WCC 2000, ICCT 2000, International conference on Communication Technology Proceedings, Aug. 21-25, 2000, Beijing, China
A Method for Modeling Autocorrelation Functions of Asymptotically LRD Traffic and Its Verification Ming Li, Weijia Jia, Wei Zhao* Dept. of Computer Science, City University of Hong Kong, Kowloon, Hong Kong, SAR China Email: {mingli, wjia}@cs.cityu.edu.hk *Dept. of Computer Science, Texas A&M University, College Station, USA *Email:
[email protected]
Abstract This paper points out that there exists a
However, for a specific traffic trace, the various estimation methods might yield different values of H’s substantially for the same trace as remarked in [7]. This makes it difficult to model traffic with ACFs. Thus, an optimal representation of ACFs of LRD processes is well worth discussing. Mathematically, optimally modeling ACFs of realtraffic traces can be abstracted as follows. For a given r indicating a target ACF of a real traffic, find a function rˆ of a LRD process which best fits it in the sense of F(e) = min, where e = ( rˆ −r).
unique optimal approximation of autocorrelation function of an asymptotically second-order selfsimilar process. A simple type of autocorrelation functions is used to explain the method of optimal solution and verify the long-range dependence of traffic in the view of optimal approximation.
KeywordsModeling, long-range dependence
1. Introduction
2. Spaces RH, R1H and l N2
Recent researches have shown that the behaviors of the traffic on LAN and WAN are well modeled by second-order self-similar processes with long-range dependence (LRD) [1-2]. Second-order self-similar processes are classified into two classes [1-2]. One is exactly second-order self-similar model and the other asymptotically second-order self-similar model. [1] pointed out that exactly second-order self-similar model is not enough to model real traffic. Hence, asymptotically secondorder self-similar processes are considered in the paper. Throughout the paper, the term LRD processes means second-order self-similar processes unless otherwise stated. LRD processes are defined by autocorrelation functions (ACFs). As ACFs can be used to study queuing systems [3], it is significant to study how to find a function that best fits the autocorrelation sequence of a real-traffic trace (target ACF). Because the ACFs of LRD processes are characterized by a single parameter H [2], the estimation of H is paid attention to [1-2], [5-7].
0-7803-6394-9/00/$10.002000 IEEE
Our aim is to solve the issue as: for a given r(k) to find a rˆ (k) which best fits it in the sense of F(e) = min and verify whether the result is of LRD. To achieve this, we have to complete two tasks. • To construct the structure of functions belonging to the space R1H such that R1H ⊆ RH ⊆ l N2 , where RH is the set of ACFs for LRD processes, R1H is the subset of RH, containing a type of ACFs of asymptotically LRD processes and l N2 is the set of ACFs of second-order processes. • To construct the space l N2 .
2.1 Main Properties of ACFs Let X = (Xt: t = 1, 2, …) be a covariance stationary second-order stochastic process with mean µ = E(Xt), variance σ2 = Var(Xt) and ACF
62
WCC 2000, ICCT 2000, International conference on Communication Technology Proceedings, Aug. 21-25, 2000, Beijing, China
E ( X t + k − µ )( X t − µ ) , |k| = 0, 1, 2, L, σ2 where E is the expectation operator and Var is the variance operator. The properties P1 and P2 have to be satisfied by all types of ACFs. P1: r(k) is an even function (|k| = 0, 1, 2, L). P2: r(0) ≥ |r(k)|. If X is a LRD process, it must satisfy the properties P3 and P4 [1], [2]. r (k) = ∞. P3:
In fact, (2-5) implies that X is asymptotically LRD [1]. Each function of (2-5) meets P3 and P4 and decays hyperbolically. One of difficulties for applying self-similar processes is that the exact whole structure of r(k) ~ ck2H−2 (k → ∞) has not been achieved exactly. Thus, Statement 2.1 defines a simple type of asymptotically LRD processes.
r(k) =
Statement 2.1: ACF (|k|+1)2H−2 is asymptotically equivalent to k2H−2 under the limit k → ∞: k2H−2 ~ (|k| + 1)2H−2 (k → ∞), k∈ I+. (2-6) The proof is obvious and omitted.
∑ k
P4: r(k) decays hyperbolically. And P5 is met when X is a short-range dependent process (SRD) [1], [2]. P5: r (k) < ∞.
With Statement 2.1, (2-5) can be equivalently expressed by (2-7).
∑ k
Statement 2.2: A process X is said to exhibit LRD if its ACF is with the structure below r(k) ~ c(|k| + 1)2H−2 (k → ∞), (2-7) where c > 0 is a constant. Proof: According to Statement 2.1, the following holds r(k) ~ c k2H−2 ~ c(|k| + 1)2H−2 (k → ∞). (2-8)
If an approximation meets P3 and P4, it is of LRD otherwise it is of SRD.
2.2 Spaces RH and R1H Definition 2.1 [1]: A process X is called exactly second-order self-similar with parameter H∈(0.5, 1), if its ACF is 1 r(k) = (k + 1) 2 H − 2k 2 H + (k − 1)2 H , k ∈ I+, (2-1) 2 where I+ = 0, 1, 2, L.
[
Equation (2-7) defines a set of functions as r(k) = (|k| + 1)2H−2. (2-9) Let RH be the set that contains the ACFs of secondorder self-similar processes. Thus, all denoted by (21) and (2-7) are characterized by the parameter H and they are elements of RH. Emphasis is given to H by the expression r(k) = r(k; H). Let R1H = {r; r = c(|k| + 1)2H−2)}. (2-10) Then, (2-11) R1H ⊆ RH.
]
Definition 2.2 [1]: A process X is called asymptotically second-order self-similar with parameter H∈(0.5, 1), if lim r ( m ) (k ) = r ( k ), m ∈ I1= {1, 2, L}, k ∈ I+, (2-2) m→∞
where r(m)(k) is the ACF of X(m) below (m) X(m) = { X t : t = 1, 2, L}. (2-3) (m) X is the new covariance stationary time series obtained by averaging the original series X over nonoverlapping blocks of size m. That is, X(m)(k) =
1 X t( m ) , t ∈ I1. ∑ m
Statement 2.3: R1H is closed and convex. Proof: For numerical computation, H∈[0.5+, 1−] and any rˆ ∈ R1H is continuous at H. Hence, R1H contains all its limit points and R1H is closed. R1H is convex since ∀rˆ1 , rˆ2 ∈R1H there exist λ ≥ 0, µ ≥ 0 and λ + µ = 1 such that λrˆ1 + µrˆ2 ∈ R1H.
(2-4)
Definition 2.3 [2, Definition 2.1]: Let X be a stationary process for which the following holds. There exists a real number H∈(0.5, 1) and a constant c > 0 such that r(k) ~ c k2H−2 (k → ∞). (2-5) Then, X is called a stationary process with long-range dependence.
0-7803-6394-9/00/$10.002000 IEEE
2.3 Space l N2 2
Let l N be the set that contains ACFs of secondorder processes, including any autocorrelation sequences of real-traffic traces. For real traffic, any concerned trace is a sequence with finite length.
63
WCC 2000, ICCT 2000, International conference on Communication Technology Proceedings, Aug. 21-25, 2000, Beijing, China
Besides, its length is finite when practical numerical computation is performed. Without loss of generality, the maximum possible length of x is assumed as p. Let N be a positive integer that is much greater than p, e.g., p / N = 10 −302 . Hence, N is “infinite” in the engineering sense. Define the norm of r as inner product N
∑| r |
r = < r, r > =
2
H:
o
[
]
5. Model Observations Approximation is obtained in R1H by investigating a real-traffic trace in [8] to demonstrate the method. A key point is to obtain Hˆ o for rˆ such that F(e) = min. The raw data file name of traffic trace in [8] is pAug.TL. Fig. 5.1 (a) indicates a segment of pAug.TL, Fig. 5.1 (b) shows its target ACF.
(2-13)
Hence, l N2 is a Hilbert space [4] and 2
(4-2)
Thus, (4-3) expresses the optimal approximation of an ACF of a LRD process in R1H 2 Hˆ − 2 rˆ(k ; Hˆ ) = k + 1 o . (4-3)
(2-12)
R1H ⊆RH ⊆ l N .
Hˆ o = arg min δ ( H ). H ∈( 0.5,1)
k =0
∑
(4-1)
Then
, k ≤ p
