistic for large N, since the latter derivation also includes the `body', but leads to the same result as the one above, which only uses the tail. Actually, the formula for E(X [N] ) in [Lipsky et al., 996] was derived for a Pareto function with mean. The same argument, that the tail is taking over responsibility for the largest order statistics in case of large N, suggests that it is also valid for the TPT-function. Simulations such as in Figure 2.4 conrmed that. 2.6 Self Similar Behavior and Long-Range Correlation The innite variance of our PT distributions with < < 2 also causes another peculiarity. The Central Limit Theorem (CLT) does not apply in its standard form: Given a set fx i g; i = ; 2; ::: of i.i.d. random variables with nite variance 2 and mean, the `average' of n of them, A n := n approaches in the limit n! a Normal Distribution with mean and variance 2 = =n, 2 which has the pdf " (x) = p exp 2 # x : (32) 2 2 The Normal Distribution is symmetrical about its mean and has exponential drop-os on both sides of it. For n!,!. Thus we use a scaling factor p n to get constant variance 2. Also, the mean is moved to. Then the random variable nx i= X i ; Z n := p n(a n ) approaches the Normal Distribution with mean and variance 2. When using a PT with < < 2 we do not have nite variance any more. The CLT then has to be modied (see [Feller, 97] and [Samorodnitsky & Taqqu, 994]): A dierent scaling factor comes in, Z n := n (A n ) where = =:

23 2.6 Self Similar Behavior and Long-Range Correlation 7 The square-root of n thus is replaced by the factor n, where = :28574 for = :4, so substantially smaller than :5. That implies, that the distribution of the A n gets narrow about its mean much more slowly for the PT-distributions than for nite variance distributions. Furthermore, in the limit n! the distribution of Z n does not approach the Normal Distribution any more, but a so called -stable distribution instead. In our case (X i > ), these distributions are asymmetric with a power-tail to the right and an exponential dropo to the left (see [Klinger, 997]). For deeper insight into the theory of -stables see [Samorodnitsky & Taqqu, 994]. It seems that Z n for our SM-models also converges towards an -stable distribution. However, this is not investigated here. Further research needs to be done in that eld. But we will look at the counting process here and there we have a similar situation as for the Z n : It also is an averaging process, though not with a xed number, n, of random variables, but with an over-all time limit P N i= X i < P N+ i= X i. The eects, however, are similar: While increasing, the distribution of the counting process gets narrow about its mean much more slowly than for classical (nite variance) processes. That is what usually is called Self-Similarity or high variability, one of the desired properties of our models. The other desired property that the PT-distributions in our SM models deliver, is the autocorrelation in the IA-times. Not only is there correlation but it also drops down very slowly with increasing lag-k. In classical models without PT-distributions the autocorrelation r(k) for large k drops down exponentially with increasing k (i.e. r(k)! c (c 2 ) k ). When using PT-distributions the correlation curve also has a power-tail with power ( ): r(k)! c r k That behavior is called Long-Range Correlation in the IA-times and is another goal of our modeling. The eect of long-range correlation also delivers a feasible way of determining the parameter for measured IA-times of real trac: Calculate the autocorrelation, draw the curve of r(k) on log-log scale and then determine the slope of the linear part.

24 8 3 Arrival Processes with Long-Range Correlation Chapter 3 Arrival Processes with Long-Range Correlation This chapter introduces three semi-markov processes for modeling network trac. All of them use PT distributions and can be calculated by our LAQT-methods. They all turn out to have long-range correlation in their IA-times, though the underlying PTrenewal process is used in dierent ways. The PT-distribution in principle has as its only varying parameter and therefore its mean, since we assume = :4 to be xed based on other research ([Leland et al., 994]). The parameter does not really seem to have much inuence on the results (see page 2), so it is assumed to be xed to :5 as well. Generally, the mean of our processes is nothing to worry about, since it can be set by scaling anyways, i.e. multiplying the inter-arrival times by some factor (therefore we look at the coecient of variation and the autocorrelation coecient, since they are not aected by multiplication of the random variable). 3. Merging of Power-Tail and Poisson Process The following model was brought up and is further discussed in [Fiorini et al., 995] and [Fiorini & Lipsky, 996]. Poisson + t Power-Tail t = Merged t Figure 3.: The Merged Process is an overlap of two independent renewal processes. In our case, one process has Poisson, the other one PT-distributed IA-times. A pure PT-renewal process does not appear to be a good trac model for two reasons: First, there are occasionally long periods with no trac at all (large PT IA-times), which does not really happen in real network trac. Secondly, as a renewal process it does not have any correlation in its IA-times.

25 3. Merging of Power-Tail and Poisson Process 9 Both problems are overcome when combining the PT-renewal process with other processes. That other process is chosen here to be Poisson; with most other processes, the resulting eects (i.e. long-range correlation in IA-times) are similar. Two dierent ways of combining will be investigated: The rst one merges the streams of arrival points of the renewal process, while the second one uses the PT-process to modulate the rate of the Poisson Process (see next section's Burst Process). The merging is done by independently generating a stream of PT-arrivals and Poissonarrivals, marked as arrival points on the time-axis. Then these two streams of arrival points are superimposed to build one single sorted stream (see Fig. 3.). The IA-times nally are the time dierences between the arrivals, which could be generated by dierent processes. So what are the eects? The long gaps will not happen anymore, since Poisson arrivals would ll them up. And there is going to be correlation in the inter-arrival times, because these lled up gaps result in periods of pure Poisson, which have a higher mean than the merged process. So having an IA-time longer than the mean increases the probability that some of the following ones are longer as well within the possible PT-gap. That is what positive correlation means. The only new parameter this merging process introduces, is the percentage p the Poisson arrivals have in terms of all arrivals. Since the overall arrival rate is the sum of both rates as it will be proven below, choosing p as the rate for the Poisson-process and p for the PT process, leads to an overall rate of with p being the percentage as claimed above. The means of the individual processes then are: PT : E(X ) = p ; Poisson : E(X ) = p =: So for example if p = : (thus % Poisson-arrivals), the IA-times during those long PT-gaps have a mean of, that is times as big as the overall mean. We next look at the LAQT matrices for this SM-Process: The TPT-T renewal process itself can be described as an SM-process. The L -matrix then has to take over the role of the entrance vector p. The state leaving rate r i (see (28)) is then spread out over the ith row of the L -matrix according to the probabilities in p : L = r p r 2 p. r T p B is the same matrix as for the GI-Process (3) = r p : The additional Poisson departures in the merged process do not extend the state space, but only add their rate = p to the diagonal of L, because a Poisson departure leaves the PT process in the same state. The M = B matrix also needs modication by adding the Poisson-rate to the state-leaving rate. So we get: L = L + I; B = M = M + I:

26 2 3 Arrival Processes with Long-Range Correlation B is still a diagonal matrix, so it can easily be inverted by hand to get V = B. The steady state entrance vector p can be shown ([Fiorini et al., 995], p. 3) to be p = + x p (I + V ): Putting in all the matrices, we get for the rst and second moments: E(X) = pv x = = + x + The last expression clearly shows that the rate of the merged process is the sum of the rates of the individual processes. E(X 2 ) = 2pV 2 = 2 ( T )( + x ) x T X i= ( 2 ) i + i where ; and are the parameters of the TPT-T distribution (see p. 3). The matrixformula for the correlation was already given in (2). The correlation-curve, as shown in. Auto-Correlation of Merged Process, p=.5 slope -.4 TPT-64 TPT-27 TPT-23 TPT-2 TPT-7 r(k)... e+6 e+7 k Figure 3.2: The autocorrelation coecient r(k) lag-k of the merged process with 5% Poisson departures. For high truncations T, the curve looks like a straight line (for bigger k > ) on a log-log scale. The smaller T, the earlier the correlation drops o, without having much inuence on r(k) for small k. Figure 3.2, shows the expected linear behavior on log-log-scale (i.e. long-range dependencies) if the used TPT-truncation, T, is high enough. The smaller T, the earlier r(k) drops o exponentially. For T = 64, the curve looks like a straight line with slope = :4 over the whole plotted range for k. Figure 3.3 shows how the coecient of variation, C, and the correlation lag-, r(), vary when changing p. The curves are calculated for a TPT-64 distribution. Increasing T further does not change the results. The calculation of C and r() for the two dierent sets of million inter-arrival times from Leland's Ethernet measurements ([Trace, 989]) gave the results: C = 3:22, C 2 = 3:3, r () = :2, r 2 () = :8, i.e. 3:2 C 3:3

27 3.2 A Single Source On/O-Model: -Burst Merged Process - coefficient of variation & correlation lag TPT-64: C TPT-64: 4*r() 2 C, 4*r() percentage p of poisson Figure 3.3: The coecient of variation and the autocorrelation lag- (multiplied by 4 to put it in the same scale) for the merged process over varying proportions, p, of Poisson arrivals. The mean of the whole process is always kept constant to. For p! the process goes to a PT-renewal process, which has innite variance (since = :4). Though r() = for the renewal process, [Fiorini & Lipsky, 996] show, that for a TPT-64 distribution, r() does not drop to until p = 6. For smaller p it seemed to be a linear drop-o to. So one could bring up the conjecture: lim p! r() > for an untruncated PT. and :8 r() :2. This would imply p = 9% and p = 33% respectively. For a more detailed discussion of this process and the correlation in particular, see [Fiorini & Lipsky, 996]. 3.2 A Single Source On/O-Model: -Burst The merged model has some nice characteristics that match those of real trac data. However, it would be dicult to explain physically, how such trac patterns could come up in a real network. We already mentioned that several researchers found evidence that le sizes are distributed according to a PT-distribution. When using xed size packets, this implies that the number of packets during one single le transfer is also PT-distributed. The following model, called -Burst, describes a series of non-overlapping le transfers with exponentially distributed gaps (OFF-times with mean B) in between. It belongs to the class of ON/OFF-models. However, there is only one source that is sending packets. This restriction will be abolished in the next section. During the PT-distributed ON-time (i.e. the le transfer), packets are generated with a constant rate (see Fig. 3.4). Let the mean length of the ON-times be L. The -Burst process with truncated PT distributions belongs to the bigger class of Markov Modulated Poisson Processes: The departures are always generated by a Poisson Process. However, the rate of that varies and is determined by the current state of an underlying Markov chain (Markov modulated). Our merged process from the last section does not

28 22 3 Arrival Processes with Long-Range Correlation -Burst Model PT M PT M PT M PT M ON ON ON ON OFF OFF OFF OFF Mean L B λ λ λ λ t Figure 3.4: The -Burst Process consists of alternating ON and OFF-periods. During the PTdistributed (with mean L) ON-periods, packets are generated by a Poisson Process with rate. The length of the OFF-periods is taken from a negative exponential distribution; no trac is generated during the OFF-periods (for now, later on there will be some low trac with rate l ). belong to that class, since there are also the departures from the PT-process. µ/γ ζ ζ 2 µ/γ OFF /B ζ T µ/γ T- ON: /L Figure 3.5: The underlying Markov chain of the -Burst process. The states on the right hand side model the TPT-T distributed ON-time, during which departures with rate occur. The process alternates between ON and OFF periods. Figure 3.5 shows that underlying Markov chain for our -Burst process. It has T + states: The single state on the left-hand side takes care of the OFF-period. Since it is exponentially distributed with rate =B, one state is enough for it. The column of states on the right is the familiar representation of the TPT-T distribution (compare with Fig. 2.2), so it stands for the ON-time. From the OFF-state (state ) the process goes to one of the T ON-states (numbered 2; : : : ; T + ) according to the PT entrance probabilities: P (m) = ;i+ i. Since ON and OFF times strictly alternate and there is only one OFF-state, there is only one choice for going back, so: P (m) k; =, k = 2; : : : ; T +. Remember that this is only the underlying process, which modulates the rate of the Poisson departures (therefore the superscript m) by describing the length of the ON-periods, so there are no

29 3.2 A Single Source On/O-Model: -Burst 23 departures (B (m) = ). The rate matrix is M (m) = diag( B ; ; : : : ); T whereby is chosen according to (29), so that the TPT-T distribution has mean L. These matrices now can easily be modied to include the modulated Poisson process; the state space is exactly the same. Since the departures do not change the state of the underlying chain, the L-matrix is diagonal with the corresponding Poisson rates (here only or ): 2 3 : : : L = : : : Since in this case L also is a diagonal matrix, and the Poisson-departures are a second possibility of leaving a phase of the SM-process, the matrix L adds to the state leaving rate: M = M (m) + L: Also, the state transition probabilities have to be changed: A transition to another state of the underlying Markov chain only occurs if the negative exponential state time (rate M (m) ii for state i) is smaller than the departure time (rate L ii ); compare with (7): P ij = P (m) ij M (m) ii M (m) ii + L ii (33) We could write down all the matrices now and do the calculations with them. However, we want to modify the model slightly before that: As it is, during the OFF-times, nothing at all happens. So again, there will be gaps with no trac at all, which is not really the case for real network trac. In addition, as we will see later, these gaps would lead to very high coecients of variation, higher than the values, we are aiming for. To avoid that, some amount of Poisson trac with relatively low rate l is superimposed (comparable to the merged process, here it is not a PT-renewal process, but the ON-OFF process, that gets merged). That additional Poisson trac could be seen as low background trac for the bursty le transfers of the ON-OFF model. Adding that background trac to the model is fairly easy, since the departure process is already Poisson. The underlying chain does not change at all: The rate l is added to the diagonal of the L-matrix, the calculations of M and P stay as they are, using the `new' L-matrix. Finally we get: L = l : : : l ; : : : + l

30 24 3 Arrival Processes with Long-Range Correlation P = M = B l p q l + B + l l + T where (q ) i = ; = i ; i = ; : : : ; T: = i + ( + l ) Let p := p =(+B l ). Since only the rst row and column of the matrix P have non-zero elements, P can be written as a sum of two rank- matrices Q ; Q 2 : P = Q + Q 2 ; Q = From these equations it is easy to show: Also, [; p ]; Q 2 = Q 2 = Q 2 2 = ; Q 2 := Q Q 2 = Q 2 := Q 2 Q = q p q : : :. : : :. q p P 2 = (Q + Q 2 ) 2 = Q 2 + Q 2 : : [; ; : : : ; ] Using these matrices helps a lot for getting the inverse V = B = (I P) M. It turns out that (I P) is a linear combination of the ve matrices I; Q ; Q 2 ; Q 2 ; Q 2. So assuming (I P) = I + aq + bq 2 + cq 2 + dq 2 ; it follows for the unknowns a; b; c; d from looking at the components of the following matrix equation: I = (I P) (I P) = (I Q Q 2 ) (I + aq + bq 2 + cq 2 + dq 2 ) Using the special structure of the participating matrices, the scalar equations for the single elements of the matrix equation give: a = b = c = d = p q Then V comes out as V = (I P) M = I + p q (Q + Q 2 + Q 2 + Q 2 ) M