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Analysis of the Leaky Bucket Output Processes for Bursty Sources in Atm Networks

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Analysis of the Leaky Bucket Output Processes for Bursty Sources in Atm Networks __________________________________________________________________________www.paper.edu.cn ANALYSIS OF THE LEAKY BUCKET OUTPUT PROCESSES FOR BURSTY SOURCES IN ATM NETWORKS Liao Jianxin (National Laboratory of Switching Technology and Telecommunication Networks BUPT of China, Beijing 100088.) Li Lemin Sun Hairong (National Key Lab. of optical Fiber Commun. UEST of China, Chengdu 610054) Abstract In ATM networks, bursty sources can be described as the Interrupted Bernoulli Pro-cess(IBP) . With the use of the thin process theory , the Probability Generating Function(PGF) of the IBP is obtained. An iterative algorithm, which can be used to calculate the IBP probability distribution, is presented. The bursty source’s equivalent description is discussed. It is proposed that the leaky bucket output process can be approximately described as the IBP. The accuracy of the analytical results has been largely validated by means of the simulation approach. Moreover, how to improve its accuracy is discussed. The smoothing function of the leaky bucket algorithm is quantitatively analyzed. Key words ATM, Bursty source , IBP, Leaky bucket algorithm , Output process, Traffic smoothing 1. Introduction Traffic control is a key technique in an ATM network. Recommendation I.371 specified by ITU-T[1] : When a new connection request is received at the network, the call admission procedure is executed to decide whether to accept or reject the call. A call is accepted if the network has enough resources to provide the quality-of-service requirements of the connection request without affecting the quality of service provided to existing connections. The procedure referred to as Call Admission Control(CAC) . The traffic is high burstiness in an ATM network. Call admission is not sufficient to prevent congestion mainly because users may not stay within the connection parameters negotiated at the call set-up phase. Accordingly,the network must ensure that sources stay within their connection parameters negotiated at the connection set-up phase. The function referred to as traffic policing or usage parameter control (UPC). The large amount of study show that the leaky bucket algorithm is a kind of effective UPC algorithm. After the leaky bucket scheme was introduced by Akhtar S. , a number of its variants has been proposed[2]. The leaky bucket algorithm has been studied in [3-6]. The smoothing effect is characterized by the squared coefficient of variation of the interdeparture time from the leaky bucket. Very little work has been done to obtain the smoothing effect of the leaky bucket. The analysis of the output process of the leaky bucket was performed under the hypothesis of Poisson model [7]. But the ATM bursty traffic cannot always be described in term of Poisson model. The average rate and the variance of the output process of the leaky bucket were analyzed by the fluid-flow ways in [8]. The paper is organized as follows. In Section 2, the Probability Generating Function(PGF) of the IBP is obtained with the use of the thin process . An iterative algorithm, which can be used to calculate the IBP probability distribution, is presented. The bursty source’s equivalent description is discussed. In Section 3, it is proposed that the leaky bucket output process can be approximately described as the IBP. In Section 4. The accuracy of the analytical results has been largely validated by means of the simulation approach. Moreover, how to improve its accuracy is discussed. The smoothing function of the leaky bucket algorithm is quantitatively analyzed.Section 5 provides the conclusions. 2. Bursty Source Model The bursty source was described as the IBP. The IBP is defined over a slotted (discrete-time) time axis and it comprises two states, an “ON” state and an “OFF” state, which alternate. The time the process spends in each state is geometrically distributed. The mean values are Ton and Toff, respectively.Cells occur in a Bernoulli fashion when the process is in the “ON” state. No cell occurs if the process is in the “OFF” state. A slot contains a cell with probability fs if the process is in the 1 中国科技论文在线_________________________________________________________________________www.paper.edu.cn “ON” state. Given that the process is in the “ON” states (or “OFF”state) at slot i , it will change to “OFF” state (or “ON” state) in the next slot i+1 with probability α =1/TOFF (orβ =1/ TON ). The bursty source traffic can be effectively described as the three parameters IBP model[9]. IBP is a special thin process which can be expressed by the three parameters, α , β and fs [10]. The reverse thin process N occur in the active state fashion . Az1 () represents the PGF of the reverse thin process N, we have 2 A1 ()z =(1－ β ) z + αβ z / [1－(1－α ) z] (1) AIBP (z) represents the PGF of the IBP ,we have 2 2 AIBP (z) = fs A1 (z) / [1－(1－fs ) A1 (z)] = (a2z + a1z) / (b2 z + b1 z +1) (2) Where a21=−f s (),()11 −α −β a = f s − β b21= ()(),[11− f s −−α β b = − 1− α + ()() 11− f s − β ] AIBP (z) has been defined as z～transform for the interarrival times of the IBP～stream cells. The probability distribution of the interarrive time can be obtained inverting AIBP (z). Let Pi be the probability that the interarrive times is equal to i service slots. We can then solve for Pi , i =0,1…, recursively as follows: min{2 ,i } － － … Pi =ai ∑bi Pi j , i = 0,1,2 (3) j=1 Where ai = 0 , i >2. From equation (2), the average arrival rate, fa and the squared coefficient of variation of the 2 interarrival time, C IBP are as follows: fa = fs α /(α + β ), (4) 2 2 CIBP = 1＋fs [ β (2—α − β ) / ()αβ+−1] (5) 2 C IBP can be seen as a measure of the burstiness. A complete description of a bursty source is given by the three parameters: the peak rate fs , the average rate fa and the average bursty length BL β = fBLs /; (6) α =−ffsa/[ BLf ( s f a )] (7) 2 C IBP can be obtained by (5),(6) and (7) as 2 2 CIBP = 1+ fa —2 fs ＋2BL (1—fa / f s ) (8) Obviously, by decreasing the bursty length, we can decrease the burstiness . In addition, by decreasing the peak rate, keeping the average load and the average bursty length constant, we can decrease the burstiness. 3. Output Process Analysis The basic idea of the leaky bucket algorithm is that a cell, before entering the network, must obtain a token from consume one token and immediately depart from the leaky bucket if there is at least one token available in the token pool. Tokens are generated at a constant rate R and placed in a token pool. Although variations exist, there are generally two parameters of a leaky bucket: the token pool size B, and the token generation rate R. A token size equals to a cell size. An additional parameter is buffer M, the cell is buffered and only allowed to enter the network after the tokens are obtained. Here the leaky bucket performs a shaping function. We model the leaky bucket by making use the fictitious queue illustrated in Fig1. The peak rate, the average rate and the average burst length of the burst sources are represented by Fs , Fa and BLi, respectively. The fictitious queue length Qf is introduced to easy analysis. In fact, the fictitious queue 2 中国科技论文在线_________________________________________________________________________www.paper.edu.cn length Qf is not exist. Fig.1 Fictitious queue leaky bucket model Let us define the following joint probability distribution: F0 (q) = Pr {off state: Qf ≤ q } ( 9 ) F1 (q) = Pr {on state: Qf ≤ q } (10 ) The fictitious queue length distribution is given by Pf (q) = Pr { Qf ≤ q } (11) Obviously, Pf (q) =F0 (q) + F1 (q) (12) Making use of the fluid-flow method , F0 (q) and F1 (q) are given by [9] ε0q ε1q F0 (q) = c0 (1－γ )e + c1 (ρ − γ )e (13) ε0q ε1q F1 (q) = c0 γ e + c1 γ e (14) Where, γ =Fa / Fs , ρ =Fa / R , ε 0 = 0, ε 1 = － ρ (1－ ρ ) / [BL(1－γ ) ( ρ − γ )], (B+M) -1 c0 = {1－( ρ − γ ) e /(1－γ ) } , c1 =－c0 . Note that for R.>Fs , the leaky bucket fictitious queue remains empty at all times, and hence no cell are rejected, marked or buffered. Therefore, we are only interested in the case when R<Fs and hence ρ > γ . The probabilities of the output rate at Fs , R , and 0 are F1 (B) , 1－Pf (B) , and F0 (B) , respectively. The average output rate is ’ Fa = FsF1 (B) + R [1－P f (B)] (15) ’ Obviously, the peak output rate FS still is the peak rate FS of bursty source. However, this duration is the average sojourn time of the output at the peak rate FS , given by [12] ’ Tpeak = BL·(1－γ )·F1 (B) / [Fa·γ ·F0 (B)] (16) The output cell stream from the leaky bucket fed by the bursty source is a very complicated stochastic process which cannot described by a exact mathematical model. Based on the analysis of output cell stream characteristic, we propose that the stochastic process can be approximately described as the IBP. The accuracy of the analytical results has been largely validated by the simulation approach. The IBP is a slotted random point process . One slot s equals to a cell transmission time. Let s = cell / F , where the cell is 53 byte and F is the output link rate. At most ’ ‘ one cell can occur in a slot, let F> Fs . The output IBP can be described by the average traffic f a (=F a / ’ F) , the peak traffic f s (=FS / F) , the average bursty length BL’(= T’peak / s ) .
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