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13 ANALYSIS OF TRANSIENT LOSS PERFORMANCE IMPACT OF LONG-RANGE DEPENDENCE IN NETWORK TRAFFIC G UANG -L IANG L I AND V ICTOR O. K. L I Department of Electrical & Electronic Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China 13.1 INTRODUCTION To support multimedia applications, high-speed networks must be able to provide quality-of-service (QoS) guarantees for connections with drastically different traf®c characteristics. Some of the characteristics fall beyond the conventional framework of Markov traf®c modeling. For instance, recent studies have demonstrated convin- cingly that there exists long-range dependence or self-similarity in packet video, which is an important traf®c component in high-speed networks. Essentially, long- range dependence cannot be captured by Markov traf®c models. Although long- range dependence in network traf®c has been widely recognized [1, 2, 4, 8, 11, 15, 17, 18], QoS impact of long-range dependence is still an open issue. For example, there are different opinions regarding whether Markov traf®c models can still be used to predict loss performance in the presence of long-range dependence. This and other related issues are also discussed in Chapter 12 of this book. QoS guarantee for long-range dependent (LRD) traf®c is the topic of Chapters 16 and 19 as well. The issue of congestion control for self-similar traf®c is addressed in Chapter 18. 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. 319 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 In this chapter, we present an analysis of transient loss performance impact of long-range dependence in network traf®c. This work is only the ®rst step of our exploration. But we hope that it will still be helpful for understandingloss performance impact of long-range dependence in the transient state, although much further work needs to be done in the future. A different transient analysis in the context of capacity planningand recovery time is given in Chapter 17. In general, the transient analysis of queueing models is very challenging. A transient solution to a queueingmodel is a function of a time index, either continuous or discrete, de®ned over an in®nite range. It is very dif®cult to ®nd an explicit, closed-form solution if the system is not Markovian. For a Markov model, the probabilistic evolution of the system state is governed by the Chapman± Kolmogorov equation. In the presence of long-range dependence, the model is essentially non-Markovian. So the Chapman±Kolmogorov equation does not hold anymore. Due to this and other dif®culties involved in transient analysis, most of the existing work on QoS impact of long-range dependence is limited to asymptotic analysis in the steady state [4, 6, 9, 10, 15±17] (see also Chapters 4±10 in this volume). Although certain insights have been gained through investigation carried out in steady-state, we feel that it is still necessary to extend the investigation beyond the region of steady-state. First, due to the high variability caused by long-range dependence, the conver- gence of an LRD traf®c process toward steady-state can be very slow. In contrast, Markov traf®c processes converge to steady-state exponentially fast. Consequently, for a link carryingLRD traf®c, the discrepany between steady-state performance and transient performance can be more signi®cant, compared with the situations in which traf®c can be modeled by Markov processes. Second, to guarantee QoS for traf®c with the high variability caused by long-range dependence, dynamic and adaptive resource allocation may be necessary to account for the effect of the current system state. Performance analysis based on the steady state may not be appropriate for this purpose, since in steady-state, any initial effect will disappear eventually. Because steady-state performance may differ from transient performance signi®cantly for LRD traf®c, the image regarding loss behavior of LRD traf®c in the transient state still largely remains vague. In this chapter, an approach different from conventional transient analysis is used, which allows us to investigate the transient performance impact of long-range dependence in traf®c without ®rst seekinga closed-form transient solution. That is, we limit our analysis to some short period of time, and even to a single state of a traf®c process. The reasons for us to adopt this approach are as follows. First, it is relatively easy, and may also be suf®cient, to consider transient solutions de®ned only for a relatively short time period, since a short time period may actually cover the time span in which we are interested for transient performance analysis. A large time span may be less interestingfrom a point of view of transient analysis, since the difference between the steady state and the transient state may diminish signi®cantly after a longtime has elapsed. Second, if the arrival process is renewal type, then the behavior of the system is probabilistically periodic. So it may be suf®cient to focus only on a ``typical'' probabilistic period to study the transient performance of the 320 TRANSIENT LOSS PERFORMANCE IMPACT OF LRD IN NETWORK TRAFFIC system. Finally, for a multiple-state arrival process, we may further limit the analysis to an arbitrary single state of the arrival process and compare transient performance measures computed under different modelingassumptions. With this approach, we have gained some insights into QoS impact of long-range dependence in network traf®c. The rest of the chapter is organized as follows. In Section 13.2, we ®rst introduce a framework for traf®c modeling that captures the essential property of long-range dependence. Within this framework, traf®c is modeled by multistate, ¯uid-type stochastic processes. When such a process is in a given state, the underlying traf®c source generates traf®c at a constant rate. The time spent by the process in a state is a random variable. For the purpose of this chapter, we let the distribution of the random variable be arbitrary. As a result, we can construct Markov and LRD traf®c models as we wish. Then we de®ne loss performance measures in the transient state. In Section 13.3, we compare transient loss performance between the traditional Markov models and the LRD models. To keep the comparison reasonable, for the Markov and LRD models, except for the distributions of the times spent by the traf®c processes in their respective states, we let all other traf®c parameters be the same. By doingso, the difference in loss behavior between Markov and LRD traf®c is only due to the modelingassumption on the underlyingtraf®c process. We then compare transient loss of Markov and LRD traf®c for two cases. In the ®rst case, we assume that both traf®c processes are in the same state with the same initial condition characterized by the amount of traf®c left in the system when the processes enter the state. In the second case, we consider two-state Markov and LRD ¯uids. To examine whether it is appropriate to predict loss performance computed accordingto Markov models in steady state for LRD traf®c, in Section 13.4, we show how to compute steady-state limits of transient loss measures for general two-state ¯uids, and compare transient loss against loss in steady state. In Section 13.5, we discuss the impact of long-range dependence in network traf®c, based on the analytical and numerical results obtained. We conclude this chapter in Section 13.6, with a summary of the ®ndings of our study, and a brief discussion on the challenge posed by transient performance guarantee in the presence of long- range dependence and some extension of this work. Section 13.7 contains two appendixes. 13.2 TRAFFIC MODELS AND TRANSIENT LOSS MEASURES We adopt general ¯uid-type stochastic processes with multiple states as a framework for traf®c modeling. The state of such a process is associated with the bit rate of the underlying traf®c source. When the process is in a given state, the source generates traf®c at a constant bit rate. The bit rates are different for different states. Such ¯uid- type traf®c models have been used in many previous studies for traf®c engineering. A well-known example is the Markov-modulated ¯uid model [5]. However, the traf®c model in our study is essentially different from traditional ¯uid traf®c models: our traf®c model is not necessarily Markovian, which allows us to capture the 13.2 TRAFFIC MODELS AND TRANSIENT LOSS MEASURES 321 property of long-range dependence in traf®c. A special case of our traf®c model is the general two-state ¯uid, which can capture the most important traf®c properties such as long-range dependence and burstiness. An important part of this work is based on the two-state ¯uid model. Various on=off ¯uid models are special cases of the general two-state ¯uid and have widely been used for traf®c modeling. For example, on=off sources with heavy-tailed on=off periods are proposed to explain long-range dependence or self-similarity in traf®c [18]. For an on=off ¯uid, no traf®c is generated in the off state. In this book, on=off traf®c models are also considered in Chapters 5, 7, 11, and 17. Let us denote a ¯uid-type traf®c process by Rt. The physical meaningof Rt is the time-dependent bit rate of the underlyingtraf®c source. Denote Rt by Rn for t Pt n ; t n1 , where t n is the instant at which the nth transition of the state of Rt occurs. Accordingly, t n ; t n1  is an interval duringwhich Rt remains unchanged. Suppose that the bit rate of the traf®c source is r duringthe interval, that is, Rnr. Denote the length of the interval t n ; t n1  by Dt n . Clearly Dt n is a random variable, representingthe time spent by Rt in the state in which the bit rate of the traf®c source is r. Suppose that Dt n obeys a distribution F Dt n sPfDt n sg. We assume that the distributions of Dt n are the same when the traf®c process is in the same state but may differ for different states. For a two-state process, we use on and off to refer to the states. When the state is on, the bit rate is denoted by r 1 , and the bit rate correspondingto the off state is r 0 , where r 1 > r 0 ! 0. Denote the lengths of the nth on and off intervals, respectively, by S n and T n . We assume that for n ! 1, S n are independent and identically distributed (i.i.d.) random variables as are T n . Since both S n and T n are i.i.d., we can drop the subscript n in S n and T n . The general two-state ¯uid model is appealingfrom an analysis point of view, since it can capture the essential property of long-range dependence in network traf®c while still permitting an exact analysis without approximation. For a Markov ¯uid, Dt n is of course exponentially distributed. To capture the property of long-range dependence in traf®c, we can assume that Dt n obeys some heavy-tailed distribution. Readers can ®nd a simple formal proof in Grossglauser and Bolot [9] for a special case of the general ¯uid traf®c model, which shows that if for all n ! 1, Dt n are i.i.d. with respect to both n and Rn, and are drawn from a common heavy-tailed distribution, then the correspondingtraf®c process Rt is an LRD process or, more exactly, an asymptotically second-order self-similar process, with autocorrelation function Ct$t Àa1 as t 3I, where the symbol $ repre- sents an asymptotic relation. To de®ne transient loss measures, we assume that traf®c loss is caused only by buffer over¯ow. For a multistate ¯uid, the loss measures are the expected traf®c loss ratio and the probability that loss of traf®c occurs in interval t n ; t n1 , conditioned on w, the amount of traf®c left in the system at t n , where n ! 1. The quantity w is a random variable. In general, for a multistate ¯uid, it is dif®cult to obtain the distribution of w. Therefore, we have to treat w as a given condition. However, in the special case of a two-state ¯uid, we only need to treat w 1 as a given condition, where w 1 is the initial amount of traf®c in the system when n  1. For any n > 1, we can compute the distribution of w by recurrence. So for the special case of a two-state 322 TRANSIENT LOSS PERFORMANCE IMPACT OF LRD IN NETWORK TRAFFIC ¯uid, it is not necessary to treat w for n > 1 as a given condition in the transient loss measures. Instead, we can account for the impact of w by its distribution. In the next section, we discuss how to compute the above transient loss measures, and compare the transient loss behavior of ¯uid traf®c based on the loss measures under different modelingassumptions on the traf®c process, which will provide useful insights into loss performance impact of long-range dependence in the transient state. 13.3 TRANSIENT LOSS OF MARKOV AND LRD TRAFFIC Now let us consider a link with ®nite buffer B and bandwidth C. Suppose that the link carries a ¯uid traf®c process Rt. Recall that t n ; t n1  is the nth interval between transitions of the state of Rt. The modelingassumption on Rt is determined by the distribution of Dt n , the length of the interval t n ; t n1 .Weare curious about the transient loss behavior of Rt at the link under two con¯icting assumptions in traf®c modeling:  Rt is a Markov process. Consequently, Dt n is exponentially distributed.  Rt is an LRD process, which implies that Dt n obeys some (asymptotically) heavy-tailed distribution. To compare the transient loss behavior of Markov and LRD traf®c, we consider the followingtwo cases. 13.3.1 Loss Behavior in Single States To compare the loss behavior of multistate Markov and LRD ¯uids in single states, we use the traf®c loss probability and the expected traf®c loss ratio de®ned in the interval t n ; t n1  where n ! 1, as the transient loss measures. Both the above loss measures are conditioned on the amount of traf®c left in the system at t n . Suppose Rnr. for convenience of exposition, we simply let t n  0 and t n1  S, where S is a random variable with distribution F S sPfS sg that may depend on r. The amount of traf®c in the system at time t is represented by wt, where t P0; S. Denote by w0 the initial amount of traf®c left in the system at the beginning of interval 0; S. Accordingly, the loss probability and the expected loss ratio are denoted, respectively, by PflossjRnr, w0wg and EljRnr, w0w, where l is the fraction of traf®c lost in 0; S. The followingtwo lemmas show how to compute the loss measures. Lemma 13.3.1 PflossjRnr; w0wg 0; r C; 1; r > C; w  B; PfS > twg; r > C; w < B; 8 < : 13:1 13.3 TRANSIENT LOSS OF MARKOV AND LRD TRAFFIC 323 where tw B À w r À C : 13:2 Proof. Clearly, if r C, then PflossjRnr, w0wg0, and r > C together with w  B implies that PflossjRnr, w0wg1. On the other hand, if r > C and w < B, then the random event that traf®c loss due to buffer over¯ow occurs in the interval is equivalent to existing t P0; S such that wtB for t Pt; S and hence PflossjRnr, w0wgPfS > tg. It is easy to see that t  B À w r À C : Since t depends on w, we denote t by tw. j C OMMENT 13.3.2. The physical meaningof tw is the instant in 0; S after which traf®c loss due to buffer over¯ow begins immediately in the interval. Since tw depends on bandwidth and buffer allocated to the underlyingtraf®c, it can be viewed as a control parameter. Lemma 13.3.3 EljRnr; w0w 0; r C;  u 0 0 PS> u 0 tw u 0 À u  du; r > C; 8 > < > : 13:3 where tw is given by Eq. (13.2) and u 0  1 À C r : 13:4 Proof. From Lemma 13.3.1 we know that traf®c loss due to buffer over¯ow will not occur during 0; S if r C,sol  0 when r C, and as a result, EljRnr; w0w0; r C: In the following, we consider r > C.IfS tw, then l  0, since in 0; S, traf®c loss due to buffer over¯ow begins only after t P0; S reaches tw.IfS > tw, then the amount of traffic lost in 0; S the amount of traffic arrived in 0; S Àthe amount of traffic accepted in 0; S  rS ÀCS  B À wr À CS À tw: 324 TRANSIENT LOSS PERFORMANCE IMPACT OF LRD IN NETWORK TRAFFIC For S > tw,wehave l  r À CS À tw rS  1 À C r  1 À tw S  : Therefore, l  0; S tw; u 0 1 À tw S  ; S > tw; 8 > < > : 13:5 where u 0 is given by Eq. (13.4). Since l depends on S, we can express l by lS. Recall that F S s is the distribution of S.Wehave ElSjRnr; w0w  I to ls dF S s lsF S s I stw À  I tw F S s dls  u 0 À  I tw 1 À PfS > sg dls  u 0 À  I tw dls  I tw PfS > sg dls   I tw PfS > sg dls: In the second line of the above equations, since F S I  PfS Ig1by de®nition, lim s3I lsu 0 , and lim s3tw ls0 from Eq. (13.5), we see that lsF S s I tw  u 0 . The proof is completed by changing the integral variable as follows. Denote ls by u;wehave u  u 0 1 À tw s  and s  u 0 tw u 0 À u : When s  tw,wehaveu  0, and s Iis equivalent to u  u 0 . j C OMMENT 13.3.4. If PflossjRnr, w0wg1, then it only means that traf®c loss due to buffer over¯ow will occur for certain in 0; S, while not 13.3 TRANSIENT LOSS OF MARKOV AND LRD TRAFFIC 325 necessarily implyingthat all traf®c arrived in 0; S is lost. As we can see from Lemmas 13.3.1 and 13.3.3, when PflossjRnr, w0wg1, we still have EljRnr, w0w1 À C=r < 1, given r > C. C OMMENT 13.3.5. For loss behavior of ¯uid traf®c in single states, the only nontrivial case is r > C and w < B. Otherwise, the loss probability equals either 0 or 1, and the expected loss ratio is either 0 or a constant equal to u 0 . So it is suf®cient to consider only r > C and w < B for the purpose of this study. As shown above, both the conditional loss probability and the conditional expected loss ratio depend explicitly on the distribution of S, which is in turn determined by the assumption on the underlyingtraf®c process Rt. For example, if we assume that Rt is a Markov process, then S is exponentially distributed. On the other hand, if we assume that Rt is an LRD process, then the distribution of S is heavy-tailed. To compare the two con¯ictingmodelingassumptions, we consider the followingscenario. Suppose that a Markov ¯uid model, denoted by Mt, is used for modelinga ¯uid-type traf®c process Rt. But, in fact, the underlyingtraf®c process Rt is an LRD process, denoted by Lt, which has the same state space as that of the Markov process Mt. The essential difference between Mt and Lt lies in the way to characterize Dt n , the length of the time interval t n ; t n1  for arbitrary n ! 1. We still use 0; S to represent t n ; t n1 , so the interval length Dt n can be denoted simply by S. As we have already mentioned, for Markov model M t, S is exponentially distributed, but for LRD model Lt, the distribution of S is heavy tailed or asymptotically heavy tailed; that is, the functional form of the distribution possesses the property of heavy tail if the value of S is suf®ciently large. For an asymptotically heavy-tailed distribution, it is only necessary for us to consider the case that the value of S is large enough to be in the heavy tail, since only the heavy-tailed effect appears essentially different from Markov traf®c modelingand hence is of great interest for the purpose of this study. To be speci®c, we consider the following heavy-tailed distribution: PfS sg1 Àgs  1 Àa ; 0 s < I; g > 0; 1 < a < 2; 13:6 which is a variant of the conventional Pareto distribution. The reason for us to consider this variant is that the range of the random variable of interest in our study is 0; I while for the conventional Pareto distribution, the range of the random variable is o; I, where o > 0. As we can see, the tail of the distribution becomes heavier and heavier as a decreases toward 1. In fact, a smaller a corresponds to a stronger LRD effect [18]. We are concerned with transient loss performance of the underlyingtraf®c process Rt predicted by Mt. In other words, we want to know the impact of long-range dependence on the transient loss performance predicted by the Markov model. For convenience of exposition, when necessary, M and L, representing respectively the Markov and LRD traf®c models, will substitute for R in the notation 326 TRANSIENT LOSS PERFORMANCE IMPACT OF LRD IN NETWORK TRAFFIC Rn for distinction of the use of the notation. For example, Ln represents the bit rate of Lt in the nth interval between transitions of the state of Lt and PflossjLnr, w0wg is the conditional loss probability of Lt. Our approach is to compare PflossjMnr, w0wg and EljM nr, w0w with PflossjLnr, w0wg and EljLnr, w0w, respec- tively, for given B < I, C < I, r > C, and w < B, under the assumption that ES is the same for Markov model Mt and LRD model Lt. That is, the comparison is made such that Mt and Lt are in the same state with the same initial amount of traf®c left in the buffer. With such a comparison, we believe that the difference in transient loss performance predicted by the Markov model and the LRD model is only due to the different modelingassumptions on the underlyingtraf®c process Rt. Accordingto Lemmas 13.3.1 and 13.3.3, the comparison is straightforward. To exclude the trivial cases, we consider only r > C and w < B. Theorem 13.3.6. Supose that S obeys the Pareto distribution (13.6) for Lt, and ES is the same for Lt and Mt. For given B < I,C< I,r> C, and w < B, we have PflossjM nr, w0wg!PflossjLnr, w0wg if tw g À1 z and PflossjMnr, w0wg < PflossjLnr, w0wg otherwise, where tw is given by Eq. (13.2) and z > 0 is the solution of e bx  x  1 for x P0; I and b  1 À a À1 . Proof. Under the assumptions that S obeys the Pareto distribution (13.6) for Lt, and ES is the same for Mt and Lt,wehaveESa À 1 À1 g À1 for both Lt and Mt, and accordingto Eq. (13.1), PflossjLnr; w0wg PflossjMnr; w0wg  e aÀ1gtw gtw1 a  e aÀ1gtw=a  a gtw1 a  e 1Àa À1 gtw gtw1 "# a : Let b  1 À a À1 and denote gtw by x; then PflossjLnr, w0wg PflossjMnr, w0wg is equivalent to e bx x  1. De®ne yx def e bx À x À 1 for x P0; I. We see that yx 0  is the only extreme of yx, where x 0  b À1 ln b À1 > 0 satisfying dy=dx  be bx À 1  0. In fact, yx 0  is a minimum of yx since d 2 y=dx 2 x 0 b > 0, and yx 0  cannot be nonnegative since y00 and yx 0  < y0.Sowehaveyx 0  < 0. On the other hand, it is evident that yx will become and remain positive after x reaches a suf®ciently large value. Thus, there must exist one and only one zero z > x 0 of yx for x P0; I such that yx 0 for x z and yx > 0 otherwise. The result to be proved then follows. j Theorem 13.3.6 shows that if an LRD ¯uid is modeled by a Markov process, then the Markov model may indeed underestimate the loss probability of the underlying LRD traf®c. A similar result holds for the conditional expected loss ratio. Theorem 13.3.7. Suppose that S obeys the Pareto distribution (13.6) for Lt, and ES is the same for Lt and M t. For given B < I,C< I,r> C, and w < B, 13.3 TRANSIENT LOSS OF MARKOV AND LRD TRAFFIC 327 we have EljM nr, w0w!EljLnr, w0w if y g À1 z and EljM nr, w0w < EljLnr, w0w otherwise, where y  u 0 tw u 0 À x ; x P0; u 0 , tw is given by Eq. (13.2), u 0 is given by Eq. (13.4), and z > 0 is the solution of e bx  x  1 for x P0; I and b  1 À a À1 . Proof. We ®rst recall a well-known result (the generalized mean value theorem) in elementary analysis. Suppose that Fu and Gu are continuous on a; b and differentiable on a; b, and G H u T 0 for a < u < b, where H indicates derivation with respect to u. Then there exists at least one x Pa; b such that FbÀFa GbÀGa  F H x G H x : Now let a  0; b  u 0 and denote  PS> u 0 tw u 0 À u  du by Fu if S obeys the Pareto distribution (13.6) and by Gu if S is exponentially distributed. Both Fu and Gu are continuous on 0; u 0  and differentiable on 0; u 0 . Accordingto the above result and Eq. (13.3), EljLnr; w0w EljM nr; w0w  Fu 0 ÀF0 Gu 0 ÀG0  F H x G H x ; where x P0; u 0 , F H x g u 0 tw u 0 À x  1  Àa ; and G H xexp Àa À 1g u 0 tw u 0 À x  > 0 for 0 < x < u 0 . Letting y  u 0 tw u 0 À x ; we have F H xgy  1 Àa and G H xe ÀaÀ1gy . Replacing tw in Theorem 13.3.6 by y, then usingthe same arguments as that used in the proof of Theorem 13.3.6, we see that the result to be proved follows directly. j C OMMENT 13.3.8. The assumption of Pareto distribution is not restrictive. Similar results hold for any other heavy-tailed distributions. One numerical example is given in Section 13.3.2. 328 TRANSIENT LOSS PERFORMANCE IMPACT OF LRD IN NETWORK TRAFFIC