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Performance bounds for rate envelope multiplexing
Performance Evaluation 48 (2002) 87–101
Performance bounds for rate envelope multiplexing Z. Heszberger∗ , J. Zátonyi, J.J. B´ıró High Speed Networks Laboratory, Department of Telecommunications and Telematics, Budapest University of Technology and Economics, Magyar Tudósok Körútja 2, H-1117 Budapest, Hungary
Abstract In QoS guaranteed communication networks, such as ATM, several classes of traffic streams with widely varying characteristics share common transmission resources. To achieve high utilization of these networks, while providing appropriate grade of service for all connections, the development of powerful traffic management algorithms is a central issue. Due to scalability reasons traffic control functions like flow, congestion and admission control often need simple and efficient characterization of traffic using mainly aggregate characteristics instead of using information about all the individual flows. In this paper, the saturation probability as a possible performance measure of aggregate traffic on a communication link is discussed. This performance metric, also referred to as the tail distribution of aggregate traffic, is essential in traffic control and management algorithms of high-speed networks including also the prospective QoS Internet. In this paper, using the Chernoff bounding method, efficient closed-form bounds have been derived for the saturation probability for the case when little information is available on the aggregate traffic. The performance of these estimates is also shown by means of numerical examples. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Quality of Service; ATM; QoS IP; Bufferless statistical multiplexing
1. Introduction In broadband QoS (Quality of Service) networks, such as ATM (asynchronous transfer mode), widely varying type of traffic sources should be handled which poses new challenges in developing appropriate bandwidth allocation and traffic management techniques. In the development of powerful management algorithms simple and efficient characterization of network traffic plays central role. A basic problem in the performance analysis of the networks with statistical multiplexing feature is to efficiently estimate or bound the overflow probability of aggregate traffic on a single link with small or large buffer. The performance of statistical multiplexing is greatly depends on the multiplexing strategy. One of the most understood strategies is the rate envelope multiplexing (REM) [9] often referred to as bufferless statistical multiplexing [6,7]. This approach can provide low delay and delay jitter while the analysis of loss performance remains tractable. In case of buffered multiplexing the delay and loss performance ∗
Corresponding author. Tel.: +36-1-463-2764; fax: +36-1-463-3107. E-mail addresses: [email protected] (Z. Heszberger), [email protected] (J. Z´atonyi), [email protected] (J.J. B´ır´o). 0166-5316/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 5 3 1 6 ( 0 2 ) 0 0 0 3 2 - 9
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may depend on complex flow characteristics prohibiting the design of efficient admission control [11]. In the case of REM, the task is to ensure that the combined instantaneous arrival rate of multiplexed traffic sources (under the continuous fluid assumption) does not exceed the link capacity C. For this the saturation probability of aggregate traffic arrival rate on a single link is an important performance parameter. This probability can be used, e.g. for calculating the probability of time-congestion of resources, i.e., the fraction of time when the traffic offered to a transmission link exceeds the capacity of that link. Moreover, it is also useful for the estimation of the traffic congestion (traffic blocking) probability. In recently developed measurement-based admission control algorithms estimates for effective bandwidth [10] are often used also closely related to the overflow probability of aggregate traffic [1,2,5,6,15,16]. In networking context, the theory of large deviations as a general tool for queuing analysis has been thoroughly discussed in several research papers [12,13]. Firm results of applying of this theory are the large buffer and many sources asymptotics which can be used for approximating the buffer overflow probability when the buffer size or the number of sources get large enough [3,4]. As opposed to this, in this paper we investigate the effect of cell scale fluctuation in case of small buffer. An appropriate modeling framework for this is the bufferless statistical multiplexing. In this scheme, a widely accepted and used method for overflow probability estimation of aggregate traffic is the Chernoff bounding method for upper tail distribution of sums of random variables. Although the use of the central limit theorem (CLT) might also be applied to similar problems [7] providing us a general estimation, for service level guarantees conservative upper bounds are often desired. In this paper, after introducing some basic inequalities we derive a new upper bound for the overflow probability of aggregate traffic which requires only the mean arrival rate of aggregate traffic, the individual peak rates and the number of traffic sources. Although the optimal solution of the new formula obtained cannot be drawn in closed form, we present some simplification to resolve this difficulty, resulting, however, in weaker upper bounds. Even these bounds by means of extensive numerical investigations seem to be better then the well-known Hoeffding inequality which uses the same information on the traffic sources. The rest of the paper is organized as follows. In the next section first we shed light on the idea behind the Chernoff bounding method, then present the use of this method for computing overflow probability of aggregate traffic on a communication link. In subsequent sections we perform the derivation of our new bound and its closed-form approximations. The performance evaluation of these bounds through numerical examples are also shown. 2. Tail estimates via basic inequalities Let us start with the introduction of the Markov inequality. Lemma 1. Let X be a non-negative random variable. Then, for any C > 0, M P (X ≥ C) ≤ , C where M = E(X), the expectation value of X.
(1)
A well-known upper bound for the tail distribution is Chernoff’s bound, i.e., for C > M, E(esX ) , s,s>0 esC
P (X ≥ C) ≤ inf
(2)
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which comes from the Markov inequality if we apply Lemma 1 for the random variable esX with parameter esC . Now it can be clearly seen that Chernoff’s idea was to compute the moment generation function E(esX ) and optimizing the formula above with respect to s. Although this approach has turned out to provide appropriate estimates for several problems, in many cases the exact distribution of X is not known, hence the formula in the equation above should be approximated. 2.1. Two essential theorems providing bounds for the sum of random variables In this section, we present fundamental results on bounding the tail distribution of the sum of bounded and independent random variables using only limited information on them. Theorem 2 introduces Hoeffding’s inequality. The proof of the theorem can be found in [8]. Theorem 2 (Hoeffding [8]). Let X1 , . . . , Xn be independentnon-negative random variables with unit ¯ and peaks, i.e., 0 ≤ Xk ≤ 1 ∀k. Let S = k Xk , X¯ = (1/n) k Xk , M = E( k Xk ), p = E(X) q = 1 − p, i.e., S = nX¯ and C = n(p + t). Then for 0 ≤ t < q, p+t q−t n p q P (X¯ − p ≥ t) ≤ , (3) p+t q −t or equivalently P (S ≥ C) ≤
M C
C
n−M n−C
n−C
.
(4)
To achieve this result it is enough to realize that substituting the original random variables Xk with Bernoulli distributed ones having the same mean values maximizes the left-hand side of (3), i.e., the tail probability. Note that the optimal parameter s ∗ in the transformed formula comes out to be C(n − M) s ∗ = log . (5) M(n − C) It is worth noting that the right-hand side of (3) is the best possible bound that can be obtained when the estimation is originated from Chernoff formula (2), and only the aggregate mean M, and the number of sources n are known about the traffic mix. We present the exact theorem in Appendix A, and a proof is also included. The subsequent inequality is concerned with Xk random variables having different upper bounds (peak values), i.e., 0 ≤ Xk ≤ pk . A slightly reformulated version of the original inequality states the following theorem. Theorem 3 (Hoeffding [8]). (C − M)2 P (S ≥ C) ≤ exp −2 2 . k pk
(6)
Remarks. This latter bound cannot be reduced to the first Hoeffding inequality when pk = 1 for all k. It is due to the different approximation technique used in the derivations. It can also be shown that for this case the first Hoeffding bound (3) is never worse then the second one (6).
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At this state, the natural question arises that how to develop an upper bound for the case of different pk s which can be the real counterpart of the first Hoeffding inequality in the sense that it would give back this bound for the case when pk = 1, for all k. In the next section we derive this bound for random variables with different peaks, and closed-form approximations are also performed. 3. Improved resource utilization formulae 3.1. Modeling principles Let us suppose that a multiservice communication network serves N users belonging to J different service classes defined as each of them having n1 , n2 , . . . , nJ users, so that N=
J
nj .
(7)
j =1
The only traffic descriptor of a service class is the maximum instantaneous data arrival rate (peak rate). Each source within a service class has identical peak rates. We model the generation of information of traffic sources in the service classes by stationary stochastic processes X11 (t), . . . , X1n1 (t), . . . , XJ 1 (t), . . . , XJnJ (t), where Xji (t) denotes the instantaneous arrival rate of the ith traffic source of type j at time t. We also assume that the arrival rate Xji (t) varies between 0 and pj the peak rate of the traffic sources in class j , i.e., 0 ≤ Xji (t) ≤ pj ∀t, j . The sources are nj considered as independent random variables for all t. Let S(t) = Jj=1 i=1 Xji (t) and M(t) = E(S(t)) the expected value of S(t). Due to stationarity we can omit the time dependency, so the tail distribution of the aggregate traffic can simply be denoted as P (S ≥ C). 3.2. Improved bound for link saturation probability The following contribution introduces a new formula that can be used for bounding the tail probability of the link saturation with (bufferless) statistically multiplexed traffic. Theorem 4. Let Xji (j = 1, . . . , J ; i = 1, . . . , nJ ) are independent (not necessarily identically distributed) random variables where 0 ≤ Xji ≤ pj ; then N J ∗ J s ∗ pj es pj − 1 nj M + n (p /(e − 1)) j j ∗ j =1 −s C P (S ≥ C) ≤ e , (8) pj N j =1 where s ∗ is the solution of the following equation: J N Jj=1 nj pj2 (espj /(espj − 1)2 ) nj pj − C = 0. − 1 − e−spj M + Jj=1 nj (pj /(espj − 1)) j =1
(9)
We sketch the proof of this theorem in Appendix B. The optimal parameter s ∗ can be numerically computed in a straightforward manner using any standard root-finding algorithm. Note, that the equation
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above contains only the aggregate mean M, meaning that there is no need to know the individual mean values of the random variables (i.e., the mean arrival rates of the traffic flows). The upper bound presented in Theorem 4 can be considered as the real counterpart of the first Hoeffding inequality (3), because in case of identical peak rates we get back this inequality. Also, in this case the optimal s ∗ from Eq. (9) becomes that in (5). Regarding the optimal property of the upper bound (8) the following statement can be formed. Theorem 5. Bound (8) is the best possible one under the following conditions: (1) the first step of the estimation process is the Chernoff formula, (2) all the information known about the traffic are J the number of service classes, the vector n of nj s, the aggregate mean rate M and the vector p of the individual peak rates of the service classes, (3) and the following inequality holds: nj M − (Npi /(espi − 1)) + Jj=1 k=1 (pk /(espk − 1)) 0< < pi ∀i. (10) N Extensive numerical investigation shows that (10) is not a strict condition in case of real applications (e.g. it is worth thinking over how it behaves when we have only one service class). For further discussion of the theorem see Appendix C. 3.3. Closed-form approximations Although the numerical evaluation of the upper bound in (8) can be straightforward, a closed-form expression can be often more useful and expressive, even if it is not optimal with respect to the transformation parameter s. The problem with the result in (8) is that although the optimal parameter s ∗ can be numerically computed, the optimization may be a time consuming process and can hardly be accomplished in real time. In the following, we present two ways of how to find suboptimal but closed-form formulae for s which can be substituted into (8) obtaining closed-form expressions for the improved bound. Let us take the first three terms in the Taylor series expansion of the fraction pj /(espj − 1) with respect to s at s = 0. Now putting them into the logarithm in (B.8), then taking the first two terms of the Taylor series expansion of the received formula in the same one can obtain 2 1 1 1 sM + s 2 − sC, P2 − M− P (11) 8 2N 2 nj where M = Jj=1 i=1 mji , P = Jj=1 nj pj , P2 = Jj=1 nj pj2 . The expression above should be minimized to obtain a suboptimal value for s. Thus we get 1 s˜1∗ = 1
1 P 4 2
C−M . − (1/N)(M − 21 P )2
(12)
This derivation is similar to that in [14] for closed-form equivalent capacity formula. However, the resulting approximate optimizer and the closed-form formula in [14] and ours provided above are completely different (indeed they are not even comparable as such) because of the different objective of the optimization tasks.
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Substituting s˜1∗ into (8), we finally obtain P (S ≥ C) ≤ T (˜s1∗ , C, M, n, p) N n J ((C−M)/K)pj M + Jj=1 (nj pj /(e((C−M)/K)pj − 1)) e −1 j ◦ −(C−M)C/K = e , N pj j =1 (13) 1 P 4 2
1 )P )2 . 2
− (1/N)(M − where K = Our second (partly heuristic) suggestion for a suboptimal s has a similar form to the optimal parameter (5) performed at the first Hoeffding inequality. Substituting each peak rate with the arithmetic mean of the peak rate of all the sources, that is p = pj = (1/N) Jj=1 nj pj ∀j , we formulate a possible suboptimal parameter s as 1 C(Np − M) s ∗ = log . (14) p M(Np − C) Taking into account that (2) is true for all s we may apply (14) for (8) achieving a new closed-form bound, the performance of which, however, should be investigated carefully. Apparently, we may use other types of means as well, e.g. when J 1
p= nj pj2 ∀j, N j =1 we get s˜2∗ =
C(P − M) N log , P2 M(P − C)
or in general ∗ s˜2k = k J
N
k j =1 nj pj
log
C(P − M) . M(P − C)
(15)
(16)
Through extensive numerical investigation the use of s˜2∗ seems to show the best properties. This choice still satisfies the requirement that s˜2∗ should remain positive since neither M nor C can exceed the sum of the peak rates. Another important property of s˜2∗ is that as opposed to s˜1∗ it becomes the original optimal value of s if all the peak rates are equal to each other. From this we can expect that if the deviation of the peak rates is small this latter solution gives better results than (13). In the next section we compare the new bounds derived from (8), (13) and (15) to each other and to the second Hoeffding bound (6), numerically. 4. Comparison via numerical examples From our large number of numerical examples hereafter we show four different cases to demonstrate the performance of the different estimation formulae for the saturation probability. Table 1 shows the
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Table 1 The parameter sets of different traffic situations J
n
p
P
M
DP
M/P
2 8 12 12
(10,1) (10,8,8,5,5,3,2,2) (3,5,3,5,3,2,5,2,5,2,5,2) (10,8,8,6,3,5,5,5,3,2,5,2)
(1,10) (1,3,5,7,10,13,15,17) (1,2,3,4,5,10,11,12,13,14,15,18) (3,5,8,8,10,12,14,15,17,19,25,28)
20 262 360 687
6 100 70 220
0.13 0.018 0.015 0.01
0.3 0.38 0.194 0.32
most important parameters of the different traffic situations, namely the number of service classes J , the vector of the number of the sources in the service classes n, the vector of the peak rates p, the sum of the peak rates P , the aggregate mean arrival rate M, and a normalized deviation-like parameter DP for comparison purposes defined as 2 J 1 1 1
DP = nj pj − P . P N j =1 N
Fig. 1. The logarithm of the saturation probability for four traffic situations.
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The mean to peak ratio M/P is also noted which may give an outline about the total traffic activity in the different cases. In Fig. 1, the logarithm of different estimation of the saturation probability is plotted for the proposed traffic examples. It can be seen that the exact (simulated) tail probability (drawn with bullets) considerably differs from the others, which is reasonable if we take into account that the used information about the traffic is rather few (J , M, n and p). It can be recognized that the first closed-form solution usually very close to the line of the exact formula. This also confirms the statement that the bound (13) tightly approximates the exact formula in (8). The Hoeffding bound (6) is considerably worse than the others, i.e., for a given saturation probability requirement transmission capacity can be saved by applying the new bounds. For further comparison the negative exponent of the bounds derived from (8), (13) and (15) are considered. This means that the progress of the difference functions Diff o,s1 (C) = −log T (s ∗ , C, M, n, p) + log T (˜s1∗ , C, M, n, p),
(17)
Diff o,s2 (C) = −log T (s ∗ , C, M, n, p) + log T (˜s2∗ , C, M, n, p),
(18)
Diff s1 ,s2 (C) = −log T (˜s1∗ , C, M, n, p) + log T (˜s2∗ , C, M, n, p),
(19)
2(C − M)2 Diff s1 ,ho2 (C) = −log T (˜s1∗ , C, M, n, p) − J 2 j =1 nj pj
(20)
and
Fig. 2. The difference function Diff o,s1 (C). (a) M = 6, P = 20, (b) M = 100, P = 262, (c) M = 70, P = 360, (d) M = 220, P = 687.
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Fig. 3. The difference function Diff o,s2 (C). (a) M = 6, P = 20, (b) M = 100, P = 262, (c) M = 70, P = 360, (d) M = 220, P = 687.
are to be analyzed, where log T (s, C, M, n, p) = N log
M+
J
j =1 (nj pj /(e
N
spj
− 1))
+
J j =1
nj log
espj − 1 − sC. pj
(21)
Figs. 2–4 show the functions Diff o,s1 , Diff o,s2 and Diff s1 ,s2 with respect to the capacity C. To better understand the meaning of the curves let us take for example the third plot of the first comparison in Fig. 2. At C = 200, it shows that the difference of the negative exponents of the probability counted with the optimal method and that of counted with (13) is about 0.3, which means that (13) gives e0.3 (≈ 1.35) times larger (i.e., worse) upper bound the optimal technique does. The difference between the optimal and the other two methods in Figs. 2 and 3 gives us a general overview about the accuracy of using the suboptimal s˜1∗ and s˜2∗ instead of s ∗ . Notice that in Fig. 3 in most of the cases the difference Diff o,s2 remains under 0.5, which turns out to be a negligible error, considering that these bounds have around the middle of the interval [M, P ] a value of 0.01 with a slope of about −10 as capacity increases, resulting in a waste of less than 0.1% of the total capacity. Investigating the relation between the two suboptimal methods we can first draw the conclusion that the one using s˜2∗ gets worse as DP increases (as we would expect), while the other one shows little dependence on it. However, considering the above-mentioned facts, this often small deviation of about 0.5 has little significance. On the other hand, changing the value M/P has considerable impact on the performance of the estimates. While the formula using s˜2∗ becomes more precise as M/P decreases, the one with s˜1∗ shows strong degradation. Note that in the above examples this tendency is mostly overshadowed due to the effect of the change of deviation DP .
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Fig. 4. The difference function Diff s1 ,s2 (C). (a) M = 6, P = 20, (b) M = 100, P = 262, (c) M = 70, P = 360, (d) M = 220, P = 687.
Fig. 5. The difference function Diff s1 ,ho2 (C). (a) M = 6, P = 20, (b) M = 100, P = 262, (c) M = 70, P = 360, (d) M = 220, P = 687.
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Finally, we present comparisons between our first closed-form formula using (13) and the second Hoeffding inequality (6) via the difference function Diff s1 ,ho2 . Fig. 5 illustrates strict increase in the difference with increasing capacity C. It means that our first closed-form upper bound seems to be always better (which is also experienced in several other examples) then the second Hoeffding inequality. To sum up the results of our comparisons it can be stated that among the approximation techniques presented in this paper, the method using (13) seems to be the most appropriate one for applications. 5. Conclusions In this paper, we have developed a new upper bound for tail distribution of aggregate traffic using the well-known Chernoff bounding technique. This bound uses the aggregate mean arrival rate, the number of sources and the individual peak rates. This estimation can be considered as a generalized version of the Hoeffding bound assuming identical and normalized peak rates of traffic sources, because that can be obtained as a special case. Under certain conditions this new upper bound can be the best possible one. Due to the fact, however, that it cannot be expressed in closed-form, we have also presented weaker but closed-form upper bounds for the tail distribution of aggregate traffic. Several numerical examples have also shown which of the closed-form bounds is the most appropriate one for practical use. Appendix A. Notes on the performance of the Hoeffding estimation Theorem A.1. be independent random variables with unit peaks, i.e., 0 ≤ Xk ≤ 1 ∀k, nLet X1 , . . . , Xn and let S = k=1 Xk , M = E[ nk=1 Xk ]. Then the estimation C n − M n−C M (A.1) P (S ≥ C) ≤ C n−C is the smallest possible upper bound under the following conditions: (1) the first step of the estimation process is the Chernoff formula, (2) all the information known about the flows is M and n. Proof. To proof the above theorem we use the indirect approach. Let us consider all the possible situations when we have n number of sources the aggregate mean of which is M. We denote the set of these cases with L. In our theorem we state that the bound in (A.1)—throughout the proof denoted by H —is the tightest one for all the elements of L. According to the indirect way, we assume that there exists a bound I , so that I < H and I is also an upper bound for all the elements of L. In the following we try to find an element which cannot be bounded above by I . Let us consider the special case in L when we have X1 , . . . , Xn independent identically Bernoulli distributed n random variables and choose P [Xk = 1] = M/n = r, since we know that E[Xk ] = r and M = k=1 mk = nr. Hence n
n E[es k=1 Xk ] (E[esXk ])n (1 − r + r es )n P Xk ≥ C ≤ = = . (A.2) esC esC esC k=1
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Now, according to the Chernoff method to get the best bound we should optimize in s, obtaining C (1 − r + r es )n n − M n−C M = . (A.3) inf s,s>0 esC C n−C Notice that the result in (A.3) is exactly H . Now if I were smaller than H then it would not be an upper bound of our special choice in L, so that it could not be an upper bound on all the elements of L, which is a contradiction to our initial statement. Appendix B. Deriving the improvement of the Hoeffding bound Proof of Theorem 4. For the proof of Theorem 4 we start with the Chernoff formula (2): J nj P (S ≥ C) ≤ E es j =1 i=1 Xji e−sC .
(B.1)
Due to independence we obtain nj J E[esXji ] e−sC . P (S ≥ C) ≤
(B.2)
j =1 i=1
The exponential function is strictly convex, thus it can be easily shown that the expectation value of the (transformed) random variables esXji can be bounded such as espj − 1 E(esXji ) ≤ 1 + mji , (B.3) pj where mji is the mean value of Xji . In this way, the tail distribution of the sum of the random variables can be bounded above by nj J espj − 1 −sC 1 + mji P (S ≥ C) ≤ e . (B.4) pj j =1 i=1 The product on the right-hand side can be reformulated as J nj J spj pj e − 1 nj −sC mji + spj . e e −1 pj j =1 j =1 i=1 Here, the last term can be further approximated by N J nj nj J spj − 1)) pj j =1 i=1 (mji + pj /(e mji + spj ≤ e − 1 N j =1 i=1
(B.5)
(B.6)
because of the relation between geometric and arithmetic mean of non-negative real numbers. Now an upper bound of the tail distribution can be expressed as N J J spj espj − 1 nj ◦ M + n (p /(e − 1)) j j j =1 −sC =T (s, C, M, n, p), (B.7) P (S ≥ C)≤e N pj j =1 where n is the vector of nj ’s and p comprises the peak rates.
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What still remains to be done is to determine the optimal s, which minimizes the right-hand side of (B.7), in other words, which gives the lowest upper bound in s. Unfortunately, a closed-form expression for the optimal s can not be derived, however, an essentially non-algebraic equation can be formulated, the solution of which serves as the optimal s. The upper bound in (B.7) is minimized if its logarithm is minimized, i.e., the optimal s should minimize the function J M + Jj=1 (nj pj /(espj − 1)) espj − 1 nj log + N log − sC. (B.8) pj N j =1 Taking the derivative of this with respect to s we get J N Jj=1 nj pj2 (espj /(espj − 1)2 ) nj pj − C = 0. − 1 − e−spj M + Jj=1 nj (pj /(espj − 1)) j =1 The solution of (B.9) gives the optimal s ∗ .
(B.9)
Appendix C. Introducing a condition to guarantee the optimal behavior of the improved Hoeffding-like bound In Theorem A.1, we have presented a result about the optimal property of the Hoeffding bound under certain conditions. On the other hand, with the introduction of an extra condition a similar statement can be formed for our bound (8) as well. To investigate the performance of the improved bound let us recall the proof in Appendix B. In the derivation process, basically, after applying the Chernoff formula as a start, at each step of the proof we subsequently obtain new upper bounds on previous ones till we get to the concluding form, which depends only on known parameters. In the following, we attempt to find a particular traffic situation with given M, n and p so that at each bounding step throughout the proof the inequalities become equalities (apparently except the first one with the Chernoff estimation). If such a case exists it follows that an upper bound on all the traffic situations cannot be smaller than (8) (since there exists at least one special case when the saturation probability is just equal to it). The first bounding step in the proof is the application of the Chernoff formula, which corresponds to the first condition in the Theorem 5. It is easy to see that, the subsequent bounding step (B.3) changes to an equality simply if the traffic flows are On/Off types. The situation is a bit more complicated in the last approximation (B.6). It is known that the geometric and arithmetic mean of non-negative real numbers are equal if and only if the numbers are equal, so in our case mji +
pj pl = mlk + sp , l −1 e −1
espj
(C.1)
where i = 1, . . . , nj ; j = 1, . . . , J ; k = 1, . . . , nl ; l = 1, . . . , J should be satisfied. Summing up such equations for k = 1, . . . , nl ; l = 1, . . . , J , with i, j fixed, we get nj M − (Npi /(espi − 1)) + Jj=1 k=1 (pk /(espk − 1)) mji = ∀i, j. (C.2) N
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Since
J
j =1
nj
k=1
0 < mji < pj
mjk = M. Now, if ∀j, i
(C.3)
is also satisfied then (C.2) provides us the missing information to construct the traffic mix, in which case the saturation probability of the aggregate flow cannot be bounded by a smaller value than formula (8) gives us. On the other hand, if (C.3) does not hold then upper bound (8) is not guaranteed to be the tightest one. References [1] F. Brichet, A. Simonian, Conservative models for measurement-based admission control, in: Proceedings of the ITC 16, 1999, pp. 581–592. [2] F. Brichet, A. Simonian, Measurement-based CAC for video applications using SBR service, in: Proceedings of the PMCCN’97, 1997, pp. 285–304. [3] C. Courcoubetis, R. Weber, Buffer overflow asymptotics for a buffer handling many traffic sources, J. Appl. Probab. 33 (1996) 886–903. [4] N.G. Duffield, Economies of scale for long-range dependent traffic in short buffers, Telecommun. Syst. 7 (1997) 267–280. [5] S. Floyd, Comments on measurement-based admission control algorithms for Internet, Technical Report. [6] R.J. Gibbens, F.P. Kelly, Measurement-based connection admission control, in: Proceedings of the International Teletraffic Congress, June 1996, pp. 879–888. [7] R. Guerin, H. Ahmadi, M. Naghshineh, Equivalent capacity and its applications to bandwidth allocation in high-speed networks, in: IEEE J. Selected Areas Commun. 9 (7) (1991) 968–981. [8] W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Am. Statist. Assoc. 58 (1963) 13–30. [9] ITU-T Recommendation E.736, Methods for cell level traffic controls in B-ISDN, May 1997. [10] F.P. Kelly, Notes on effective bandwidths, in: Stochastic Networks: Theory and Applications, Vol. 4, Oxford University Press, Oxford, 1996, pp. 141–168. [11] J. Roberts, Engineering for quality of service, in: K. Park, W. Willinger (Eds.), Self-similar Network Traffic and Performance Evaluation, Wiley, New York, 2000 (Chapter 16). [12] J.W. Roberts (Ed.), Final Report on COST 224, Performance Evaluation and Design of Multiservice Networks, October 1991. [13] J.W. Roberts, U. Mocci, J. Virtamo (Eds.), Final Report on COST 242, Methods for the Performance Evaluation and Design of Broadband Multiservice Networks, June 1996. [14] Z. Turányi, A. Veres, A. Oláh, A family of measured-based admission control algorithms, in: Proceedings of the IFIP PICS’98, Lund, Sweden, May 1998. [15] M. Villen-Altamirano, M.F. Sanchez-Canabate, Effective bandwidth dependent of the actual traffic mix: an approach for bufferless CAC, in: Proceedings of the ITC 15, 1997, pp. 47–57. [16] M. Villen-Altamirano, M.F. Sanchez-Canabate, A tight admission control technique based on measurements, in: Proceedings of the ITC 16, 1999, pp. 603–612.
Z. Heszberger received his M.Sc. degree in electrical engineering from Budapest University of Technology and Economics (BUTE), Budapest, Hungary in 1997. Since 1997, he has been working in the Department of Telecommunications and Telematics. He is about to finish his Ph.D. dissertation. His research focuses mainly on design and control of QoS guaranteed networks.
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J. Zátonyi received his M.Sc. degree in electrical engineering from Budapest University of Technology and Economics (BUTE), Budapest, Hungary in 2000. Since September 2000, he has been a Ph.D. student in the High-Speed Networks Laboratory at the Department of Telecommunications and Telematics at BUTE. His research interests include measurement-based traffic management, admission control, traffic description techniques, QoS provisioning in IP networks.
J.J. B´ıró received his M.Sc. and Ph.D. degrees in electrical engineering from Budapest University of Technology and Economics (BUTE), Budapest, Hungary in 1993 and 1998, respectively. In 1996, he joined the Department of Telecommunications and Telematics at BUTE, where he is working as a research fellow. His main research interests are in optimization, random processes, queuing theory, in particular in performance analysis and evaluation of telecommunication and computer networks.
Performance bounds for rate envelope multiplexing Z. Heszberger∗ , J. Zátonyi, J.J. B´ıró High Speed Networks Laboratory, Department of Telecommunications and Telematics, Budapest University of Technology and Economics, Magyar Tudósok Körútja 2, H-1117 Budapest, Hungary
Abstract In QoS guaranteed communication networks, such as ATM, several classes of traffic streams with widely varying characteristics share common transmission resources. To achieve high utilization of these networks, while providing appropriate grade of service for all connections, the development of powerful traffic management algorithms is a central issue. Due to scalability reasons traffic control functions like flow, congestion and admission control often need simple and efficient characterization of traffic using mainly aggregate characteristics instead of using information about all the individual flows. In this paper, the saturation probability as a possible performance measure of aggregate traffic on a communication link is discussed. This performance metric, also referred to as the tail distribution of aggregate traffic, is essential in traffic control and management algorithms of high-speed networks including also the prospective QoS Internet. In this paper, using the Chernoff bounding method, efficient closed-form bounds have been derived for the saturation probability for the case when little information is available on the aggregate traffic. The performance of these estimates is also shown by means of numerical examples. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Quality of Service; ATM; QoS IP; Bufferless statistical multiplexing
1. Introduction In broadband QoS (Quality of Service) networks, such as ATM (asynchronous transfer mode), widely varying type of traffic sources should be handled which poses new challenges in developing appropriate bandwidth allocation and traffic management techniques. In the development of powerful management algorithms simple and efficient characterization of network traffic plays central role. A basic problem in the performance analysis of the networks with statistical multiplexing feature is to efficiently estimate or bound the overflow probability of aggregate traffic on a single link with small or large buffer. The performance of statistical multiplexing is greatly depends on the multiplexing strategy. One of the most understood strategies is the rate envelope multiplexing (REM) [9] often referred to as bufferless statistical multiplexing [6,7]. This approach can provide low delay and delay jitter while the analysis of loss performance remains tractable. In case of buffered multiplexing the delay and loss performance ∗
Corresponding author. Tel.: +36-1-463-2764; fax: +36-1-463-3107. E-mail addresses: [email protected] (Z. Heszberger), [email protected] (J. Z´atonyi), [email protected] (J.J. B´ır´o). 0166-5316/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 5 3 1 6 ( 0 2 ) 0 0 0 3 2 - 9
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may depend on complex flow characteristics prohibiting the design of efficient admission control [11]. In the case of REM, the task is to ensure that the combined instantaneous arrival rate of multiplexed traffic sources (under the continuous fluid assumption) does not exceed the link capacity C. For this the saturation probability of aggregate traffic arrival rate on a single link is an important performance parameter. This probability can be used, e.g. for calculating the probability of time-congestion of resources, i.e., the fraction of time when the traffic offered to a transmission link exceeds the capacity of that link. Moreover, it is also useful for the estimation of the traffic congestion (traffic blocking) probability. In recently developed measurement-based admission control algorithms estimates for effective bandwidth [10] are often used also closely related to the overflow probability of aggregate traffic [1,2,5,6,15,16]. In networking context, the theory of large deviations as a general tool for queuing analysis has been thoroughly discussed in several research papers [12,13]. Firm results of applying of this theory are the large buffer and many sources asymptotics which can be used for approximating the buffer overflow probability when the buffer size or the number of sources get large enough [3,4]. As opposed to this, in this paper we investigate the effect of cell scale fluctuation in case of small buffer. An appropriate modeling framework for this is the bufferless statistical multiplexing. In this scheme, a widely accepted and used method for overflow probability estimation of aggregate traffic is the Chernoff bounding method for upper tail distribution of sums of random variables. Although the use of the central limit theorem (CLT) might also be applied to similar problems [7] providing us a general estimation, for service level guarantees conservative upper bounds are often desired. In this paper, after introducing some basic inequalities we derive a new upper bound for the overflow probability of aggregate traffic which requires only the mean arrival rate of aggregate traffic, the individual peak rates and the number of traffic sources. Although the optimal solution of the new formula obtained cannot be drawn in closed form, we present some simplification to resolve this difficulty, resulting, however, in weaker upper bounds. Even these bounds by means of extensive numerical investigations seem to be better then the well-known Hoeffding inequality which uses the same information on the traffic sources. The rest of the paper is organized as follows. In the next section first we shed light on the idea behind the Chernoff bounding method, then present the use of this method for computing overflow probability of aggregate traffic on a communication link. In subsequent sections we perform the derivation of our new bound and its closed-form approximations. The performance evaluation of these bounds through numerical examples are also shown. 2. Tail estimates via basic inequalities Let us start with the introduction of the Markov inequality. Lemma 1. Let X be a non-negative random variable. Then, for any C > 0, M P (X ≥ C) ≤ , C where M = E(X), the expectation value of X.
(1)
A well-known upper bound for the tail distribution is Chernoff’s bound, i.e., for C > M, E(esX ) , s,s>0 esC
P (X ≥ C) ≤ inf
(2)
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which comes from the Markov inequality if we apply Lemma 1 for the random variable esX with parameter esC . Now it can be clearly seen that Chernoff’s idea was to compute the moment generation function E(esX ) and optimizing the formula above with respect to s. Although this approach has turned out to provide appropriate estimates for several problems, in many cases the exact distribution of X is not known, hence the formula in the equation above should be approximated. 2.1. Two essential theorems providing bounds for the sum of random variables In this section, we present fundamental results on bounding the tail distribution of the sum of bounded and independent random variables using only limited information on them. Theorem 2 introduces Hoeffding’s inequality. The proof of the theorem can be found in [8]. Theorem 2 (Hoeffding [8]). Let X1 , . . . , Xn be independentnon-negative random variables with unit ¯ and peaks, i.e., 0 ≤ Xk ≤ 1 ∀k. Let S = k Xk , X¯ = (1/n) k Xk , M = E( k Xk ), p = E(X) q = 1 − p, i.e., S = nX¯ and C = n(p + t). Then for 0 ≤ t < q, p+t q−t n p q P (X¯ − p ≥ t) ≤ , (3) p+t q −t or equivalently P (S ≥ C) ≤
M C
C
n−M n−C
n−C
.
(4)
To achieve this result it is enough to realize that substituting the original random variables Xk with Bernoulli distributed ones having the same mean values maximizes the left-hand side of (3), i.e., the tail probability. Note that the optimal parameter s ∗ in the transformed formula comes out to be C(n − M) s ∗ = log . (5) M(n − C) It is worth noting that the right-hand side of (3) is the best possible bound that can be obtained when the estimation is originated from Chernoff formula (2), and only the aggregate mean M, and the number of sources n are known about the traffic mix. We present the exact theorem in Appendix A, and a proof is also included. The subsequent inequality is concerned with Xk random variables having different upper bounds (peak values), i.e., 0 ≤ Xk ≤ pk . A slightly reformulated version of the original inequality states the following theorem. Theorem 3 (Hoeffding [8]). (C − M)2 P (S ≥ C) ≤ exp −2 2 . k pk
(6)
Remarks. This latter bound cannot be reduced to the first Hoeffding inequality when pk = 1 for all k. It is due to the different approximation technique used in the derivations. It can also be shown that for this case the first Hoeffding bound (3) is never worse then the second one (6).
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At this state, the natural question arises that how to develop an upper bound for the case of different pk s which can be the real counterpart of the first Hoeffding inequality in the sense that it would give back this bound for the case when pk = 1, for all k. In the next section we derive this bound for random variables with different peaks, and closed-form approximations are also performed. 3. Improved resource utilization formulae 3.1. Modeling principles Let us suppose that a multiservice communication network serves N users belonging to J different service classes defined as each of them having n1 , n2 , . . . , nJ users, so that N=
J
nj .
(7)
j =1
The only traffic descriptor of a service class is the maximum instantaneous data arrival rate (peak rate). Each source within a service class has identical peak rates. We model the generation of information of traffic sources in the service classes by stationary stochastic processes X11 (t), . . . , X1n1 (t), . . . , XJ 1 (t), . . . , XJnJ (t), where Xji (t) denotes the instantaneous arrival rate of the ith traffic source of type j at time t. We also assume that the arrival rate Xji (t) varies between 0 and pj the peak rate of the traffic sources in class j , i.e., 0 ≤ Xji (t) ≤ pj ∀t, j . The sources are nj considered as independent random variables for all t. Let S(t) = Jj=1 i=1 Xji (t) and M(t) = E(S(t)) the expected value of S(t). Due to stationarity we can omit the time dependency, so the tail distribution of the aggregate traffic can simply be denoted as P (S ≥ C). 3.2. Improved bound for link saturation probability The following contribution introduces a new formula that can be used for bounding the tail probability of the link saturation with (bufferless) statistically multiplexed traffic. Theorem 4. Let Xji (j = 1, . . . , J ; i = 1, . . . , nJ ) are independent (not necessarily identically distributed) random variables where 0 ≤ Xji ≤ pj ; then N J ∗ J s ∗ pj es pj − 1 nj M + n (p /(e − 1)) j j ∗ j =1 −s C P (S ≥ C) ≤ e , (8) pj N j =1 where s ∗ is the solution of the following equation: J N Jj=1 nj pj2 (espj /(espj − 1)2 ) nj pj − C = 0. − 1 − e−spj M + Jj=1 nj (pj /(espj − 1)) j =1
(9)
We sketch the proof of this theorem in Appendix B. The optimal parameter s ∗ can be numerically computed in a straightforward manner using any standard root-finding algorithm. Note, that the equation
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above contains only the aggregate mean M, meaning that there is no need to know the individual mean values of the random variables (i.e., the mean arrival rates of the traffic flows). The upper bound presented in Theorem 4 can be considered as the real counterpart of the first Hoeffding inequality (3), because in case of identical peak rates we get back this inequality. Also, in this case the optimal s ∗ from Eq. (9) becomes that in (5). Regarding the optimal property of the upper bound (8) the following statement can be formed. Theorem 5. Bound (8) is the best possible one under the following conditions: (1) the first step of the estimation process is the Chernoff formula, (2) all the information known about the traffic are J the number of service classes, the vector n of nj s, the aggregate mean rate M and the vector p of the individual peak rates of the service classes, (3) and the following inequality holds: nj M − (Npi /(espi − 1)) + Jj=1 k=1 (pk /(espk − 1)) 0< < pi ∀i. (10) N Extensive numerical investigation shows that (10) is not a strict condition in case of real applications (e.g. it is worth thinking over how it behaves when we have only one service class). For further discussion of the theorem see Appendix C. 3.3. Closed-form approximations Although the numerical evaluation of the upper bound in (8) can be straightforward, a closed-form expression can be often more useful and expressive, even if it is not optimal with respect to the transformation parameter s. The problem with the result in (8) is that although the optimal parameter s ∗ can be numerically computed, the optimization may be a time consuming process and can hardly be accomplished in real time. In the following, we present two ways of how to find suboptimal but closed-form formulae for s which can be substituted into (8) obtaining closed-form expressions for the improved bound. Let us take the first three terms in the Taylor series expansion of the fraction pj /(espj − 1) with respect to s at s = 0. Now putting them into the logarithm in (B.8), then taking the first two terms of the Taylor series expansion of the received formula in the same one can obtain 2 1 1 1 sM + s 2 − sC, P2 − M− P (11) 8 2N 2 nj where M = Jj=1 i=1 mji , P = Jj=1 nj pj , P2 = Jj=1 nj pj2 . The expression above should be minimized to obtain a suboptimal value for s. Thus we get 1 s˜1∗ = 1
1 P 4 2
C−M . − (1/N)(M − 21 P )2
(12)
This derivation is similar to that in [14] for closed-form equivalent capacity formula. However, the resulting approximate optimizer and the closed-form formula in [14] and ours provided above are completely different (indeed they are not even comparable as such) because of the different objective of the optimization tasks.
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Substituting s˜1∗ into (8), we finally obtain P (S ≥ C) ≤ T (˜s1∗ , C, M, n, p) N n J ((C−M)/K)pj M + Jj=1 (nj pj /(e((C−M)/K)pj − 1)) e −1 j ◦ −(C−M)C/K = e , N pj j =1 (13) 1 P 4 2
1 )P )2 . 2
− (1/N)(M − where K = Our second (partly heuristic) suggestion for a suboptimal s has a similar form to the optimal parameter (5) performed at the first Hoeffding inequality. Substituting each peak rate with the arithmetic mean of the peak rate of all the sources, that is p = pj = (1/N) Jj=1 nj pj ∀j , we formulate a possible suboptimal parameter s as 1 C(Np − M) s ∗ = log . (14) p M(Np − C) Taking into account that (2) is true for all s we may apply (14) for (8) achieving a new closed-form bound, the performance of which, however, should be investigated carefully. Apparently, we may use other types of means as well, e.g. when J 1
p= nj pj2 ∀j, N j =1 we get s˜2∗ =
C(P − M) N log , P2 M(P − C)
or in general ∗ s˜2k = k J
N
k j =1 nj pj
log
C(P − M) . M(P − C)
(15)
(16)
Through extensive numerical investigation the use of s˜2∗ seems to show the best properties. This choice still satisfies the requirement that s˜2∗ should remain positive since neither M nor C can exceed the sum of the peak rates. Another important property of s˜2∗ is that as opposed to s˜1∗ it becomes the original optimal value of s if all the peak rates are equal to each other. From this we can expect that if the deviation of the peak rates is small this latter solution gives better results than (13). In the next section we compare the new bounds derived from (8), (13) and (15) to each other and to the second Hoeffding bound (6), numerically. 4. Comparison via numerical examples From our large number of numerical examples hereafter we show four different cases to demonstrate the performance of the different estimation formulae for the saturation probability. Table 1 shows the
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Table 1 The parameter sets of different traffic situations J
n
p
P
M
DP
M/P
2 8 12 12
(10,1) (10,8,8,5,5,3,2,2) (3,5,3,5,3,2,5,2,5,2,5,2) (10,8,8,6,3,5,5,5,3,2,5,2)
(1,10) (1,3,5,7,10,13,15,17) (1,2,3,4,5,10,11,12,13,14,15,18) (3,5,8,8,10,12,14,15,17,19,25,28)
20 262 360 687
6 100 70 220
0.13 0.018 0.015 0.01
0.3 0.38 0.194 0.32
most important parameters of the different traffic situations, namely the number of service classes J , the vector of the number of the sources in the service classes n, the vector of the peak rates p, the sum of the peak rates P , the aggregate mean arrival rate M, and a normalized deviation-like parameter DP for comparison purposes defined as 2 J 1 1 1
DP = nj pj − P . P N j =1 N
Fig. 1. The logarithm of the saturation probability for four traffic situations.
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The mean to peak ratio M/P is also noted which may give an outline about the total traffic activity in the different cases. In Fig. 1, the logarithm of different estimation of the saturation probability is plotted for the proposed traffic examples. It can be seen that the exact (simulated) tail probability (drawn with bullets) considerably differs from the others, which is reasonable if we take into account that the used information about the traffic is rather few (J , M, n and p). It can be recognized that the first closed-form solution usually very close to the line of the exact formula. This also confirms the statement that the bound (13) tightly approximates the exact formula in (8). The Hoeffding bound (6) is considerably worse than the others, i.e., for a given saturation probability requirement transmission capacity can be saved by applying the new bounds. For further comparison the negative exponent of the bounds derived from (8), (13) and (15) are considered. This means that the progress of the difference functions Diff o,s1 (C) = −log T (s ∗ , C, M, n, p) + log T (˜s1∗ , C, M, n, p),
(17)
Diff o,s2 (C) = −log T (s ∗ , C, M, n, p) + log T (˜s2∗ , C, M, n, p),
(18)
Diff s1 ,s2 (C) = −log T (˜s1∗ , C, M, n, p) + log T (˜s2∗ , C, M, n, p),
(19)
2(C − M)2 Diff s1 ,ho2 (C) = −log T (˜s1∗ , C, M, n, p) − J 2 j =1 nj pj
(20)
and
Fig. 2. The difference function Diff o,s1 (C). (a) M = 6, P = 20, (b) M = 100, P = 262, (c) M = 70, P = 360, (d) M = 220, P = 687.
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Fig. 3. The difference function Diff o,s2 (C). (a) M = 6, P = 20, (b) M = 100, P = 262, (c) M = 70, P = 360, (d) M = 220, P = 687.
are to be analyzed, where log T (s, C, M, n, p) = N log
M+
J
j =1 (nj pj /(e
N
spj
− 1))
+
J j =1
nj log
espj − 1 − sC. pj
(21)
Figs. 2–4 show the functions Diff o,s1 , Diff o,s2 and Diff s1 ,s2 with respect to the capacity C. To better understand the meaning of the curves let us take for example the third plot of the first comparison in Fig. 2. At C = 200, it shows that the difference of the negative exponents of the probability counted with the optimal method and that of counted with (13) is about 0.3, which means that (13) gives e0.3 (≈ 1.35) times larger (i.e., worse) upper bound the optimal technique does. The difference between the optimal and the other two methods in Figs. 2 and 3 gives us a general overview about the accuracy of using the suboptimal s˜1∗ and s˜2∗ instead of s ∗ . Notice that in Fig. 3 in most of the cases the difference Diff o,s2 remains under 0.5, which turns out to be a negligible error, considering that these bounds have around the middle of the interval [M, P ] a value of 0.01 with a slope of about −10 as capacity increases, resulting in a waste of less than 0.1% of the total capacity. Investigating the relation between the two suboptimal methods we can first draw the conclusion that the one using s˜2∗ gets worse as DP increases (as we would expect), while the other one shows little dependence on it. However, considering the above-mentioned facts, this often small deviation of about 0.5 has little significance. On the other hand, changing the value M/P has considerable impact on the performance of the estimates. While the formula using s˜2∗ becomes more precise as M/P decreases, the one with s˜1∗ shows strong degradation. Note that in the above examples this tendency is mostly overshadowed due to the effect of the change of deviation DP .
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Fig. 4. The difference function Diff s1 ,s2 (C). (a) M = 6, P = 20, (b) M = 100, P = 262, (c) M = 70, P = 360, (d) M = 220, P = 687.
Fig. 5. The difference function Diff s1 ,ho2 (C). (a) M = 6, P = 20, (b) M = 100, P = 262, (c) M = 70, P = 360, (d) M = 220, P = 687.
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Finally, we present comparisons between our first closed-form formula using (13) and the second Hoeffding inequality (6) via the difference function Diff s1 ,ho2 . Fig. 5 illustrates strict increase in the difference with increasing capacity C. It means that our first closed-form upper bound seems to be always better (which is also experienced in several other examples) then the second Hoeffding inequality. To sum up the results of our comparisons it can be stated that among the approximation techniques presented in this paper, the method using (13) seems to be the most appropriate one for applications. 5. Conclusions In this paper, we have developed a new upper bound for tail distribution of aggregate traffic using the well-known Chernoff bounding technique. This bound uses the aggregate mean arrival rate, the number of sources and the individual peak rates. This estimation can be considered as a generalized version of the Hoeffding bound assuming identical and normalized peak rates of traffic sources, because that can be obtained as a special case. Under certain conditions this new upper bound can be the best possible one. Due to the fact, however, that it cannot be expressed in closed-form, we have also presented weaker but closed-form upper bounds for the tail distribution of aggregate traffic. Several numerical examples have also shown which of the closed-form bounds is the most appropriate one for practical use. Appendix A. Notes on the performance of the Hoeffding estimation Theorem A.1. be independent random variables with unit peaks, i.e., 0 ≤ Xk ≤ 1 ∀k, nLet X1 , . . . , Xn and let S = k=1 Xk , M = E[ nk=1 Xk ]. Then the estimation C n − M n−C M (A.1) P (S ≥ C) ≤ C n−C is the smallest possible upper bound under the following conditions: (1) the first step of the estimation process is the Chernoff formula, (2) all the information known about the flows is M and n. Proof. To proof the above theorem we use the indirect approach. Let us consider all the possible situations when we have n number of sources the aggregate mean of which is M. We denote the set of these cases with L. In our theorem we state that the bound in (A.1)—throughout the proof denoted by H —is the tightest one for all the elements of L. According to the indirect way, we assume that there exists a bound I , so that I < H and I is also an upper bound for all the elements of L. In the following we try to find an element which cannot be bounded above by I . Let us consider the special case in L when we have X1 , . . . , Xn independent identically Bernoulli distributed n random variables and choose P [Xk = 1] = M/n = r, since we know that E[Xk ] = r and M = k=1 mk = nr. Hence n
n E[es k=1 Xk ] (E[esXk ])n (1 − r + r es )n P Xk ≥ C ≤ = = . (A.2) esC esC esC k=1
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Now, according to the Chernoff method to get the best bound we should optimize in s, obtaining C (1 − r + r es )n n − M n−C M = . (A.3) inf s,s>0 esC C n−C Notice that the result in (A.3) is exactly H . Now if I were smaller than H then it would not be an upper bound of our special choice in L, so that it could not be an upper bound on all the elements of L, which is a contradiction to our initial statement. Appendix B. Deriving the improvement of the Hoeffding bound Proof of Theorem 4. For the proof of Theorem 4 we start with the Chernoff formula (2): J nj P (S ≥ C) ≤ E es j =1 i=1 Xji e−sC .
(B.1)
Due to independence we obtain nj J E[esXji ] e−sC . P (S ≥ C) ≤
(B.2)
j =1 i=1
The exponential function is strictly convex, thus it can be easily shown that the expectation value of the (transformed) random variables esXji can be bounded such as espj − 1 E(esXji ) ≤ 1 + mji , (B.3) pj where mji is the mean value of Xji . In this way, the tail distribution of the sum of the random variables can be bounded above by nj J espj − 1 −sC 1 + mji P (S ≥ C) ≤ e . (B.4) pj j =1 i=1 The product on the right-hand side can be reformulated as J nj J spj pj e − 1 nj −sC mji + spj . e e −1 pj j =1 j =1 i=1 Here, the last term can be further approximated by N J nj nj J spj − 1)) pj j =1 i=1 (mji + pj /(e mji + spj ≤ e − 1 N j =1 i=1
(B.5)
(B.6)
because of the relation between geometric and arithmetic mean of non-negative real numbers. Now an upper bound of the tail distribution can be expressed as N J J spj espj − 1 nj ◦ M + n (p /(e − 1)) j j j =1 −sC =T (s, C, M, n, p), (B.7) P (S ≥ C)≤e N pj j =1 where n is the vector of nj ’s and p comprises the peak rates.
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What still remains to be done is to determine the optimal s, which minimizes the right-hand side of (B.7), in other words, which gives the lowest upper bound in s. Unfortunately, a closed-form expression for the optimal s can not be derived, however, an essentially non-algebraic equation can be formulated, the solution of which serves as the optimal s. The upper bound in (B.7) is minimized if its logarithm is minimized, i.e., the optimal s should minimize the function J M + Jj=1 (nj pj /(espj − 1)) espj − 1 nj log + N log − sC. (B.8) pj N j =1 Taking the derivative of this with respect to s we get J N Jj=1 nj pj2 (espj /(espj − 1)2 ) nj pj − C = 0. − 1 − e−spj M + Jj=1 nj (pj /(espj − 1)) j =1 The solution of (B.9) gives the optimal s ∗ .
(B.9)
Appendix C. Introducing a condition to guarantee the optimal behavior of the improved Hoeffding-like bound In Theorem A.1, we have presented a result about the optimal property of the Hoeffding bound under certain conditions. On the other hand, with the introduction of an extra condition a similar statement can be formed for our bound (8) as well. To investigate the performance of the improved bound let us recall the proof in Appendix B. In the derivation process, basically, after applying the Chernoff formula as a start, at each step of the proof we subsequently obtain new upper bounds on previous ones till we get to the concluding form, which depends only on known parameters. In the following, we attempt to find a particular traffic situation with given M, n and p so that at each bounding step throughout the proof the inequalities become equalities (apparently except the first one with the Chernoff estimation). If such a case exists it follows that an upper bound on all the traffic situations cannot be smaller than (8) (since there exists at least one special case when the saturation probability is just equal to it). The first bounding step in the proof is the application of the Chernoff formula, which corresponds to the first condition in the Theorem 5. It is easy to see that, the subsequent bounding step (B.3) changes to an equality simply if the traffic flows are On/Off types. The situation is a bit more complicated in the last approximation (B.6). It is known that the geometric and arithmetic mean of non-negative real numbers are equal if and only if the numbers are equal, so in our case mji +
pj pl = mlk + sp , l −1 e −1
espj
(C.1)
where i = 1, . . . , nj ; j = 1, . . . , J ; k = 1, . . . , nl ; l = 1, . . . , J should be satisfied. Summing up such equations for k = 1, . . . , nl ; l = 1, . . . , J , with i, j fixed, we get nj M − (Npi /(espi − 1)) + Jj=1 k=1 (pk /(espk − 1)) mji = ∀i, j. (C.2) N
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Since
J
j =1
nj
k=1
0 < mji < pj
mjk = M. Now, if ∀j, i
(C.3)
is also satisfied then (C.2) provides us the missing information to construct the traffic mix, in which case the saturation probability of the aggregate flow cannot be bounded by a smaller value than formula (8) gives us. On the other hand, if (C.3) does not hold then upper bound (8) is not guaranteed to be the tightest one. References [1] F. Brichet, A. Simonian, Conservative models for measurement-based admission control, in: Proceedings of the ITC 16, 1999, pp. 581–592. [2] F. Brichet, A. Simonian, Measurement-based CAC for video applications using SBR service, in: Proceedings of the PMCCN’97, 1997, pp. 285–304. [3] C. Courcoubetis, R. Weber, Buffer overflow asymptotics for a buffer handling many traffic sources, J. Appl. Probab. 33 (1996) 886–903. [4] N.G. Duffield, Economies of scale for long-range dependent traffic in short buffers, Telecommun. Syst. 7 (1997) 267–280. [5] S. Floyd, Comments on measurement-based admission control algorithms for Internet, Technical Report. [6] R.J. Gibbens, F.P. Kelly, Measurement-based connection admission control, in: Proceedings of the International Teletraffic Congress, June 1996, pp. 879–888. [7] R. Guerin, H. Ahmadi, M. Naghshineh, Equivalent capacity and its applications to bandwidth allocation in high-speed networks, in: IEEE J. Selected Areas Commun. 9 (7) (1991) 968–981. [8] W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Am. Statist. Assoc. 58 (1963) 13–30. [9] ITU-T Recommendation E.736, Methods for cell level traffic controls in B-ISDN, May 1997. [10] F.P. Kelly, Notes on effective bandwidths, in: Stochastic Networks: Theory and Applications, Vol. 4, Oxford University Press, Oxford, 1996, pp. 141–168. [11] J. Roberts, Engineering for quality of service, in: K. Park, W. Willinger (Eds.), Self-similar Network Traffic and Performance Evaluation, Wiley, New York, 2000 (Chapter 16). [12] J.W. Roberts (Ed.), Final Report on COST 224, Performance Evaluation and Design of Multiservice Networks, October 1991. [13] J.W. Roberts, U. Mocci, J. Virtamo (Eds.), Final Report on COST 242, Methods for the Performance Evaluation and Design of Broadband Multiservice Networks, June 1996. [14] Z. Turányi, A. Veres, A. Oláh, A family of measured-based admission control algorithms, in: Proceedings of the IFIP PICS’98, Lund, Sweden, May 1998. [15] M. Villen-Altamirano, M.F. Sanchez-Canabate, Effective bandwidth dependent of the actual traffic mix: an approach for bufferless CAC, in: Proceedings of the ITC 15, 1997, pp. 47–57. [16] M. Villen-Altamirano, M.F. Sanchez-Canabate, A tight admission control technique based on measurements, in: Proceedings of the ITC 16, 1999, pp. 603–612.
Z. Heszberger received his M.Sc. degree in electrical engineering from Budapest University of Technology and Economics (BUTE), Budapest, Hungary in 1997. Since 1997, he has been working in the Department of Telecommunications and Telematics. He is about to finish his Ph.D. dissertation. His research focuses mainly on design and control of QoS guaranteed networks.
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J. Zátonyi received his M.Sc. degree in electrical engineering from Budapest University of Technology and Economics (BUTE), Budapest, Hungary in 2000. Since September 2000, he has been a Ph.D. student in the High-Speed Networks Laboratory at the Department of Telecommunications and Telematics at BUTE. His research interests include measurement-based traffic management, admission control, traffic description techniques, QoS provisioning in IP networks.
J.J. B´ıró received his M.Sc. and Ph.D. degrees in electrical engineering from Budapest University of Technology and Economics (BUTE), Budapest, Hungary in 1993 and 1998, respectively. In 1996, he joined the Department of Telecommunications and Telematics at BUTE, where he is working as a research fellow. His main research interests are in optimization, random processes, queuing theory, in particular in performance analysis and evaluation of telecommunication and computer networks.