11 TRAFFIC AND QUEUEING FROM AN UNBOUNDED SET OF INDEPENDENT MEMORYLESS ON=OFF SOURCES P HILIPPE J ACQUET INRIA, 78153 Le Chesnay CeÂdex, France 11.1 INTRODUCTION 11.1.1 Long-Term Dependence and Packet Loss in Telecommunication Traf®c In early literature about the performance of telecommunication systems, traf®c was generally modeled as memoryless Poisson streams of packets. In these models the packet arrival processes show no time interdependence. Recent measurements on Web traf®c show that this hypothesis is wrongand that Web traf®c actually experiences what we now call long-term dependence. Long-term dependence is interesting not only because it contradicts Poisson's law, but also because it signi®cantly impacts the performance of networks. One effect is that it dramatically increases packet loss in data networks. For example, let us focus on an Internet router. In a simple model, the router can be seen as a buffer served by a single server. When the buffer over¯ows, some packets are lost. The lost packets must be re-sent followingTCP=IP, thus addingextra delay and traf®c. If we simply model the router by a M=M =1 queue with an in®nite buffer, input rate l, and service rate 1, then the probability p n that the queue length is greater than n is exactly l n . In a ®rst-order approximation, quantity p n can be identi®ed with the packet loss rate in a buffer of size n. Therefore, to keep packet loss below some acceptable level e, it suf®ces to make the buffer capacity greater than log e=log l), Self-Similar Network Traf®c and Performance Evaluation, Edited by KihongPark and Walter Willinger ISBN 0-471-31974-0 Copyright # 2000 by John Wiley & Sons, Inc. 269 Self-Similar Network Traf®c and Performance Evaluation, Edited by KihongPark and Walter Willinger Copyright # 2000 by John Wiley & Sons, Inc. Print ISBN 0-471-31974-0 Electronic ISBN 0-471-20644-X that is, a logarithmic function of 1=e. In general, telecommunication designers work on e on the order of 10 À6 and l < 0:8. Long-term dependent traf®c can make loss and retry rates p n  Bn Àb for some b > 0 [1]. In other words, the queue size distribution has a heavy tail. Under this condition it is clear that buffer capacity would need to be raised to b=e 1=b to keep packet loss rate under the acceptable level e, which no longer leads to a logarithmic function of 1=e, but to a polynomial function of 1=e. Indeed, this minimal size would be several orders of magnitude higher than the capacity obtained with the Poisson model. In fact, actual router capacities are dangerously underestimated with regard to this new traf®c condition. 11.1.2 Contribution of this Chapter The Poisson law is the natural consequence of the law of large numbers, best describes the cumulated effect of several independent, identically distributed (i.i.d) sources in parallel. Assume, for example, N sources, each of them producingon average l=N events per time unit accordingto a stationary random process. Then when N tends to in®nity, the interevent times T tend to be i.i.d with a distribution function characterizingthe Poisson law, PrfT > xge Àlx : 11:1 The convergence to a Poisson distribution still holds when the sources are not quite identical, as long as they have similar pro®les. Feller [2] gave pretty general conditions for this convergence: basically, ®rst moments of the source interevent generation time must be O1=N  and second moments must be o1=N. In this chapter we are interested in the case where traf®c is created from a large set of independent sources that do not satisfy Feller's conditions. In particular, we focus on certain sets of on=off sources that produce long-term dependence when their sizes tend to in®nity. It will also be shown that queues submitted to such sets of sources will experience buffer occupation with a polynomially decayingtail distribution. In the other chapters of this book it is assumed that some of the sources, taken individually, already produce long-term dependence. For example, some sources have heavy-tailed ``on'' periods. In this case the cumulated traf®c shows long-term dependence and creates a heavy-tailed queue size distribution [3, 4]. The challenge in the present chapter is that none of the sources, taken separately, produces long- term dependence and a heavy-tailed queue size distribution, and that those phenomena eventually take place when the number of sources increases. To insist on this point, we will focus on individual on=off sources with memoryless pro®les (exponentially distributed on periods and off periods). It has already been shown by Beran [5] and Jacquet [6] that such sources can create long-term dependence when their number increases. The contribution of the present chapter is to show that such sources can also create a polynomially decayingqueue size distribution. We do not claim that pure memoryless on=off sources are necessarily realistic models for Web 270 TRAFFIC AND QUEUEING FROM AN UNBOUNDED SET sources of traf®c, but they have the advantage of being simple and analytically more tractable than more general models. Our model stands halfway between the Boxma±Dumas models [7] with ®nite sets of on=off sources and the very excitingmodel of Tsybakov et al. [8], which assumes a continuum of on=off sources from which ``on'' periods are activated accordingto a Poisson process. A last technical point: we don't use ¯uid approximation in this chapter, although this feature has very interesting aspects. 11.2 TOOLS AND MODELS 11.2.1 The Mellin Transform Applied to Performance Analysis The technical novelty of this chapter is in the extensive use of the Mellin transform. The Mellin transform is particularly well adapted to capturingthe polynomial effects in function asymptotics with an unprecedented accuracy [9]. It has similar features to those of the Laplace transform, the latter beingknown for more than a century to be a good tool for capturing exponential effects in functions. The Mellin transform f *s of a function f x, de®ned for real x > 0, is f *s  I 0 f xx sÀ1 dx: 11:2 Note that the Mellin transform of function f x is nothingmore than the Laplace transform of f e x . The Mellin transform is de®ned for s in the fundamental strip fs; RsPa; bg of function f x. The constants a and b are the lower and upper bounds of the real numbers c such that f xox Àc  for both x 3 0 and x 3I. The fundamental strip of function e Àx is fs; Rs > 0g: in other words, a  0, b I. In passing, the Mellin transform of e Àx is the celebrated Euler Gamma function, denoted Gs. The fundamental strip of any polynomial function is the empty set: there is no Mellin transform of pure polynomial functions. The inverse Mellin transform is f x 1 2ip  ciI cÀiI f *sx Às ds 11:3 valid for any c contained in the fundamental strip. The Mellin transform has been introduced primarily to handle harmonic sums. A harmonic sum of function f x is a series of the followingkind: P i!0 a i f o i x 11.2 TOOLS AND MODELS 271 for some sequences a i and o i , which make the sum properly convergent. The Mellin transform of the harmonic sum is P i!0 a i o Às i ! f *s; where f *s is the Mellin transform of function f x. This latter expression is sometimes much easier to handle than the harmonic sum itself. The analysis of function f x asymptotics, when x tends to both limit 0 and I,is equivalent to the singularity analysis of the Mellin transform f *s on the boundaries of its fundamental strip. The right boundary corresponds to the asymptotics when x 3I; the left boundary corresponds to the asymptotics when x 3 0. For example, if f *s has one single pole on the right boundary at s  b, this pole has residue m, and function f *s can be extended for ResPb; b  e. Then in Eq. (11.3) the integration line can be moved on the right (see Fig. 11.1). Applyingthe residue theorem, we ®nd f xmx Àb  x ÀbÀe 2ip  I ÀI x Àit f b  e  it dt: 11:4 Using a majorization under the integral sign, we get a second-order estimate: f xmx Àb  Ox ÀbÀe  when x 3I. The Mellin transform can provide a more accurate estimate and can even capture the case where the constant before the x Àb estimate is a ¯uctuatingfunction of x. Fig. 11.1 Movement of the integration line in the reverse Mellin transform. 272 TRAFFIC AND QUEUEING FROM AN UNBOUNDED SET As an illustration, consider the followingfunction f x satisfyingthe functional equation: f x 1 2 f x=2e Àx =2: 11:5 This equation arises in the analysis of divide and conquer algorithms [10]. The solution is obtained by iteration: f x P i!0 2 Ài e x=2 i : 11:6 By virtue of the harmonic sum expression, the Mellin transform f *s is immediately identi®ed by f *s2 À1s f *sÀGs,or f *s Gs 1 À 2 À1s 11:7 for s in the fundamental strip fs; RsPÀ0; 1g. Note that there is a sequence of single poles 1  2ikp=log2, for k integers, that are all located on the strip boundary fs; Rs1g. Usingthe reverse Mellin transform and catchingall the residues of these poles by moving the integration line on the right (see Fig. 11.2), we get f xPlog xx À1  Ox À1Àe 11:8 with Py P k G 1  2ikp log2  exp 2ikpy log2  : 11:9 Fig. 11.2 Reverse Mellin transform with multiple poles. 11.2 TOOLS AND MODELS 273 It is clear that Px is of period log2 with Fourier coef®cients identi®ed in Eq. (11.9). To make a more general statement, the Mellin transform is the ideal tool for capturingthe slowly varyingfunctions in front of polynomial factors. Here the term analytical information theory is used to describe problems of information theory that are solved by analytical methods borrowed from complex analysis [11, 12]. Refer to Szpankowski [12] for a survey and to Jacquet [6] for a detailed description of the tools and proofs used in the present chapter. 11.2.2 Memoryless On=Off Sources As mentioned earlier we focus on the simpli®ed case where each source has a memoryless pro®le. A memoryless on=off source is described by only two statesÐ the ``on'' state and the ``off'' stateÐand has the followingproperties:  In the ``off'' state, the source does not generate any packet.  In the ``on'' state, the source generates packets as a Poisson stream with constant rate l.  The transition from the ``on''state to the ``off'' state occurs with constant rate n 0 .  The transition from the ``off'' state to the ``on''state occurs with constant rate n 1 . Note that the ``on'' periods and ``off'' periods are both exponentially distributed; that is, the state transition times follow a memoryless process. We introduce the matrix T, which we call the transition matrix: T  Àn 1 n 0 n 1 Àn 0  11:10 The eigenvalues of T are 0 and Àn 1  n 0 . In most of this chapter we consider a general system where the arrival process comes from several independent memoryless on=off sources. We consider a denumerable set of on=off sources indexed from 1 to N, N tendingto in®nity in the analysis. See Fig. 11.3 for an illustration. 11.3 QUEUEING UNDER ON=OFF SOURCES In the following, the network is modeled via several queues with single servers and an in®nite buffer, which receive input from a set of on=off sources. The service time at each queue is exponential. Our aim is to ®nd the steady-state distribution of the queue length and, in particular, the asymptotics of the probabilities p n that the queue length exceeds n customers, when n 3I. 274 TRAFFIC AND QUEUEING FROM AN UNBOUNDED SET 11.3.1 Queueing with aSingle On=Off Source We consider a single queue with service mean equal to 1 time unit. We refer to Neuts [13] for the followingtheorem. Theorem 11.3.1. The queue length generating function with the exponential server and an on=off input source satis®es qz1 À z À 1lz 1 n 1 n 1  n 0 z À z 1  11:11 with z 1  1 2 1  l  n 1  n 0   1 À l 2  2n 1  n 0 2ln 1 À n 0 n 1  n 0  2 q 1  n 1 l 0 @ 1 A : 11:12 Proof. This a straightforward adaptation of that in Neuts [13]. j Corollary 11.3.2. When n 3I, quantities p n exponentially decrease with p n  l n 1 v 1  v o z 1Àn 1 Fig. 11.3 Aggregation of a denumerable set of independent on=off sources. 11.3 QUEUEING UNDER ON=OFF SOURCES 275 11.3.2 Parallel Queues Under Self-Similar On=Off Sources Here we consider several memoryless on=off sources, each of them served by a server. Each service time is exponential with a mean of 1 time unit, and the peak rate of each source is l > 1. Therefore, the probability generating function gz of the queue length distribution is simply gz P N i1 1 N q i z; 11:13 where q i z is the probability generating function of the queue assigned to source number i. To simplify our analysis we look at a set of self-similar on=off sources. The vector n 1 ; n 0  of each source is collinear to a ®xed vector v 1 ; v 0 : for source i, n 1 ; n 0 e i v 1 ; e i v 0 . Referringto Theorem 11.3.1, we see that the probability that queue i contains more than n packets is exactly lv 1 v 1  v 0 z 1Àn 1 e i  with z 1 x given by Eq. (11.12) with substitution of n 1 ; n 0  by xv 1 ; xv 0 .Ifwe denote by re the asymptotic density of sources with vector n 1 ; n 0  equal to ev 1 ; ev 0 , then we have the followingresult [14]. Theorem 11.3.3. Let 0 < b < 1. If function rxmx bÀ1  Ox bÀ1e  when x 3 0, then when N 3Iquantity p n converges to the following expression: lv 1 v 1  v 0 v 1 l À 1 À v 0  Àb mn Àb  On ÀbÀe : 11:14 Proof. Note that the conditions of the theorem are basically equivalent to the fact that the Mellin transform of function rx has a single pole at s  1 À b, with residue m, and can be analytically continued on the strip fs; RsP1 À b À e; 1 À bg. For any ®xed N we have p n  1 N P N i1 lv 1 v 1  v 0 z 1Àn 1 e i : 11:15 At the limit we have lim M3I p n  lv 1 v 1  v 0  I 0 z 1 y 1Àn r y dy: 11:16 276 TRAFFIC AND QUEUEING FROM AN UNBOUNDED SET In the above expression, substitutinga real number x for integer n, we obtain a function px whose Mellin transform p*s satis®es p*s lv 1 v 1  v 0  I 0 z 1 y log z 1 y Às r y dy: 11:17 Since z 1  y1  v 1 l À 1 À v 0  y  Oy 2 ; log z 1  y Às  v 1 l À 1 À v 0  Às y Às 1  sOy when y 3 0. Therefore, we have the estimate p*s lv 1 v 1  v 0 v 1 l À 1 À v 0  Às  I 0 ryy Às 1  sOy dy  lv 1 v 1  v 0 v 1 l À 1 À v 0  Às r*1 À ssOr*2 À Rs; where function r*s is the Mellin transform of function rx. Therefore, the Mellin transform of px is related to the Mellin transform of rx. More precisely, if r*s has a single pole at s  1 À b with residue m and can be extended further to the right, then the ®rst encountered singularity of p*s is at s  b. Note that the singularity boundary of r*2 À Rs is shifted by 1 on the right from that of r*1 À s, p n  lv 1 v 1  v 0 v 1 l À 1 À v 0  Àb mn Àb  On ÀbÀe : 11:18 j As an application, we choose rxbx bÀ1 valid for x 1 and rx0 for x > 1. In this case, r*sb=b À 1  s and we get p n  lv 1 v 1  v 0 v 1 l À 1 À v 0  Àb bn Àb  On ÀbÀ1 : Polynomial Tail with ¯uctuating Coef®cients. By tuningsystem parameters, one can give rise to oscillating coef®cients in the asymptotics of p n when n 3I. Indeed, we can have p n  Pnn Àb with Px oscillatingbetween two values: lim inf Px T lim sup Px. For example, one can take rxax À2 2 À P i!9 2 Ài=2 expÀ2 i x 21b , the constant term a is here to make the density function rx sum to 1 and is equal here to À21  b1 À 2 Àb=21b =GÀ1=21  b. 11.3 QUEUEING UNDER ON=OFF SOURCES 277 Therefore, r*sÀ a 21  b G s À 2 21  b  1 À 2 À 1 2 ÀsÀ2=21b : 11:19 The Mellin transform p*s therefore has a singularity set made of simple poles s k b  4i1  bkp=2, for k integer. As we saw in the tutorial section about the Mellin transform, this kind of set creates periodic ¯uctuatingterms in front of the polynomial expansion. This periodic ¯uctuation is re¯ected in the asymptotics of p n , namely: p n  lv 1 v 1  v 0 v 1 l À 1 À v 0  Àb a log2 Â P k Gs k  exp À4ikp 1  b log2  log n  n Àb  On ÀbÀe :  In other words, we have proved that p n  Plog nn Àb , where PÁ is a periodic function, of period log2=1 À b, whose Fourier coef®cients are proportional to Gs k . Since function PÁ is not constant (indeed Fourier coef®cients are all nonzero), we don't have lim inf Pxlim sup Px. Figure 11.4 displays function rx computed for b  0:5, which causes ¯uctuat- ingpolynomial coef®cients. 11.3.3 A Multiplexer Queue Under an In®nite Number of On=Off Sources In the previous section we considered on=off sources served by separate parallel queues. In this section we consider that all on=off sources forward their packet to a Fig. 11.4 Density function rx, which leads to ¯uctuatingasymptotics with b  0:5: 278 TRAFFIC AND QUEUEING FROM AN UNBOUNDED SET