An admission control approach for multifractal network traffic flows using effective envelopes

Int. J. Electron. Commun. (AEÜ) 64 (2010) 629 – 639 www.elsevier.de/aeue

An admission control approach for multifractal network traffic flows using effective envelopes Flávio H.T. Vieiraa,∗ , Luan L. Leeb a School of Electrical and Computer Engineering of Federal University of Goiás (UFG), Av. Universitária, n. 1488 Postal Code 74605-010 – Setor Leste Universitário, Goiânia, GO, Brazil b Department of Communications, School of Electrical and Computer Engineering, Av. Albert Einstein, 400, State University of Campinas, P.O. Box 6101 13083-852, Campinas, SP, Brazil

Received 10 July 2008; accepted 22 April 2009

Abstract In this paper, we present a novel admission control scheme that takes into account the multifractal properties of network traffic flows. To this end, we mathematically determine the variance of the aggregated traffic processes as well as their wavelet energy across time scales, considering a multiplicative cascade based multifractal model for the network traffic traces. In addition, we derive an analytical expression for the effective bandwidth estimation of the multifractal modeled traffic process. We use the proposed effective bandwidth equation to determine the effective envelope for multifractal traffic flows. The development on the multifractal traffic effective envelopment allows us to design an admission control scheme capable of efficiently guaranteeing quality of service in terms of effective envelope violation and delay probabilities for multifractal traffic flows. Through simulations, we compare the admissible number of flows in connections under different types of scheduling algorithms as well as different traffic models, such as the monofractal fBm and the proposed multifractal approach. 䉷 2009 Elsevier GmbH. All rights reserved. Keywords: Admission control; Multifractal traffic flows; Effective bandwidth; Network calculus; Quality of service

1. Introduction Network applications such as voice over IP and video conference that demand quality of service (QoS) guarantees have become popular and are frequently encountered in the Internet. These growing new applications have motivated many studies and proposals of providing QoS for Internet traffic flows. Among them, the most popular QoS architectures are the Intserv (integrated services) and Diffserv (differentiated services) [1,2]. However, one major drawback of these QoS mechanisms is that they can greatly reduce the ∗ Corresponding author. Tel.: +55 62 3209 6070.

E-mail addresses: [email protected] (F.H.T. Vieira), [email protected] (L.L. Lee). 1434-8411/$ - see front matter 䉷 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2009.04.011

network utilization, especially when traffic becomes bursty within multiple time scales [3]. In order to provide efficient multiplexing of network traffic flows and service guarantees, the concept of service curve derived from network calculus was introduced [4]. Under the deterministic framework of network calculus, a service curve can only guarantee that all packets of a traffic flow attain a worst case bound of end-to-end delay under the restriction of zero packet loss [3]. As a consequence, frequently a significant percentage of network resources is not used, resulting therefore in very conservative service curves (low network utilization). Aiming at overcoming such a problem, some approaches for providing QoS statistical guarantees were proposed [5,6]. These QoS providing approaches have led to significant increases of link utilization by tolerating

630

F.H.T. Vieira, L.L. Lee / Int. J. Electron. Commun. (AEÜ) 64 (2010) 629 – 639

some non-zero loss or delay, in contrast to the deterministic cases [7]. Besides network calculus, there is also an alternative way of determining traffic requirements based on the concept of effective bandwidth [8,9]. The effective bandwidth theory, introduced a decade ago, has the goal of simplifying network resource allocation tasks by associating a bandwidth value to the corresponding connection. Since its introduction, the notion of effective bandwidth has found wide acceptance and applicability to the issues related to broadband network management and performance analysis [10,11]. Closely related to the network resource allocation, admission control has been regarded as an important scheme in stipulating the correct amount of traffic allowed to be inserted into networks on a given time interval. Several research works have been carried out to establish deterministic traffic envelopes and regulators to the cases in which traffic flows are statistically independent [12,13]. However, it has been demonstrated that statistical envelopes can provide higher link utilization as well as a larger number of admitted flows with guaranteed QoS parameters in comparison to their deterministic counterparts [5]. Statistical bounds on network performance can be adequately determined for end-to-end QoS provisioning schemes even without having traffic correlation types explicitly specified [13] or through delay variation control implemented in each network connection [14]. In [6], the authors defined statistical service curves that permit to establish statistical guarantees for the delay and queueing length size in the buffer for simple traffic models. Afterwards, in [5] Boorstyn et al. introduced the concept of effective envelope, a generalization of some results of previous works on network calculus. In this work, firstly we obtain an expression for the variance of aggregated multifractal traffic by assuming that the traffic process can be described by a pairwise product of a multiplicative cascade and a lognormal random variable. Furthermore, we determine the second moment of wavelet coefficients of the network traffic flows through the Haar wavelet analysis. The knowledge of both the variance of the traffic flows and the wavelet energy across time scales allows us to compute the effective bandwidth of multifractal cascade based traffic processes. Next, we investigate the integration between the network calculus formalism and the effective bandwidth theory by obtaining a formula for the effective envelope for multifractal processes. As as result, we are able to analyze different traffic flow scheduling schemes with multifractal traffic input that can be considered as a pioneer work. Through the obtained effective service curves, we verify the performance of the following scheduling disciplines: SP (static priority) [7], EDF (earliest-deadline-first) [15] and GPS (generalized processor sharing) [16]. Finally, we propose an admission control scheme that makes use of the multifractal effective envelope. In other words, we consider a more general situation than those found in the literature where simpler traffic models are used in the estimation of QoS parameters. We also present a comparison among

different admission control approaches using the monofractal traffic fBm (fractional Brownian motion) model [17] and the proposed multifractal approach.

2. Network traffic characteristics For many real network traffic processes, their wavelet energy-scale or variance-time plots usually are far from tending to straight lines. Instead, many of them have piecewise fractal behaviors with varying Hurst parameters over a range of time scales [18]. When this is the case, such processes are usually referred to as multifractal processes [19]. Multifractal processes are defined by a scaling law for the statistical moments of the processes’ increments over finite time intervals. This means the multimedia network traffic has complex and strong dependence structures inherently, appearing very bursty and the burstiness looks similar over many scales [20]. A stochastic process X (t) is called multifractal if it satisfies E(|X (t)|q ) = c(q)t (q)+1

(1)

for t ∈ T and q ∈ Q, where T and Q are intervals on the real line, and (q) and c(q) are functions with domain Q, (q) is the scaling factor and c(q) is the moment factor of the multifractal process. Furthermore, we assume that T and Q have positive lengths, and that 0 ∈ T , [0, 1] ⊆ Q. If (q) is linear in q, the process X (t) is called monofractal; otherwise, it is multifractal [19].

2.1. Capture of multifractal traffic properties The simplest multifractal, the binomial measure can be generated through an iterative procedure, called multiplicative cascade, on the compact interval [0, 1]. Let m 0 and m 1 be two positive numbers adding up to 1. At stage j = 0, we start the construction with the uniform probability measure 0 on [0, 1]. In the step j = 1, the measure 1 uniformly spreads mass equal to m 0 on the subinterval [0, 1/2] and mass m 1 on [1/2, 1]. This process is iterated for j levels, and at each stage the total measure is preserved. If the multipliers used have the same fixed value m 0 then the multiplicative cascade is deterministic [21]. Allowing the cascade multipliers to be random variables, we get a stochastic multiplicative cascade. Denoted by A j,k , these multipliers are chosen to be independent random variables with probability distribution function f A (x). Discrete wavelet transforms can be used for multiscale representation of the process X (t) through the wavelet (t) and scaling (t) functions [22]. Multiplicative cascades in the wavelet domain are capable of characterizing network traffic by computing the corresponding wavelet coefficients W j,k as [23,22] W j,k = U j,k A j,k

(2)

F.H.T. Vieira, L.L. Lee / Int. J. Electron. Commun. (AEÜ) 64 (2010) 629 – 639

where W j,k =

2.2. Variance of aggregated traffic data

 

U j,k =

X (t) j,k (t) dt

(3)

X (t) j,k (t) dt

(4)

and A j,k is a random variable in the range [−1, 1]. Further, some additional conditions are frequently assumed: the multipliers A j,k are statistically independent and identically distributed (i.i.d) within each scale, independent of U j,k and symmetric at the origin. A traffic model can capture multifractal characteristics by choosing the multipliers A j,k in or2 ). Thus, der to control the wavelet coefficient energy E(W j,k the following relations can be established [23]: 2 E(W j−1,k ) 2 E(W j,k )

=

2pj + 1 p j−1 + 1

(5)

In this section, we derive explicit equations that allow direct computation of the variance of the aggregated process in terms of the time scale of aggregation considering multifractal characteristics. Let X (k) be a discrete time process corresponding to the network traffic volume per unit time interval. When the representation of a process is done in the Haar wavelet domain, two general relations for the scaling and wavelet coefficients can be stated as follows, respectively [28]: U j,k j = 2

− j/2

j−1   U0,0 [1 + (−1)ki Ai,ki ]

W j,k j = 2− j/2 U0,0

i=0 j−1 

(9)



[1 + (−1)ki Ai,ki ]Ai,ki

(10)

i=0

and 2 2 ) = E(U0,0 ) (2 p0 + 1)E(W0,0

(6)

From Eqs. (5) and (6), it can be seen that the parameter p j can be used to capture the decay of the wavelet energy in scale, where U0,0 is the scaling coefficient at the coarsest scale. The multifractal properties of a real traffic data trace are characterized by its corresponding scaling function (q) and moment factor function c(q) as described by Eq. (1). Therefore, a multifractal model is expected to capture these two multifractal properties, which can be fulfilled by the pairwise product of a cascade process (t j )( j = 1, . . . , N ) and i.i.d. samples of a positive random variable Y, where t j represents a time interval in the stage j of the cascade [24]. The scaling function (q) is accurately modeled by choosing the cascade multipliers A j,k as a symmetric beta random variable with beta(,) distribution for  > 0 and the random variable Y as a lognormal process whose moment function is 2 2 E(Y q ) = eq+ q /2 , defined by the parameters  and . Under these assumptions, the scaling function 0 (q) = (q) + 1 and the moment factor c(q) can be respectively written as the following [25]:   () (2 + q) (7) 0 (q) = log2 (2) ( + q) and c(q) = e

631

q+2 q 2 /2

where here U0,0 is assumed to be a Gaussian random variable. The shift k j of scaling coefficients is related to the shift of one of its two direct descendents in a dyadic cascade as the following [23]: k j + 1 = 2k j + k j

where k j = 0 corresponds to the left-hand side descendent and k j = 1 to the right-hand side descendent. It can be easily demonstrated that the discrete time traffic process X (k) is related to the scaling coefficients U j,k at the finest scale, scaled by a factor, i.e.: X (k) = 2− j/2 U j,k

where (.) is the Gamma function. A method for estimating the scaling function (q) and the moment factor c(q) based on Eq. (1) is described in [25]. In this work, we include a recursive least squares algorithm [26] to adaptively obtain the updated values of 0 (q) and log c(q). Moreover, after obtaining the values for 0 (q) and log c(q), we apply an optimization algorithm (the Levenberg–Marquardt algorithm) to estimate the parameters ,  and  of functions 0 (q) and log c(q) [27].

(12)

Assuming that X (k) is a pairwise product process and through Haar wavelet properties, we determine the variance of the aggregated network traffic traces across scales as follows. Lemma 1. Let X (k) be a multifractal process whose (q) and c(q) functions are respectively given by (7) and (8) and the aggregated process X m of X (k) defined as d

(8)

(11)

X m (i) =

1 m

im 

X (k)

(13)

k=(i−1)m+1

where i = 1, 2, . . . , L, k = 1, 2, . . . , 2 N and L = 2 N /m. The variance of the aggregated process var [X m ] for m = 2 j , ( j = 1, . . . , N ), is determined by the following equation: var [X 2 ] = 2−4 j (e2+2 j

2

Proof. See Appendix A.



+1  + 1/2

j

− (e2+ )) 2

(14)

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F.H.T. Vieira, L.L. Lee / Int. J. Electron. Commun. (AEÜ) 64 (2010) 629 – 639

2.3. Second order moment of wavelet coefficients In our effective bandwidth estimation proposal, we need to estimate the second moment of the Haar wavelet coefficient of multifractal traffic process in order to generate the cascade multipliers A j,k . Again assuming that X (k) is a pairwise product process and through Haar wavelet properties, we derive an analytical expression for the second moment E(W j2 ) of the wavelet coefficients that can be adaptively computed. We will use this analytical expression to generate the multipliers A j,k which are capable of capturing the wavelet energy decay via Eq. (5). The following lemma enunciates the second moment of the wavelet coefficients E(W j2 ). Lemma 2. Assuming that Haar wavelet transform is applied to the process X (k) which has the (q) and c(q) functions respectively given by (7) and (8), then the second moment E(W j2 ) of the wavelet coefficients can be written as E(W j2 ) = e2+2

2



 +1 2−3 j − 2Z j  + 1/2

where Zj =2

j−1

2+22

[ 2 (2 j ) X

−e

2+2

−2 j

+ (e

)2



(15)

 +1 2−3 j  + 1/2

]

effective bandwidth of a traffic stream is defined as [29] (s, t, Nt ) =

1 log Eˆ Nt [es X (0,t) ] st

(19)

where 0 < s; 0 < t < Nt , X (0, t) represents the aggregate number of packet arrivals at a time interval of length t, Eˆ Nt [e X (0,t) ] is the measured moment generating function over a Nt -sample traffic trace, and s is the space parameter that is related to the loss probability P(Q > x) [11]. In the next section, we introduce an effective bandwidth estimation approach for multifractal processes.

3.1. Effective bandwidth for multifractal cascade based traffic flows In this work, we explore some statistical approaches to develop an effective bandwidth expression for multifractal traffic processes as summarized in the following proposition. Proposition 1. The effective bandwidth of multifractal cascade based traffic process X (k) can be expressed in terms of its corresponding multipliers A j,k as 1 (s, k) = k s2



2 N

k=1 | j,k | 2N

log

(20)

(16)

Moreover, the mean and variance of the scaling coefficients U j,k at the coarsest scale j = N are respectively given by E{U N ,k } = 2−N /2 (e+ /2 ) (17)    + 1 2 2 2 2−3 j + 2Z N − (2−N e2+ ) = e2+2

U N ,k  + 1/2 (18) 2

where j,k = 2− j/2 [esv 1 − esv 2 ] v1 = 2− j/2 U0,0 v2 = 2− j/2 U0,0

j−1 

(21) 

[1 + (−1)ki Ai,2ki ]

i=0 j−1 



[1 + (−1)ki Ai,2ki +1 ]

(22)

(23)

i=0

Proof. See A.2. Lemmas 1 and 2 allow the effective bandwidth estimation parameters to be updated without computing the DWT (discrete wavelet transform) for the entire traffic trace. This property is highly intended for real-time applications that demand fast and on-line processing time.

3. Effective bandwidth for traffic flows The effective bandwidth of traffic flows can be parametrically formulated requiring the application of a suitable analytical traffic model capable of fully describing the traffic source characteristics. As an alternative, the measured effective bandwidth (MEB) is an approach of effective bandwidth computing based on the process samples. The measured

and N is the number of stages of the multiplicative cascade. Proof. See A.3. Notice that the effective bandwidth estimation approach given by (20) requires the knowledge of the cascade multipliers A j,k . We propose to estimate the cascade multipliers through the proposed equations for the variance of the aggregated traffic process and the wavelet energy across time scales. Therefore, the complete procedure for computing the effective bandwidth of the traffic traces becomes as follows: Proposed Algorithm 1. Effective bandwidth computing q

(1) Let p0 = 1, aˆ 0 = [0 0], k = 1, . . . , 2 j , j = 1, 2, . . . , N and q = 1, . . . , q2 .

F.H.T. Vieira, L.L. Lee / Int. J. Electron. Commun. (AEÜ) 64 (2010) 629 – 639 q

(2) Compute aˆ k = [ˆ0 (q) log c(q)] ˆ for each q value using the following recursive equations [26]:

q aˆ k

q = aˆ k−1

−

q pk [xk xkT aˆ k−1

−

pk−1 x k ] −

q x k yk ]

x kT

pq,k−1

8

(24)

7

(25)

where yk = log E(| X¯ k |)q , X¯ k = [X 1 , X 2 , . . . , X k ] and x k = [1, log 2, . . . , log k]. (3) Estimate the  parameter of 0 using the Levenberg– Marquardt algorithm according to the following updating rule [27]: q

Effective Bandwidth x Buffer size

9

Effective Bandwidth

pq,k = pq,k−1 − pq,k−1 x k [1 +

x kT

x 105

633

MEB NEB Proposed EB

6 5 4 3 2

−1

i+1 = i − (Hes + diag(H ))

∇(i )

(26)

1 0

(4) (5) (6) (7) (8) (9)

(10) (11)

(12) (13)

where Hes is the Hessian matrix (Hes =∇ 2 (i )) of the quadratic error function  and is a control parameter at the ith iteration of the Levenberg–Marquardt algorithm. Apply again the Levenberg–Marquardt algorithm to estimate the  and  parameters of the c(q) function. Set scale j = 1, corresponding to the aggregation fraction equal to 2 j = 2. Compute the variance of the aggregated process var [X m ], (m = 2 j ) through (14). Compute Z j by using (16). Compute the second moment of the wavelet coefficients. Increment j by 1. If j = N (desired maximum number of cascade stages), compute the mean and variance of the scaling coefficients at the coarsest scale using (17) and (18); otherwise, go to step 6. Calculate p j s using (5). Here starts a new recursive procedure having the p j s values. Set j = 0 and compute the scaling coefficient U0,0 at the coarsest scale. Generate the random variables A j,k for k = 1, . . . , 2 j . Using the cascade multipliers, calculate the effective bandwidth via (20).

Fig. 1 shows the effective bandwidths computed through (20) as well as those given by the measured effective bandwidth (MEB) [29] and NEB (Norros effective bandwidth) [17] approaches versus the buffer size, under the target byte loss probability set to 10−6 . The traffic trace dec-pkt-2 was used here on purpose because it is monofractal with H =0.8, and as consequence its effective bandwidth can be appropriately determined via the NEB approach, which corresponds to a monofractal based effective bandwidth formulation [17]. The MEB and our effective bandwidth method show similar results. For small buffer size, the NEB granted a high effective bandwidth estimate value; evidently is different from the multifractal based effective bandwidth and the MEB estimates that were less sensitive to the buffer size variation.

1

2

3 4 Buffer size (bytes)

5

6 x 105

Fig. 1. Effective bandwidth × buffer size (dec-pkt-2 traffic trace).

4. Statistical network calculus: an overview The network calculus framework has provided the development of traffic flow scheduling algorithms [30,16] and of new network services [1,13]. Two important concepts such as the traffic process envelopes and the service curves derived from network calculus can be used to obtain bounds on data transfer delay and buffer occupation [4]. In the network calculus formulation, end-to-end guarantees are achieved through the min-plus algebra [31]. The packet arrivals in a network connection and its corresponding traffic output process can be represented by the non-negative, non-decreasing functions A(t) and D(t), respectively. Then, the queueing length in buffer (backlog) is given by B(t) = A(t) − D(t) and the delay by W (t) = inf{d ⱖ 0|A(t − d) ⱕ D(t)} at time instant t. In the statistical network calculus framework, the concept of effective envelope is introduced. The effective envelope of an arrival process A(t) whose effective bandwidth is represented by (s, ) is given by [6,32] 

log  (27)  () = inf (s, ) − s>0 s A probabilistic measure of the available service, namely the effective service curve, is useful to characterize the currently available service to the traffic flows, in conjunction with the effective envelope curve [6]. The service curve S  (t) for an arrival input process A(t) is a non-negative function which for a given time scale T satisfies the following equation: 

 (28) Pr D(t) ⱖ inf {A(t − ) + S () ⱖ 1 −  ⱕT

for t ⱖ 0.

634

F.H.T. Vieira, L.L. Lee / Int. J. Electron. Commun. (AEÜ) 64 (2010) 629 – 639

For workconserving schedulers, the value of the time scale T in (28) can be determined from bounding the busy period of the scheduler. For  > 0 and t,  ⱖ 0, if T is chosen such that [32] T = sup{| C > C}

(29)

where C is the server capacity. Then, T is a probabilistic bound on the busy period. Let g be the effective envelope for the arrival process A(t) in a network node and S s the effective service curve that satisfies (28) for some T < ∞. Define  as  = s + T g

(30)

where g is the probability that the arrival traffic violates the effective envelope and s is the probability that the service process violates the service curve, i.e., the condition established in (28). Then, d is a probabilistic bound on the delay once for all t ⱖ 0, Pr {W (t) ⱕ d} ⱖ 1 − , if d ⱖ 0 satisfies the following equation [32]: sup { g ( − d) − S s ()} ⱕ 0

(31)

ⱕT

A monofractal model whose effective bandwidth is widely known is the fBm. The effective envelope of a set C of the fBm monofractal traffic processes is expressed as [8,6]  Cg (t) = C t + −2 log g C t H (32) where C is the mean rate, C is the standard deviation and H is the Hurst parameter of the set C of fBm monofractal traffic processes. Let Aq be the arrival process of a class q ∈ Q and AC the group C of arrival processes of all classes types. Consider also that traffic flows belonging to Q different classes arrive at a server with capacity C. For each class q = 1, . . . , Q,  qg represents the effective envelope for the arrival process Aq . If the probability bound on the busy period T satisfy (29) for the aggregated process AC where b < 1, the effective service curves Sqs under the SP (static priorities) [7], EDF (earliest-deadline-first) [15], GPS (generalized processor sharing) [16] scheduling algorithms can be respectively expressed as [7]

  g p (t) (33) Sqs (t) = Ct − p
⎡

+

⎤  g p (t − [d p − dq ]+ )⎦ Sqs (t) = ⎣Ct − ⎛

pq

Sqs (t) = q ⎝Ct +



(34)

⎞+ 

[ p Ct − pg (t)]+ ⎠

(35)

pq

 where s =b +(q −1)T g , q =q / p  p is the guaranteed rate for class q and q is the GPS parameter that determines the proportion of service rate that traffic class q will have.

5. Effective envelope for multifractal processes In this section, we introduce an effective envelope expression for multifractal cascade based traffic models. The proposed effective envelope is also written in function of the cascade multipliers Ai, j that can be estimated through the proposed Algorithm 1. Proposition 2. Assume that the parameter  is dyadic, that is,  = 2 j , where j = 1, . . . , N . Then, the effective envelope of the multifractal cascade based traffic process X (k) can be written as   

 log g | j,k | 1 log k=1 − (36) g () = inf s>0 s  s where j,k , v1 and v2 are given by Eqs. (21)–(23), respectively. Proof. The proof of Proposition 2 is straightforward verified by combining Proposition 1 and the effective envelope given by (27).  It is worth mentioning that the effective envelope g () at time instant  given by (36) depends on the effective bandwidth updated through the process samples until this time instant. Therefore, the more precise the effective bandwidth estimation for a traffic flow, a better model for the accumulated traffic volume is obtained, i.e., the effective envelope.

5.1. Evaluation of effective envelopes In this section, we present the simulation results of effective envelope evaluation for the fBm traffic model and the proposed multifractal one. The comparison is carried out on effective envelopes normalized by the number of flows, i.e.,   Ng (t)/N , where Ng (t) is the effective envelope for N flows with equal statistical characteristics. In the effective envelope evaluation, we considered two types of traffic flows (Type 1 and Type 2) at a unit time scale of 1 ms. Table 1 lists some statistical parameters of the Type 1 and Type 2 traffic flows estimated from the real TCP/IP traffic traces dec-pkt-1 and lbl-pkt-5, respectively. These statistical parameters are: P is the peak rate,  is the mean rate, is the standard deviation, 2 is the variance and H is the Hurst parameter [33] for the Type 1 and Type 2 traffic flows. Fig. 2 shows the effective envelopes per flow considering a violation probability of g = 10−9 for the Type 1 traffic flow. We included in these plots the average rate of the traffic flows. A small value of effective envelope means that a low amount of bytes per flow is required to preserve the violation probability (10−9 ). By analyzing Fig. 2, one can observe that when the number of flows N increases, a smaller amount of bytes per flow is required for attaining the QoS parameters. Notice also that the proposed effective envelope is smaller than that of the fBm. This fact reveals that the

F.H.T. Vieira, L.L. Lee / Int. J. Electron. Commun. (AEÜ) 64 (2010) 629 – 639

635

Table 1. Traffic parameters. P (bytes)

 (bytes)

(bytes)

2

H

1 2

1536 1024

92, 26 10, 72

232, 3772 82, 5621

5.4 × 104 6.8165 × 103

0, 8084 0, 7810

5

Amount of Traffic per Flow (Kbytes)

Amount of Traffic per Flow (Kbytes)

Type

4.5 N=100

4 3.5 3

N=1000

2.5 2

N=10000

1.5 1 0.5

Average rate

0 0

100

200 300 400 Time interval (ms)

0.9 0.8 0.7 0.6 0.5

N=10000

0.4 0.3

N=100

0.2 N=1000

0.1

Average rate

0

500

0

100

200 300 400 Time interval (ms)

500

(b) Multifractal approach

(a) fBm g

Fig. 2. Effective envelope per flow ( N (t)/N ) for the Type 1 flows. (a) fBm and (b) multifractal approach.

server capacity demanded by the monofractal fBm model is more than necessary.

6. Admission control for multifractal traffic processes In this section, we propose an admission control approach for multifractal flows under different scheduling disciplines. We aim at verifying the impact on the number of admissible flows when the proposed multifractal based effective envelope is used for admission control in a single network node. Moreover, we compare the performance of our admission control scheme to an approach that takes into account a monofractal traffic modeling.

6.1. Effective envelope based admission control The proposed admission control strategy consists of determining through the multifractal effective envelope g , the maximum number of Type 1 flows that can be admitted without violating the probabilistic delay bound d1 . The admission control strategy is outlined below: Proposed Algorithm 2. Multifractal effective envelope based admission control applied to a network node (1) Let C be the server capacity and d1 the desired maximum delay. Stipulate the values of the violation probabilities g , s and b .

Table 2. Traffic parameters. Type

P (Kbytes)

 (Kbytes)

(Kbytes)

H

1 2

81, 4 45, 9

26, 5 2, 70

12, 3 3, 20

0, 8080 0, 7820

(2) Compute the cascade parameters (multipliers) A j,k through the proposed Algorithm 1. (3) In order to admit a traffic flow in the group C of admitted flows, the effective envelope for this group of flow must be obtained. Compute this effective envelope by using Proposition 2, i.e., via Eq. (36). (4) Estimate the probabilistic bound T on the busy period for the group of C flows in a network node by (29). (5) A new traffic flow is admitted without violating the probabilistic delay bound d1 . That is, it must satisfy the restriction given by (31). As a simulation scenario for evaluation of the admission control schemes, we considered a single server link with capacity of 1 Mbps. The input traffic consists of Type 1 and Type 2 traffic flows whose statistical characteristics are extracted from the dec-pkt-1 and lbl-pkt-5 traffic traces at the time scale of 100 ms (Table 2 ). Under the proposed admission control strategy, Type 1 flows must obey the maximum delay d1 . Given a fixed number of Type 2 flows in the connection, we determine through the effective envelope g , the maximum number of Type 1 flows that can be admitted without violating the

40 35 Ave

30

rage

25

ED

15

G

F

Pe

SP

0

50 100 150 Number of Type 2 Flows

Rat

e

25 20 15 10

ED

F

5

200

GPS SP

te

te

0

rage

30

Ra

Ra

5

Ave

35

ak

PS

ak

10

e

40

Pe

20

Rat

Number of Type 1 Flows

F.H.T. Vieira, L.L. Lee / Int. J. Electron. Commun. (AEÜ) 64 (2010) 629 – 639

Number of Type 1 Flows

636

0

50 100 150 Number of Type 2 Flows

(a) fBm

200

(b) Multifractal Approach

Fig. 3. Number of admitted Type 1 flows in function of the number of Type 2 flows (C = 1 Mbps) for different scheduling disciplines, g = 10−6 , d1 = 200 ms, 1 = 0.25 and 2 = 0.75. (a) fBm and (b) multifractal approach.

70 fBm Multifractal approach

60 Maximum Delay (ms)

probabilistic delay bound d1 . Notice also that it is possible to estimate the admissible number of flows through only the knowledge of the parameters ,  and  of the scaling function and moment factor. We evaluate the service offered to the Type 1 traffic flow under the three previously mentioned scheduling algorithms: SP, EDF and GPS. The simulations were carried out adopting the following additional configuration and set of parameters. The maximum delay for Type 1 flow was set to d1 = 200 ms. In the SP scheduling scheme, we assume that the Type 1 flows possess higher priority level. In the EDF scheduling algorithm, we fix the delay indexes as d1 = 200 ms and d2 = 100 ms for the Type 1 and Type 2 traffic flows, respectively. For the GPS, we chose the weights (priority GPS parameters) as 1 = 0.25 and 2 = 0.75. For comparison purposes, we also include in the simulations the number of flows that can be accommodated in the network link using the mean and peak rates as traffic flow bandwidths. Fig. 3 shows the admissible number of Type 1 flows that attains the desired probabilistic delay bound (d1 = 200 ms) and the violation probability g = 10−6 in function of the number of Type 2 flows. We also present the admitted number of flows when considering the fBm monofractal model. It can be noticed that the choice of the traffic model has significative influence on the admissible number of flows. For instance, the application of the fBm modeling results in a reduced number of admitted flows 1 in comparison to our approach. Concerning the analyzed schedulers, some interesting results can be observed. The EDF scheduling discipline provides the largest number of admitted Type 1 flows. This admissible number is reduced when the delay values d1 and d2 are decreased. As expected for the GPS scheduling scheme, the minimum number of admitted Type 1 flows is independent of the number of Type 2 flows [16]. That is, the GPS really guarantees a minimum number of Type 1 flows. Due to its largest number of admitted Type 1 flows, we chose the EDF scheduling discipline in order to compare the maximum delay obtained through the admission control

50 40 30 20 10 0 5

10 15 20 25 Number of Type 1 Traffic Flows

30

Fig. 4. Maximum delay versus admitted number of type 1 traffic flows.

approaches. Fig. 4 shows that the proposed admission control guarantees the required maximum delay bound. It can also be observed that although the fBm based approach guarantees the required delay, a smaller number of admitted flows is provided.

7. Conclusion In this paper, we proposed analytical expressions for some properties of multifractal cascade based processes such as the variance and the wavelet energy, both in terms of the time scale of aggregation. Using a convergence theory and other related statistical approaches, we derived an effective bandwidth expression for the multifractal cascade based processes. The experimental results showed that the proposed analytical effective bandwidth estimate is more precise than the Norros’ monofractal one for smaller buffer sizes.

F.H.T. Vieira, L.L. Lee / Int. J. Electron. Commun. (AEÜ) 64 (2010) 629 – 639

The proposed effective bandwidth equation allowed us to obtain the effective envelope for multifractal processes. We restrict the analysis of the effective envelopes to time scales in which a multifractal behavior is verified [34–36]. In these scales, the multifractal effective envelope is more attractive and appropriate once the traffic traces possess approximately lognormal distribution and other multifractal characteristics. In fact, the effective envelope for the Gaussian monofractal fBm model is numerically larger than our multifractal based effective envelope.

Acknowledgment The authors would like to thank Fapesp (Proc. 06/603636) for the financial support.

Appendix A. A.1. Proof of Lemma 1  jm Let m = 2 j and x( j) = ( j−1)m+1 X (k). Then, the aggregated cascade based process X m at stage j can be denoted as X m = 2− j x( j)

(A.1)

The variance of the aggregated multifractal process X m of X (k) can be written as var [X m ] = E[{2− j x( j) }2 ] − E 2 [2− j x( j) ]

(A.2)

Assuming that the process X is the product of a random lognormal variable Y and a multiplicative cascade , we have the following relation: jm 

x( j) =

Y (k)(k)

(A.3)

k=( j−1)m+1

By using the knowledge of the moments of the cascade multipliers A, the moments of the lognormal random variable Y and Eq. (A.3), the variance of the aggregated process of X (k) becomes ⎡⎧ ⎫2 ⎤ jm ⎨ ⎬  ⎢ ⎥ Y (k)(k) ⎦ var [X m ] = E ⎣ 2− j ⎩ ⎭ k=( j−1)m+1

⎡ −E

jm 

2 ⎣ −j

2

⎤ Y (k)(k)⎦

k=( j−1)m+1

 var [X ] = 2

−4 j

e2+2

2



+1  + 1/2

j

Let X m be the m-order aggregated process of X (k). For m = 2, the variance of the aggregated process X (2) can be written as  2  Nt −2 X (i) + X (i + 1) −X i=0,2, ... 2 (A.5)

2X (2) = Nt /2 where X and Nt are the mean and the number of samples of the traffic process. Now, let m = 2 j , the variance of the 2 j -order aggregated process is: 1  Nt −1 X (i)2 1 2 Nt i=0,1, ... 2 + j−1 Z j − X (A.6)

(2 j ) = X Nt 2  X (i) + X (i + j)/Nt . Using the fact where Z j = Nt −2 i=0,2, ... that 1 2 j −1   X (i)2 +1 2+22 2 j i=0,1, ... 2−3 j = e (A.7) 2j  + 1/2 Z j can be alternatively expressed as    +1 j−1 2 2+22 2−3 j

(2 j ) − e Zj =2 X  + 1/2 + (e2+ )2−2 j 2

− (e2+ ) 2

(A.8)

According to [37], the second moment of wavelet coefficients obtained from the discrete wavelet transform in the Haar wavelet domain of a traffic process can be determined as 2 j −1 j −2 2 X (i)2 X (i) + X (i + j) i=0,1, ... 2 E(W j ) = − 2 2j 2j i=0,2, ... (A.9) Thus, we can write the second moment of the wavelet coefficients E(W j2 ) as   +1 2 2−3 j − 2Z j E(W j2 ) = e2+2 (A.10)  + 1/2 where Z j is given by (A.8). The scaling coefficients U j,k at the coarsest scale j = N are computed as the following: −N /2

2 N (k+1)−1 

X (i)

(A.11)

i=2 N k

= 2−2 j E[Y 2 ]E[A2 ] j − 2−2 j E 2 [Y ] 2j

A.2. Proof of Lemma 2

U N ,k = 2

 2 j 1 2

637

where k = 0, 1, . . . , Nt − 1. Then, the mean of the scaling coefficients at the coarsest scale j = N of the multifractal process X (k) is given by (A.4)

E{U N ,k } = 2−N /2 (e+

2 /2

)

(A.12)

638

F.H.T. Vieira, L.L. Lee / Int. J. Electron. Commun. (AEÜ) 64 (2010) 629 – 639

Similarly, the variance of the scaling coefficients at the coarsest scale j = N of the process X (k) can be expressed as  Nt −1 2

U N ,k

i=0,1, ...

=

2

U =e N ,k

Nt 

2+22

X (i)2

W j,k = 2− j/2 (X (2 j k) − X (2 j k + 1))

+ 2Z N − (E{U N ,k }) (A.13)  +1 2 2−3 j + 2Z N − (2−N e2+ )  + 1/2 (A.14)

A.3. Proof of Proposition 1 Firstly, let us turn to a more precise mathematical specification of multiplicative cascades. Consider repeated splittings of T, a unit cube R d for some d ⱖ 1, into a number bn (b ⱖ 2), sub-cubes (pixels) of volume b−n , n = 1, 2, . . ., such that the n th-stage mass of the pixel n (t1 , t2 , . . . , tn ), tk ∈ {0, 1, . . . , b − 1} is given by the random measure ! n (n (t1 , t2 , . . . , tn )) = b−n nk=1 At , ...,t , where Av ’s are 1 k the cascade generators, an i.i.d family of non-negative, unit mean random variables indexed by the pixel addresses v at different fine scales b−n for n ⱖ 1. The cascade measure ∞ is obtained as the limit of the sequence n as n −→ ∞. The structure function or the modified cumulant generating function for log Av is defined as [38] (A.15)

which parameterizes the distribution of Av ’s. It has been shown that the function b (h) can be estimated by [38] nh =

 1 h logb ∞ (n ) n

(A.16)

n

That is, nh converges to b (h) as n → ∞ for all h > 0 and h (T ) > 0. cascades that E∞ Alternatively, the structure function of a cascade process ˆ with j stages, denoted as (q, j), can also be obtained by the following equations [21,37]: j

Z˘ (q, j) =

2 

|W j,k |q

(A.17)

k=1

logb ( Z˘ (q, j)) ˆ (q, j) = j

(A.18)

where W j,k represents the wavelet coefficients of the traffic process. Let the process g be the exponential version of a multifractal cascade multiplied by s. Then, we can express this new process as g∞ (n (t1 , t2 , . . . , tn )) = es ∞ (I (n (t1 ,t2 , ...,tn )))

(A.20)

2

Therefore, inserting Z N (A.8) into (A.14), we can calculate the values of the variance of the wavelet coefficients.

b (h) = logb E[Avh 1[A > 0]] − (h − 1)

In case of Haar wavelets, the wavelet coefficients W j,k of a stochastic process X are explicitly given by

(A.19)

For multifractal processes, expressions (9) and (12) hold. Then, the wavelet coefficients j,k for the process g given by (A.19), can be written as j,k = 2− j/2 [esv 1 − esv 2 ]

(A.21)

where v1 and v2 are given by (22) and (23), respectively. According to the large deviation theory, the effective bandwidth can be expressed as [8]  p  1 1 s X [,+k] log + log (A.22) e (s, t) = st p k=1

where  ⱖ 0, p = 2 N is the number of samples of X and N isthe number of cascade stages. Notice that the term p log k=1 es X [,+k] of (A.22), can be estimated using (A.16) applied to the process g. Once having the structure function (A.16) estimated via Zˆ (q, j), the second part of the right side ˆ of (s, t) can be estimated by 2 N (1, j) and consequently ⎞ ⎛ j 2  ˆ |W j,k |⎠ (A.23) (1, j) = log ⎝ k=1

Finally, inserting (A.21) into (A.23), we can compute the effective bandwidth (s, t) using the multipliers A j,k ’s by the following equation:  2 N | | 1 j,k log k=1N (A.24) (s, t) = st 2 where t is dyadic (t = 2k ) and k = 1, . . . , 2 N .

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