Tài liệu Mạng lưới giao thông và đánh giá hiệu suất P13 pdf

In this chapter, we present an analysis of transient loss performance impact oflong-range dependence in network traf®c.. To examine whether it is appropriate to predict loss performancec

GUANG-LIANGLI ANDVICTOR O K LI

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 providequality-of-service (QoS) guarantees for connections with drastically different traf®ccharacteristics Some of the characteristics fall beyond the conventional framework

of Markov traf®c modeling For instance, recent studies have demonstrated 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,

convin-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 beused to predict loss performance in the presence of long-range dependence This andother related issues are also discussed in Chapter 12 of this book QoS guarantee forlong-range dependent (LRD) traf®c is the topic of Chapters 16 and 19 as well Theissue 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

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 oflong-range dependence in network traf®c This work is only the ®rst step of ourexploration But we hope that it will still be helpful for understandinglossperformance impact of long-range dependence in the transient state, althoughmuch further work needs to be done in the future A different transient analysis inthe context of capacity planningand recovery time is given in Chapter 17.

In general, the transient analysis of queueing models is very challenging Atransient solution to a queueingmodel is a function of a time index, eithercontinuous or discrete, de®ned over an in®nite range It is very dif®cult to ®nd anexplicit, 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 isessentially non-Markovian So the Chapman±Kolmogorov equation does not holdanymore Due to this and other dif®culties involved in transient analysis, most of theexisting work on QoS impact of long-range dependence is limited to asymptoticanalysis in the steady state [4, 6, 9, 10, 15±17] (see also Chapters 4±10 in thisvolume) Although certain insights have been gained through investigation carriedout in steady-state, we feel that it is still necessary to extend the investigation beyondthe region of steady-state

First, due to the high variability caused by long-range dependence, the 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 andtransient performance can be more signi®cant, compared with the situations in whichtraf®c can be modeled by Markov processes Second, to guarantee QoS for traf®cwith the high variability caused by long-range dependence, dynamic and adaptiveresource allocation may be necessary to account for the effect of the current systemstate Performance analysis based on the steady state may not be appropriate for thispurpose, since in steady-state, any initial effect will disappear eventually Becausesteady-state performance may differ from transient performance signi®cantly forLRD traf®c, the image regarding loss behavior of LRD traf®c in the transient statestill largely remains vague

conver-In this chapter, an approach different from conventional transient analysis is used,which allows us to investigate the transient performance impact of long-rangedependence 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 atraf®c process The reasons for us to adopt this approach are as follows First, it isrelatively easy, and may also be suf®cient, to consider transient solutions de®nedonly for a relatively short time period, since a short time period may actually coverthe time span in which we are interested for transient performance analysis A largetime span may be less interestingfrom a point of view of transient analysis, since thedifference between the steady state and the transient state may diminish signi®cantlyafter a longtime has elapsed Second, if the arrival process is renewal type, then thebehavior of the system is probabilistically periodic So it may be suf®cient to focusonly on a ``typical'' probabilistic period to study the transient performance of the

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 performancemeasures computed under different modelingassumptions With this approach, wehave gained some insights into QoS impact of long-range dependence in networktraf®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-rangedependence Within this framework, traf®c is modeled by multistate, ¯uid-typestochastic processes When such a process is in a given state, the underlying traf®csource generates traf®c at a constant rate The time spent by the process in a state is arandom variable For the purpose of this chapter, we let the distribution of therandom variable be arbitrary As a result, we can construct Markov and LRD traf®cmodels 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 traditionalMarkov models and the LRD models To keep the comparison reasonable, forthe Markov and LRD models, except for the distributions of the times spent by thetraf®c processes in their respective states, we let all other traf®c parameters be thesame 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 thencompare transient loss of Markov and LRD traf®c for two cases In the ®rst case, weassume that both traf®c processes are in the same state with the same initialcondition characterized by the amount of traf®c left in the system when theprocesses enter the state In the second case, we consider two-state Markov andLRD ¯uids To examine whether it is appropriate to predict loss performancecomputed accordingto Markov models in steady state for LRD traf®c, in Section13.4, we show how to compute steady-state limits of transient loss measures forgeneral two-state ¯uids, and compare transient loss against loss in steady state InSection 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 inSection 13.6, with a summary of the ®ndings of our study, and a brief discussion onthe challenge posed by transient performance guarantee in the presence of long-range dependence and some extension of this work Section 13.7 contains twoappendixes

13.2 TRAFFIC MODELS AND TRANSIENT LOSS MEASURES

We adopt general ¯uid-type stochastic processes with multiple states as a frameworkfor traf®c modeling The state of such a process is associated with the bit rate of theunderlying traf®c source When the process is in a given state, the source generatestraf®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, thetraf®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

property of long-range dependence in traf®c A special case of our traf®c model isthe general two-state ¯uid, which can capture the most important traf®c propertiessuch as long-range dependence and burstiness An important part of this work isbased on the two-state ¯uid model Various on=off ¯uid models are special cases ofthe general two-state ¯uid and have widely been used for traf®c modeling Forexample, on=off sources with heavy-tailed on=off periods are proposed to explainlong-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 inChapters 5, 7, 11, and 17

Let us denote a ¯uid-type traf®c process by R…t† The physical meaningof R…t† isthe time-dependent bit rate of the underlyingtraf®c source Denote R…t† by R…n† for

t 2 ‰tn; tn‡1†, where tn is the instant at which the nth transition of the state of R…t†occurs Accordingly, ‰tn; tn‡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† ˆ r.Denote the length of the interval ‰tn; tn‡1† by Dtn Clearly Dtn is a random variable,representingthe time spent by R…t† in the state in which the bit rate of the traf®csource is r Suppose that Dtn obeys a distribution FDtn…s† ˆ PfDtn  sg We assumethat the distributions of Dtnare the same when the traf®c process is in the same statebut 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 r1, and the bit ratecorrespondingto the off state is r0, where r1> r0  0 Denote the lengths of the nth

on and off intervals, respectively, by Sn and Tn We assume that for n  1, Sn areindependent and identically distributed (i.i.d.) random variables as are Tn Since both

Sn and Tnare i.i.d., we can drop the subscript n in Snand Tn The general two-state

¯uid model is appealingfrom an analysis point of view, since it can capture theessential property of long-range dependence in network traf®c while still permitting

an exact analysis without approximation

For a Markov ¯uid, Dtn is of course exponentially distributed To capture theproperty of long-range dependence in traf®c, we can assume that Dtn obeys someheavy-tailed distribution Readers can ®nd a simple formal proof in Grossglauserand Bolot [9] for a special case of the general ¯uid traf®c model, which shows that iffor all n  1, Dtn are i.i.d with respect to both n and R…n†, and are drawn from acommon heavy-tailed distribution, then the correspondingtraf®c process R…t† is anLRD process or, more exactly, an asymptotically second-order self-similar process,with autocorrelation function C…t†  t a‡1 as t ! 1, where the symbol  repre-sents an asymptotic relation

To de®ne transient loss measures, we assume that traf®c loss is caused only bybuffer over¯ow For a multistate ¯uid, the loss measures are the expected traf®c lossratio and the probability that loss of traf®c occurs in interval ‰tn; tn‡1†, conditioned

on w, the amount of traf®c left in the system at tn, where n  1 The quantity w is arandom variable In general, for a multistate ¯uid, it is dif®cult to obtain thedistribution of w Therefore, we have to treat w as a given condition However, in thespecial case of a two-state ¯uid, we only need to treat w1as a given condition, where

w1is the initial amount of traf®c in the system when n ˆ 1 For any n > 1, we cancompute the distribution of w by recurrence So for the special case of a two-state

¯uid, it is not necessary to treat w for n > 1 as a given condition in the transient lossmeasures 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 measuresunder different modelingassumptions on the traf®c process, which will provideuseful insights into loss performance impact of long-range dependence in thetransient 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 thelink carries a ¯uid traf®c process R…t† Recall that ‰tn; tn‡1† is the nth intervalbetween transitions of the state of R…t† The modelingassumption on R…t† isdetermined by the distribution of Dtn, the length of the interval ‰tn; tn‡1† We arecurious about the transient loss behavior of R…t† at the link under two con¯ictingassumptions in traf®c modeling:

 R…t† is a Markov process Consequently, Dtn is exponentially distributed

 R…t† is an LRD process, which implies that Dtn obeys some (asymptotically)heavy-tailed distribution

To compare the transient loss behavior of Markov and LRD traf®c, we consider thefollowingtwo 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 theinterval ‰tn; tn‡1† where n  1, as the transient loss measures Both the above lossmeasures are conditioned on the amount of traf®c left in the system at tn

Suppose R…n† ˆ r for convenience of exposition, we simply let tnˆ 0 and

tn‡1ˆ S, where S is a random variable with distribution FS…s† ˆ PfS  sg that maydepend on r The amount of traf®c in the system at time t is represented by w…t†,where t 2 ‰0; S† Denote by w…0† the initial amount of traf®c left in the system atthe beginning of interval ‰0; S† Accordingly, the loss probability and the expectedloss ratio are denoted, respectively, by PflossjR…n† ˆ r, w…0† ˆ wg and E‰ljR…n† ˆ r,

w…0† ˆ wŠ, where l is the fraction of traf®c lost in ‰0; S† The followingtwo lemmasshow how to compute the loss measures

Proof Clearly, if r  C, then PflossjR…n† ˆ r, w…0† ˆ wg ˆ 0, and r > Ctogether with w ˆ B implies that PflossjR…n† ˆ r, w…0† ˆ wg ˆ 1 On the otherhand, if r > C and w < B, then the random event that traf®c loss due to bufferover¯ow occurs in the interval is equivalent to existing t 2 …0; S† such that w…t† ˆ Bfor t 2 ‰t; S† and hence PflossjR…n† ˆ r, w…0† ˆ wg ˆ PfS > tg It is easy to see that

t ˆB wr C:

COMMENT13.3.2 The physical meaningof t…w† is the instant in ‰0; S† after whichtraf®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

the amount of traffic lost in ‰0; S†

ˆ …the amount of traffic arrived in ‰0; S††

…the amount of traffic accepted in ‰0; S††

ˆ rS …CS ‡ B w† ˆ …r C†‰S t…w†Š:

When s ˆ t…w†, we have u ˆ 0, and s ˆ 1 is equivalent to u ˆ u0 j

COMMENT 13.3.4 If PflossjR…n† ˆ r, w…0† ˆ wg ˆ 1, then it only means thattraf®c loss due to buffer over¯ow will occur for certain in ‰0; S†, while not

necessarily implyingthat all traf®c arrived in ‰0; S† is lost As we can see fromLemmas 13.3.1 and 13.3.3, when PflossjR…n† ˆ r, w…0† ˆ wg ˆ 1, we still haveE‰ljR…n† ˆ r, w…0† ˆ wŠ ˆ 1 C=r < 1, given r > C.

COMMENT 13.3.5 For loss behavior of ¯uid traf®c in single states, the onlynontrivial 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 u0 So it is suf®cient toconsider only r > C and w < B for the purpose of this study

As shown above, both the conditional loss probability and the conditionalexpected loss ratio depend explicitly on the distribution of S, which is in turndetermined 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 theother hand, if we assume that R…t† is an LRD process, then the distribution of S isheavy-tailed To compare the two con¯ictingmodelingassumptions, we consider thefollowingscenario

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 anLRD process, denoted by L…t†, which has the same state space as that of the Markovprocess M…t† The essential difference between M…t† and L…t† lies in the way tocharacterize Dtn, the length of the time interval ‰tn; tn‡1† for arbitrary n  1 We stilluse ‰0; S† to represent ‰tn; tn‡1†, so the interval length Dtn can be denoted simply by

S As we have already mentioned, for Markov model M…t†, S is exponentiallydistributed, but for LRD model L…t†, the distribution of S is heavy tailed orasymptotically heavy tailed; that is, the functional form of the distribution possessesthe property of heavy tail if the value of S is suf®ciently large For an asymptoticallyheavy-tailed distribution, it is only necessary for us to consider the case that thevalue of S is large enough to be in the heavy tail, since only the heavy-tailed effectappears essentially different from Markov traf®c modelingand hence is of greatinterest for the purpose of this study To be speci®c, we consider the followingheavy-tailed distribution:

PfS  sg ˆ 1 …gs ‡ 1† a; 0  s < 1; g > 0; 1 < a < 2; …13:6†which is a variant of the conventional Pareto distribution The reason for us toconsider this variant is that the range of the random variable of interest in our study

is ‰0; 1† while for the conventional Pareto distribution, the range of the randomvariable is ‰o; 1†, where o > 0 As we can see, the tail of the distribution becomesheavier and heavier as a decreases toward 1 In fact, a smaller a corresponds to astronger LRD effect [18]

We are concerned with transient loss performance of the underlyingtraf®cprocess R…t† predicted by M…t† In other words, we want to know the impact oflong-range dependence on the transient loss performance predicted by the Markovmodel For convenience of exposition, when necessary, M and L, representingrespectively the Markov and LRD traf®c models, will substitute for R in the notation

R…n† for distinction of the use of the notation For example, L…n† represents the bitrate of L…t† in the nth interval between transitions of the state of L…t† andPflossjL…n† ˆ r, w…0† ˆ wg is the conditional loss probability of L…t†.

Our approach is to compare PflossjM…n† ˆ r, w…0† ˆ wg and E‰ljM…n† ˆ r,

w…0† ˆ wŠ with PflossjL…n† ˆ r, w…0† ˆ wg and E‰ljL…n† ˆ r, w…0† ˆ wŠ, tively, for given B < 1, C < 1, r > C, and w < B, under the assumption that E‰SŠ

respec-is the same for Markov model M…t† and LRD model L…t† That respec-is, the comparrespec-ison respec-ismade such that M…t† and L…t† are in the same state with the same initial amount oftraf®c left in the buffer With such a comparison, we believe that the difference intransient loss performance predicted by the Markov model and the LRD model isonly 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 Toexclude 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†, andE‰SŠ is the same for L…t† and M…t† For given B < 1, C < 1, r > C, and w < B, wehave PflossjM…n† ˆ r, w…0† ˆ wg  PflossjL…n† ˆ r, w…0† ˆ wg if t…w†  g 1z andPflossjM…n† ˆ r, w…0† ˆ wg < PflossjL…n† ˆ r, w…0† ˆ wg otherwise, where t…w† isgiven by Eq (13.2) and z > 0 is the solution of ebxˆ x ‡ 1 for x 2 …0; 1† and

b ˆ 1 a 1

Proof Under the assumptions that S obeys the Pareto distribution (13.6) for

L…t†, and E‰SŠ is the same for M…t† and L…t†, we have E‰SŠ ˆ …a 1† 1g 1for both

L…t† and M…t†, and accordingto Eq (13.1),

x 1 for x 2 ‰0; 1† We see that y…x0† is the only extreme of y…x†, where

x0ˆ b 1ln b 1> 0 satisfying dy=dx ˆ bebx 1 ˆ 0 In fact, y…x0† is a minimum

of y…x† since d2y=dx2…x0† ˆ b > 0, and y…x0† cannot be nonnegative since y…0† ˆ 0and y…x0† < y…0† So we have y…x0† < 0 On the other hand, it is evident that y…x† willbecome and remain positive after x reaches a suf®ciently large value Thus, theremust exist one and only one zero z > x0of y…x† for x 2 …0; 1† such that y…x†  0 for

x  z and y…x† > 0 otherwise The result to be proved then follows jTheorem 13.3.6 shows that if an LRD ¯uid is modeled by a Markov process, thenthe Markov model may indeed underestimate the loss probability of the underlyingLRD 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†, andE‰SŠ is the same for L…t† and M…t† For given B < 1, C < 1, r > C, and w < B,

we have E‰ljM…n† ˆ r, w…0† ˆ wŠ  E‰ljL…n† ˆ r, w…0† ˆ wŠ if y  g 1z andE‰ljM…n† ˆ r, w…0† ˆ wŠ < E‰ljL…n† ˆ r, w…0† ˆ wŠ otherwise, where

y ˆu0t…w†

u0 x;

x 2 …0; u0†, t…w† is given by Eq (13.2), u0 is given by Eq (13.4), and z > 0 is thesolution of ebxˆ x ‡ 1 for x 2 …0; 1† and b ˆ 1 a 1

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; bŠ anddifferentiable on …a; b†, and G0…u† 6ˆ 0 for a < u < b, where0 indicates derivationwith respect to u Then there exists at least one x 2 …a; b† such that

F…b† F…a†

G…b† G…a†ˆ

F0…x†

G0…x†:Now let a ˆ 0; b ˆ u0 and denote

F0…x† ˆ gu0t…w†

u0 x‡ 1

;and

G0…x† ˆ exp …a 1†gu0t…w†

u0 x

> 0for 0 < x < u0 Letting

y ˆu0t…w†

u0 x;

we have F0…x† ˆ …gy ‡ 1† a and G0…x† ˆ e …a 1†gy Replacing t…w† in Theorem13.3.6 by y, then usingthe same arguments as that used in the proof of Theorem13.3.6, we see that the result to be proved follows directly j

COMMENT13.3.8 The assumption of Pareto distribution is not restrictive Similarresults hold for any other heavy-tailed distributions One numerical example is given

in Section 13.3.2

COMMENT13.3.9 The above theorems imply that if an LRD ¯uid is modeled by aMarkov ¯uid, then in all nontrivial cases, that is, for each state with a durationinterval ‰tn; tn‡1† such that Wn< B and R…n† > C, there exists some critical value of

t…w† and y, beyond which the Markov model underestimates traf®c loss of LRD ¯uidcharacterized by both the loss probability and the expected loss ratio

COMMENT13.3.10 The critical value of both t…w† and y is g 1z, where g is one ofthe two parameters of the Pareto distribution (13.6), and z is the solution of equation

ebx ˆ x ‡ 1 Since b ˆ 1 a 1, we see that z is determined by a, while a is anotherparameter of the Perato distribution Therefore, the critical value of t…w† and y isindependent of buffer and bandwidth allocated to the traf®c source and determinedonly by the property of LRD traf®c Applications with stringent loss constraintsrequire more bandwidth and buffer, which can cause t…w† and y to increase beyondthe critical value, and therefore result in underestimation of loss performancedegradation in the transient state for LRD traf®c modeled by Markov processes.13.3.2 Loss Behavior of General Two-State Fluids

We have just shown that long-range dependence can affect transient loss mance of ¯uid traf®c in single states To extend the analysis beyond single states, weexamine further transient loss behavior of general two-state ¯uids [12] We stilldenote the two-state traf®c process by R…t† and refer to the states as on and off Thelengths of both on and off intervals are i.i.d random variables denoted, respectively,

perfor-by S and T Such traf®c models are appealingfrom a mathematical point of view,since they can capture the essential property of long-range dependence while stillkeepingthe analysis tractable In fact, as we can see later, general two-state traf®cmodels permit a complete analysis by which we can obtain exact results withoutapproximation

Let us consider a link carryingtwo-state ¯uid traf®c with bit rates r1> r0 We canmodel the link by a ®nite buffer queueingsystem with buffer B and service rate Ccorrespondingto the link bandwidth The link transmits traf®c in the queue unlessthe queue is empty We are concerned with transient loss performance de®ned by thetraf®c loss probability and the expected traf®c loss ratio in the nth on interval for

n  1

If C is greater than r0 but less than r1, then loss of traf®c due to buffer over¯owcan only occur when the traf®c process is in the on state, that is, R…t† ˆ r1 In fact,this is the only nontrivial case that should be considered If C  r0, then lossperformance will be out of control in the transient state On the other hand, if C  r1,then loss will never occur for certain Therefore, in the following, we assume

r0 < C < r1 and consider only loss measures de®ned for on intervals Sincetransient performance of the queueingsystem depends on temporal behavior ofthe system state, before we derive transient loss measures, we shall ®rst investigatethe stochastic evolution of system state variables

13.3.2.1 Stochastic Dynamics of the Queueing Model The state variables ofinterest regarding the queueing model are Wnand Qn, which represent the amounts

of traf®c in the system at the beginning and the end of the nth on interval of thetraf®c process, respectively Suppose that initially the traf®c process is on with

W1ˆ w1;where w1 is a given constant between 0 and B Accordingto ¯ow balance, theevolution of Wn and Qn is as follows:

is the probability density function of Qn conditioned on Wnˆ w, fS…s† is theprobability density function of S,

of Wn for 1 < n < 1

13.3.2.2 Transient Loss Measures To compare the loss behavior of two-stateMarkov and LRD traf®c in the transient state, we consider two transient lossmeasures de®ned for on intervals Recall that R…n† denotes R…t† for t 2 ‰tn; tn‡1† As

we have pointed out already, for a two-state process, it is only necessary to consider

on intervals, that is, ‰tn; tn‡1† duringwhich R…n† ˆ r1

Our ®rst transient loss measure is the probability that loss of traf®c occurs duringthe nth on interval due to buffer over¯ow, denoted by PflossjR…n† ˆ r1g The secondtransient loss measure is the expected traf®c loss ratio of the nth interval, denoted byE‰ljR…n† ˆ r1Š Since W1 ˆ w1is a constant, we can use PflossjR…1† ˆ r1, W1ˆ w1gand E‰ljR…1† ˆ r1, W1 ˆ w1Š directly as transient loss measures for n ˆ 1 UsingLemmas 13.3.1 and 13.3.3, we can easily prove the followingtheorem, which showshow to compute the loss measures

13.3.2.3 Investigating Transient Loss Behavior We can investigate the transientloss behavior of a general two-state ¯uid-type traf®c process, at least in principle, bycomputingthe loss measures for any given n  1 By doingso, in fact, we havealready broken, although in a ``brute-force'' way, the barrier of the Markovassumption in transient performance analysis of queueingsystems Recall that thebasis of transient performance analysis for a Markov model is the Chapman±Kolmogorov equation governing the probabilistic evolution of the system state If themodel is not Markovian, then the Chapman±Kolmogorov equation does notnecessarily hold In this case, some auxiliary Markov model may be constructedsuch that the original system can be analyzed approximately In contrast to thisconventional approach, our method does not rely on the Markov assumption, so itcan analyze non-Markov systems without approximation In addition, it can alsoanalyze Markov systems without usingthe Chapman±Kolmogorov equation.However, if we compute the transient performance measures for each n  1directly, then the computational complexity will gradually become prohibitive as nincreases But in the context of transient analysis, performance measures corre-spondingto a large n may be less interesting, since when n is large, the differencebetween transient and steady states may be less signi®cant Consequently, fortransient performance analysis, it may be suf®cient to focus only on a relativelyshort time periods spanned by a number of on and off intervals By doingso, thecomputation involved will be feasible.

In fact, there is a simple way to explore the transient loss performance impact oflong-range dependence in two-state ¯uid traf®c Recall that a two-state ¯uid processcan be viewed as an alternatingrenewal process By takingadvantage of the renewalnature of two-state ¯uids, we can simply focus only on a series of a small number of

on and off intervals Since both on and off intervals have impact on transient lossperformance, such a series must contain at least two on intervals and one off intervalbetween them, as shown in Fig 13.1

Now let us consider such an on±off±on series Without loss of generality, the two

on intervals in the series are indexed by n ˆ 1 and 2 Since loss behavior of thequeueingsystem duringthe ®rst on interval in the series cannot capture the impact ofthe off interval, we examine loss behavior of the system in the second on interval