Note on Effective Bandwidth of ATM Traffic1

Note on Effective Bandwidth of ATM Traffic 1 Costas Courcoubetis 2 and Jean Walrand 3 2 3 University of Crete, Heraklion University of California at Berkeley ABSTRACT The objective of this note is to explore the existence of an effective bandwidth for ATM (asynchronous transfer mode) traffic. We show that such an effective bandwidth exists for a class of stationary Gaussian sources. We also show that the effective bandwidth cannot be defined for general non-stationary sources by providing an elementary counter-example. 1. Introduction A number of recent papers have shown the existence of an effective bandwidth for some traffic models ([4], [5]). What this mean is that the set of acceptable numbers Ni of sources of type i , for i = 1, 2, ..., J , that can go through a common buffer with service rate c is such that J Σ Ni αi ≤ c. i =1 (1) Here, acceptable means that the resulting cell loss probability because of buffer overflow is less than some specified small value. Thus, the result says that one can consider that each source of type i uses a bandwidth αi . This result is somewhat surprising in that one might reasonably expect that the set of acceptable numbers Ni would be characterized by a nonlinear expression. Thus, the question arises of identifying the generality of the concept of effective bandwidth. In terms of small loss probability, the effective bandwidth has been justified for the following two models. In [5], at each time n ≥ 1, a source of type i produces a random batch of Xni cells. The buffer has capacity B and it serves one cell per second. The Xni are independent and their distribution depends only on i . The buffer can then be viewed as a GI/GI/1 queue where the interarrival times are deterministic. It is shown in [5] that the probability that an arriving cell finds a queue length larger than some large B is smaller than a specified value if and only if (1) holds where B logE {exp[ hh γ X i ]}. αi = hh γ B n In [4] it is shown that if the sources are two-state Markov modulated fluid sources and if the buffer serves the fluid at a constant rate c , then an effective bandwidth can also be defined for each source. In this paper, we prove the existence of an effective bandwidth for a class of stationary Gaussian sources. We also show that, in general, sources do not have an effective bandwidth. We exhibit an elementary counter-example in section 2. In section 3 we perform a simple calculation for i.i.d. Gaussian batches. This result, a particular case of that in [5], introduces the method that we will need for the next section. In section 4, we prove the existence of an effective bandwidth for a class of stationary Gaussian hhhhhhhhhhhhhhhhhh 1Research supported in part by a Nato Grant, by a grant from Pacific Bell, and by a matching grant from the state of California. This research was performed while the second author was visiting the University of Crete. -2- sources. Section 5 is an appendix that provides some details about a derivation of section 4. The results are summarized in the conclusions of section 6. 2. A counter-example A source of type 1 produces cells at rate α between time 0 and time 1 and stops producing after time 1. A source of type 2 produces cells at rate β between time 1 and time 2 and does not produce cells at other times. There are N 1 sources of type 1 and N 2 sources of type 2. The total traffic of cells produced by all these sources goes through a buffer that serves the cells at a constant rate of c cells per second. We consider a fluid model of the evolution of the number xt of cells in the buffer at time t . Assuming that the buffer is initially empty, we find that x 1 = (N 1α − c )+ x 2 = (x 1 + N 2β − c )+. Let us assume that the buffer capacity is equal to B cells. We want to find conditions on N 1 and N 2 so that very few cells are lost because of buffer overflow. Specifically, we want to find the set of vectors (N 1, N 2) so that x1 ≤ B (2) x 2 ≤ B. (3) and A straight-forward analysis shows that these inequalities imply that N 1 ≤ (B + c )/α (4) c + B − (N 1α − c )+ N 2 ≤ hhhhhhhhhhhhhhhhh . β (5) and The set of admissible vectors (N 1, N 2) is shown on Figure 1. N 2 (2 c + β)/β ( c + β)/β c /β N c /α (c + β)/α 1 (2 c + β)/α Figure 1. The set of admissible vectors. As the figure shows, the set of admissible vectors is not characterized by a linear boundary. Consequently, no effective bandwidth can be defined for these simple sources. The model can be made more complicated by randomizing the traffic, but the conclusion is rather clear: no effective bandwidth can be defined for general sources. -3- 3. I.i.d. Gaussian batches The sources of the counter-example of section 2 are highly non-stationary. We will see in section 4 that an effective bandwidth can be defined for a class of stationary Gaussian sources. Before doing this, we perform a simple calculation in the case of i.i.d. Gaussian batches. This calculation will introduce the method used in section 4 on a simple example. A source of type i produces Xki cells at time k (k ≥ 1), where the Xki are independent and N (mi , σi2). (The notation N (µ, σ2) designates a Gaussian random variable with mean µ and variance σ2.) Also, the sources are all independent. (The positive part of these random variables can be taken to avoid negative numbers, but since this does not change the analysis, we will not do it.) The cells produced by all the sources go into a buffer that serves c cells at each time k . The buffer capacity is B and we want to estimate the probability that the buffer will overflow. We assume that B is large, so that overflows are very unlikely. That is, we are interested in cases where the overflow corresponds to a large deviation of the behavior of the sources. Specifically, we assume that J Σ Ni mi < c , so that it is unlikely that the queue i =1 length in the buffer will reach a large value B before becoming empty again. To perform the analysis of the large deviations, we argue as follows. (See [1], [2], [3], [6] for related derivations.) An initially empty buffer will overflow before becoming empty again if the sources happen to produce cells at a rate larger than the average rate. Specifically, the overflow will occur if the Ni sources produce cells at a total rate Ni ai for all i in such a way that J Σ Ni ai = α + c , i =1 (6) for some α > 0. In that case, the buffer will overflow at time n := B /α since the quantity of cells stored in the buffer grows at rate α + c − c = α. It turns out that the most likely way for the Ni sources of type i to produce cells at a total rate Ni ai for a long time n is for each of these sources to produce cells at rate ai . Moreover, the probability that this will occur is of the order of (ai − mi )2 exp{− nNi Ii (ai )} where Ii (ai ) = hhhhhhhhh . 2σi2 (7) (See [1] for this result.) Thus, the probability that this occurs simultaneously for all i = 1, ..., J is approximately equal to J exp{−n Σ Ni Ii (ai )}. i =1 Consequently, the probability that the sources of type i produce at rate ai for n = B /α in such a way that J Σ Ni ai = α + c is approximately equal to i =1 J B N I (a )}. exp{− hh i i i α iΣ =1 (8) Thus, the probability that the sources will produce at a total rate α + c until the buffer overflows is approximately equal to the sum over all possible combinations of rates ai such that J Σ Ni ai = α + c i =1 of the expres- sions (7). When B is large, this sum of exponentials is approximately equal to the exponential with the smallest factor of − B , i.e., to B I * (α)}, exp{− hh α (9) where J I* (α) = min Σ Ni Ii (ai ), i =1 where the minimization is over the ai subject to (10) -4- J Σ Ni ai = α + c. i =1 (11) Thus, the probability that the buffer overflows because the sources produce cells at some rate α + c is approximately equal to the sum of expressions (9) where the sum is over different values of α > 0. When B is large, we can argue that this sum is essentially equal to exp{− BK } where 1 I * (α). K = min hh α α (12) Let us first calculate I * (α). To do this, we must solve the problem (10)-(11). One finds, by Lagrange multipliers, that the optimal ai satisfy Ni Ii ′(ai ) = 2λNi , i = 1, ..., J , for some λ. Using (7), this implies that ai − mi hhhhhhh = λ, i = 1, ..., J. σi2 Substituting into (11), we can solve for λ. This specifies the optimal ai . Substituting their values into (10), we find that J (α + c − Σ Ni mi )2 i =1 * I (α) = hhhhhhhhhhhhhhhh . J 2 Σ Ni σi2 i =1 The minimization (12) then yields J c − Σ Ni mi i =1 K = 2 hhhhhhhhhhh . J 2 N σ i i Σ i =1 The set of values of {Ni } such that the overflow probability exp{− BK } ≤ exp{− δ} is then such that KB ≥ δ, i.e., J c − Σ Ni mi i =1 2B hhhhhhhhhhh ≥ δ, J 2 N σ i i Σ i =1 i.e., J δσ 2 i } ≤ c. Σ Ni {mi + hhhh 2B i =1 (13) Comparing this expression with (1), we see that each source of type i can be thought of as having an effective bandwidth αi given by δσi2 αi = mi + hhhh . 2B (14) It can be verified that this formula is equivalent to that given in section 1. 4. Stationary Gaussian sources In this section, we apply the method of the previous section to analyze the effective bandwidth of stationary Gaussian sources. {Xki , The model is as follows. A source of type i produces Xki cells at time k ≥ 1. The sequence k ≥ 1} is stationary and Gaussian. Also, different sources are independent. The values {Xki , k ≥ 1} -5- can be correlated. It will be assumed that 1 var { n X i } → γ as n → ∞, for i = 1, ..., J. hh i Σ k n k =1 (15) This condition holds for sources that have an auto-correlation that dies out fast enough. To analyze this model, we first consider the evolution of the buffer over a fixed time interval {1, 2, ..., n }. We will then examine what happens for different values of n . Let xi = [X i1 , ..., Xni ]T where (.)T denotes the transposition. Thus, xi is a vector of jointly Gaussian random variables and we denote its mean by mi and its covariance matrix by Ki . For simplicity we assume that Ki is invertible. It can be shown that the probability that xi is close to ai , a vector very far from the mean mi , is of the order of exp{− Ii (ai )}, with 1 (a − m )T K −1(a − m ). Ii (ai ) := hh i i i i 2 i (See e.g. [1].) The probability that the Ni sources of type i produce a sequence ai of cells, for all i, is approximately equal to exp{− J Σ Ni Ii (ai )}. i =1 Thus, the probability that the sources produce a total number of cells equal to B + nc (16) is approximately equal to exp{− I * (n )} where I * (n ) := min J Σ Ni Ii (ai ) i =1 (17) where the minimization is over the vectors ai such that J Σ Ni aiT1 = B + cn i =1 (18) where 1 is the column vector with n ones. To solve this minimization problem, we use Lagrange multipliers and we find that ∇Ii (ai ) = λ1. Using the expression for Ii (.), this gives Ki −1(ai − mi ) = λ1, i.e., ai = mi + λKi 1. (19) Substituting in (18) we get J J Σ Ni miT1 + λiΣ=1Ni 1T Ki 1 = B + cn , i =1 (20) which gives J B + cn − Σ Ni miT1 i =1 h λ = hhhhhhhhhhhhhhh . J N 1 Σ i T Ki 1 (21) i =1 We now substitute the expressions (19) and (21) in the expression to be minimized in (17) to conclude that I * (n ) (B + cn − ΣNi miT1)2 i hhhhhhhhhhhhhhhhhh = . J TK 1 N 1 Σ i i i =1 Now, by stationarity, (22) -6miT1 = βi n. (23) 1T Ki 1 ∼ ∼ n γi . (24) Also, from (15), We show in the appendix that (15) enables us to replace the approximate sign in (24) by an equality. Using (23)-(24) into (22), we find (B + cn − nΣNi βi )2 i I * (n ) = hhhhhhhhhhhhhhhhhh . J n Σ Ni γi i =1 We now minimize this expression over n and find J c − Σ Ni βi i =1 K := min I * (n ) = 4B hhhhhhhhhh . J n N γ i i Σ (25) i =1 Consequently, the set {Ni , i = 1, ..., J } such that the probability of overflow is less than exp{− δ} is such that exp{− K } ≤ exp{− δ}, i.e. , k ≥ δ, which gives J Σ Ni αi ≤ c i =1 δγi where αi := βi + h hh . 4B (26) This expression establishes the existence of an effective bandwidth αi for a source of type i . 5. Appendix It remains to justify replacing the approximate sign in (24) by an equality. More precisely, we need to justify (25). Let + nD )2 hhhhhhhh f (n , B ) := h(B γn (27) where D := c − J Σ Ni βi i =1 (28) and γn := J n Σ Ni E {[kΣ=1(Xki − βi )]2}. i =1 (29) From (15), n hγhh → γ as n → ∞. n (30) 1 min f (n , B ) = h4D hh , lim hh B n γ (31) We will show that B →∞ which will justify (25). Define εn by γn = n (γ + εn ), n ≥ 1. -7Let us fix ε > 0 arbitrarily small. We will show that for B large enough, 1 minf (n , B ) − h4D 4D − hhhhh 4D . hh | ≤ hhhhh | hh B n γ γ−ε γ+ε (32) This will prove (31). To show (32), let us fix n 0 large enough so that | εn | ≤ ε for n ≥ n 0. (33) (B + D )2 for B ≥ B n 0 ≤ hhhhhhhhhhh 0 4BD (γ + α)2 (34) 1γ . γ + α = max hh n n n (35) + nD )2 ≤ f (n , B ) ≤ h(B + nD )2 , for n ≥ n . h(B hhhhhhhh hhhhhhhh 0 n (γ + ε) n (γ − ε) (36) Let us then choose B 0 large enough so that where It follows from (27) and (33) that Therefore, by taking the minimum over n , + nD )2 h4BD hhhh ≤ min f (n , B ) ≤ min h(B hhhhhhhh γ + ε n ≥n n ≥ n n (γ − ε) (37) + D )2 , for n ≤ n . hhhhhhh f (n , B ) ≥ h(B 0 n 0(γ + α) (38) + D )2 ≥ hhhhh 4BD , for B ≥ B , h(B hhhhhhh 0 n 0(γ + α) γ+α (39) 0 0 for all B . Now, trivially, But by (34). Therefore, (38) and (39) imply 4BD , for B ≥ B , min f (n , B ) ≥ hhhhh 0 γ+α (40) + nD )2 ≥ h(B + D )2 ≥ h4BD h(B hhhhhhhh hhhhhhh hhhh , n (γ − ε) n 0(γ − ε) γ−ε (41) + nD )2 = min h(B + nD )2 = h4BD hhhhhhhh hhhhhhhh hhhh , for B ≥ B 0. min h(B n (γ − ε) n n (γ − ε) γ−ε (42) n ≤ n0 Similarly, for n ≤ n 0 and B ≥ B 0, which implies that n ≥ n0 The inequalities (37), (40), and (42) show that, for B ≥ B 0, the minimum of f (n , B ) is achieved for n ≥ n 0 so that we can conclude that h4BD hhhh ≤ minf (n , B ) ≤ h4BD hhhh , γ+ε n γ−ε (43) which proves (32) and terminates our derivation. 6. Conclusions We have shown that an effective bandwidth cannot be defined for general non-stationary sources. We also showed that such an effective bandwidth can be defined for stationary Gaussian sources (modulo a weak technical assumption). It remains to determine whether an effective bandwidth can be defined for general stationary sources. -8- 7. References [1] J.A. Bucklew. Large Deviation Techniques in Decision, Simulation, and Estimation. John Wiley, New York, 1990. [2] M. Cottrell, J-C. Fort, and G. Malgouyres, "Large deviations and rare events in the study of stochastic algorithms," IEEE Trans. on Aut. Control, Vol. AC-28, no. 9, pp. 907-920, 1983. [3] C. Courcoubetis, G. Kesidis, A. Ridder, J. Walrand, and R. Weber, "Admission control and routing of ATM networks based on measured buffer occupancy," EECS, Univ. California, preprint. [4] R.G. Gibbens and P.J. Hunt, "Effective bandwidths for the multi-type UAS channel,"" Statistical Laboratory, University of Cambridge, preprint. [5] F.P. Kelly, "Effective bandwidth at multi-class queues," Statistical Laboratory, University of Cambridge, preprint. [6] S. Parekh and J. Walrand, "A quick simulation method for excessive backlogs in networks of queues," IEEE Trans. on Aut. Control, Vol. AC-34, no. 1, pp. 54-66, 1989. View publication stats