Tài liệu Mạng lưới giao thông và đánh giá hiệu suất P9 pptx

In this chapter we focus instead on the class of M=G=1 input processes aspotential traf®c models.. An M=G=1 input process is understood as the busy serverprocess of a discrete-time in®ni

INPUT PROCESSES

ARMAND M MAKOWSKI AND MINOTHI PARULEKAR

Institute for Systems Research, and Department of Electrical and Computer Engineering,University of Maryland, College Park, MD 20742

9.1 INTRODUCTION

Several recent measurement studies have concluded that classical Poisson-like traf®cmodels do not account for time dependencies observed at multiple time scales in awide range of networking applications, for example, Ethernet LANs [13, 20, 33],variable bit rate (VBR) traf®c [3, 15], Web traf®c [6], and WAN traf®c [31] As theresulting temporal correlations are expected to have a signi®cant impact on bufferengineering practices, this ``failure of Poisson modeling'' has generated an increasedinterest in a number of alternative traf®c models that capture observed (long-range)dependencies [14, 24] Proposed models include fractional Brownian motion [25]and its discrete-time analog, fractional Gaussian noise [1] Already both haveexposed clearly the limitations of traditional traf®c models in predicting storagerequirements and devising congestion controls A discussion of these issues in thecase of fractional Brownian motion is summarized in Chapter 4

In this chapter we focus instead on the class of M=G=1 input processes aspotential traf®c models An M=G=1 input process is understood as the busy serverprocess of a discrete-time in®nite server system fed by a discrete-time Poissonprocess of rate l (customers=slot) and with generic service time s distributedaccording to G As argued in Parulekar [27] and Parulekar and Makowski [29], theseM=G=1 input processes constitute a viable alternative to existing traf®c models;reasons range from ¯exibility to tractability

Self-Similar Network Traf®c and Performance Evaluation, Edited by Kihong Park and Walter Willinger ISBN 0-471-31974-0 Copyright # 2000 by John Wiley & Sons, Inc.

215

Self-Similar Network Traf®c and Performance Evaluation, Edited by Kihong Park and Walter Willinger

Copyright # 2000 by John Wiley & Sons, Inc Print ISBN 0-471-31974-0 Electronic ISBN 0-471-20644-X

First, the relevance of the M=G=1 input model to network traf®c modeling isperhaps best explained through its connection to an attractive model for aggregatepacket streams proposed by Likhanov et al [21] They show that the combinedtraf®c generated by several independent, identically distributed (i.i.d.) on=off sourceswith Pareto distributed activity periods behaves in the limit, as the number of sourcesincreases, like the M=G=1 input stream with a Paretodistributed s This provides arationale for the view that M=G=1 input processes could provide a naturalalternative to existing traf®c models, at least for certain multiplexed applications.Second, the class of M=G=1 input processes is stable under multiplexing; that is,the superposition of several M=G=1 processes can be represented by an M=G=1input process.

Third, the M=G=1 model displays great ¯exibility in capturing positivedependencies over a wide range of time scales; this is achieved very simply throughthe tail behavior of s (Proposition 9.4.1) The degree of positive correlation canfurther be characterized by the sum of the autocovariances, or index of dispersion ofcounts (IDCs), with the process being short-range dependent (SRD) (i.e., IDC ®nite)

if and only if E‰s2Š is ®nite (Proposition 9.5.1)

Insights into how temporal correlations of M=G=1 input processes will affectqueueing performance can be gained by analyzing the behavior of a multiplexer fed

by an M=G=1 input process For simplicity, we model the multiplexer as a time single server system consisting of an in®nite size buffer and a server with aconstant release rate c (cells=slot) The number of customers in the input buffer attime t is denoted by qt Our performance index is the steady-state buffer tailprobability P‰q1> bŠ, as this quantity is indicative of the buffer over¯ow prob-ability in a corresponding ®nite buffer system with b positions

discrete-Computing these tail probabilities, either analytically or numerically, represents achallenging problem in the absence of any underlying Markov property for M=G=1inputs Instead, we focus on the simpler task of determining the asymptotic tailbehavior of the queue-length distribution for large buffer size More precisely, weseek results of the form

lim

b!1

1

h…b†ln P‰q1> bŠ ˆ g …9:1†for some positive constant g and mapping h: R‡! R‡; these quantities arecharacterized by l, G, and c and should be easily computable Limits such as Eq.(9.1) suggest approximations of the form

P‰q1> bŠ  e h…b†g …b ! 1†: …9:2†Needless to say, such estimates should be approached with care [5] Nonetheless, Eq.(9.1) already provides some qualitative insights intothe queueing behavior at themultiplexer and could, in principle, be used to produce guidelines for sizing up itsbuffers

In this chapter we provide an overview of some recent work on this issue.Drastically different behaviors emerge depending on whether vt* ˆ O…t† or vt* ˆ o…t†(with t ! 1), where v* ˆ ln P‰^s > tŠ; t ˆ 1; 2; , and ^s is the forwardtrecurrence time (9.9) associated with s The case vt* ˆ O…t† is associated with theservice time s having exponential tails, while the case vt* ˆ o…t† corresponds toheavy or subexponential tails for s.

Our focus here is primarily (but not exclusively) on large deviations techniques inorder to obtain Eq (9.1) This approach has already been adopted by a number ofauthors [10, 16, 19] Applying results by Duf®eld and O'Connell [10] (and somerecent extensions thereof [11, 30]), we are able to compute h…b† and g underreasonably general conditions In fact, for a large class of distributions, we can select

h…b† ˆ vdbe* , and the asymptotics (9.1) and (9.2) then take the compact form

P‰q1> bŠ  P‰^s > bŠg …b ! 1†: …9:3†Hence, in many cases, including Weibull, lognormal, and Pareto service times, q1and ^s (thus s) belong to the same distributional class as characterized by tailbehavior

In many cases of interest, in lieu of Eq (9.1), these large deviations techniquesyield only the weaker asymptotic bounds

g lim inf

b!1

1

h…b†ln P‰q1> bŠ …9:4†and

lim sup

b!1

1

h…b†ln P‰q1> bŠ  y* …9:5†with g6ˆ g* This situation typically occurs when s is heavy tailed (more generally,subexponential) with either ®nite or in®nite E‰s2Š, in which case large deviationsexcursions are only one of several causes for buffer exceedances [19] While Eqs.(9.4) and (9.5) are still useful in providing bounds on decay rates, they will not betight in the heavy-tail case and other approaches are needed Of particular relevanceare the approaches of Liu et al [22] (summarized in Section 9.11) and of Likhanov(discussed in Chapter 8) Liu et al [22] derive bounds through direct arguments thatrely on the asymptotics of Pakes [26] for the GI=GI=1 queue under subexponentialassumptions [12] While Likhanov presents lower and upper bounds only when s isPareto, these bounds are asymptotically tight Results for the continuous-time modelcan be found in Jelenkovic and Lazar [17], and in Chapters 10 and 7

Comparison of Eq (9.3) with results from Norros [25] and Parulekar andMakowski [28] points already to the complex and subtle impact of (long-range)dependencies on the tail probability P‰q1> bŠ Indeed, in Norros [25] the inputstream to the multiplexer was modeled as a fractional Gaussian noise process (orrather its continuous-time analog) exhibiting long-range dependence (in fact, self-similarity), and the buffer asymptotics displayed Weibull-like characteristics On theother hand, by the results described above, an M=G=1 input process with a Weibull

service time also yields Weibull-like buffer asymptotics although the input process isnow short-range dependent Hence, the same asymptotic buffer behavior can beinduced by two vastly different input streams, one long-range dependent and theother short-range dependent! To make matters worse, if the pmf G were Paretoinstead of Weibull, the input process would now be long-range dependent, in factasymptotically self-similar [28], but the buffer distribution would now exhibitPareto-like asymptotics, in sharp contrast with the results of Norros [25].

To reiterate the main conclusion of Parulekar and Makowski [28], the value of theHurst parameter as the sole indicator of long-range dependence (via asymptotic self-similarity) is at best questionable as it does not characterize buffer asymptotics byitself Furthermore, buffer sizing cannot be determined adequately by appealingsolely to the short- versus long-range dependence characterization of the inputmodel used, be it of the M=G=1 type or otherwise Of course, this is not toosurprising since long-range dependence (and its close cousin, second-order self-similarity) is determined by second-order properties of the input process, whileasymptotics of the form (9.1) invoke much ®ner probabilistic properties, which areembedded here in the sequence fvt*; t ˆ 1; 2; g The ®niteness of E‰s2Š (whichcharacterizes the SRD nature of the M=G=1 input process) is obviously a poormarker for predicting the behavior of this sequence

To close, we note that the diverse queueing behavior demonstrated here is tied tothe tail behavior of s, which determines the correlation structure of M=G=1 inputs.This clearly illustrates the tremendous impact that the correlation structure of aninput stream can have on the corresponding queueing performance given that theM=G=1 inputs all have Poisson marginals! One more data point for the need of acautious approach in modeling network traf®c when time dependencies are eitherobserved or suspected

9.2 THE M=G=11 INPUT PROCESS

We summarize various facts concerning the busy server process of a discrete-timeM=G=1 system; details are available in Parulekar [27]

9.2.1 The Model

Consider a system with in®nitely many servers During time slot ‰t; t ‡ 1†; bt‡1newcustomers enter the system Customer i; i ˆ 1; ; bt‡1, is presented toits ownserver and begins service by the start of slot ‰t ‡ 1; t ‡ 2†; its service time hasduration st‡1;i (expressed in number of slots) Let bt denote the number of busyservers or, equivalently, of customers still present in the system, at the beginning ofslot ‰t; t ‡ 1† We assume that b servers are initially present in the system at t ˆ 0(i.e., at the beginning of slot ‰0; 1†) with customer i; i ˆ 1; ; b, requiring anamount of work of duration s0;i from its own server The busy server process

fbt; t ˆ 0; 1; g is what we refer toas the M=G=1 input process

The following assumptions are enforced on the R-valued random variables (rvs)

b, fbt‡1; t ˆ 0; 1; g and fst;i; t ˆ 0; 1; ; i ˆ 1; 2; g: (1) The rvs aremutually independent; (2) The rvs fbt‡1; t ˆ 0; 1; g are i.i.d Poisson rvs withparameter l > 0; (3) The rvs fst;i; t ˆ 1; 2; ; i ˆ 1; 2; g are i.i.d withcommon pmf G on f1; 2; g We denote by s a generic R-valued rv distributedaccording to the pmf G Throughout we assume this pmf G tohave a ®nite ®rstmoment, or equivalently, E‰sŠ < 1 At this point, no additional assumptions aremade on the rvs fs0;i; i ˆ 1; 2; g

For each t ˆ 0; 1; , we note the decomposition

btˆ b…0†t ‡ b…a†t ; …9:6†where the rvs b…0†t and b…a†t describe the contributions to the number of customers inthe system at the beginning of slot ‰t; t ‡ 1† from those initially present (at t ˆ 0)and from the new arrivals in the interval [0, t], respectively Under the enforcedoperational assumptions, we readily check that

9.2.2 The Stationary Version

Although the busy server process fbt; t ˆ 0; 1; g is in general not a (strictly)stationary process, it does admit a stationary and ergodic version fbt*; t ˆ 0; 1; g.This stationary version satis®es the decomposition (9.6) with the portion in (9.7) due

to the initial condition replaced by

b…0†t ˆPb

nˆ11‰^sn> tŠ; t ˆ 0; 1; ; …9:8†where (1) the rvs b and f^sn; n ˆ 1; 2; g are independent of the rvs

fbt‡1; t ˆ 0; 1; g and fst;i; t ˆ 1; 2; ; i ˆ 1; 2; g; (2) the rvsf^sn; n ˆ 1; 2; g are independent of the rv b, which is Poisson distributed withparameter lE‰sŠ; and (3) the rvs f^sn; n ˆ 1; 2; g are i.d.d rvs distributedaccording to the forward recurrence time ^s associated with s; the correspondingequilibrium pmf ^G of ^s is given by

P‰^s ˆ rŠ ˆP‰s  rŠE‰sŠ ; r ˆ 1; 2; : …9:9†

The following properties of fbt*; t ˆ 0; 1; g follow readily from this sentation [7, 18 (Theorem 3.11, p 79), 27].

repre-Proposition 9.2.1 The stationary and ergodic version fb*; t ˆ 0; 1; g of thetbusy server process has the following properties:

(i) For each t ˆ 0; 1; , the rv b* is a Poisson rv with parameter lE‰sŠ.t(ii) The process is reversible in that

…b0*; b1*; ; b*† ˆt st…b*; bt t 1* ; ; b0*†; t ˆ 0; 1; : …9:10†

9.3 THE BUFFER SIZING PROBLEM

As we shall see shortly in Sections 9.4 and 9.5, M=G=1 input processes display anextremely rich correlation structure We expect these temporal correlations to have asigni®cant impact on queueing performance when such processes are offered to

a multiplexer Togain some insights intothis basic issue we map a multiplexer intoadiscrete-time single server queue with in®nite capacity and constant release rate of ccells=slot under the ®rst-come-®rst-served discipline The cell stream is modeled by

an M=G=1 input process as de®ned above, with bt‡1 representing the number ofnew cells that arrive at the start of time slot ‰t; t ‡ 1† Let qb

t denote the number ofcells remaining in the buffer by the end of slot ‰t 1; t†, sothat qb

t ‡ bt‡1 cells areready for transmission during slot ‰t; t ‡ 1† If the multiplexer output link cantransmit c cells=slot, then the buffer content sequence fqb

t; t ˆ 0; 1; g evolvesaccording to the Lindley recursion

qb

0ˆ q; qb

t‡1 ˆ ‰qb

t ‡ bt‡1 cŠ‡; t ˆ 0; 1; ; …9:11†for some initial condition q

Conditions under which the queueing system (9.11) admits a steady-state regimeare well known and are given next

Proposition 9.3.1 If lE‰sŠ < c, then there exists an R‡-valued rv qb

Stationary M=G=1 processes being time-reversible, we have the representation

qb

1ˆstsup…Sb

t ct; t ˆ 0; 1; † …9:12†for the steady-state buffer content qb

1with

Sb

0 ˆ 0; Sb

t ˆ b1* ‡


‡ bt*; t ˆ 1; 2; : …9:13†Hereafter, by an M=G=1 input process we mean its stationary version

fb*; t ˆ 0; 1; g, which is fully characterized by the pair …l; G† Moreover, fromtnow on, we always assume the stability condition

9.4 SECOND-ORDER CORRELATIONS

Before discussing the asymptotics associated with buffer over¯ow induced byM=G=1 input processes, we make a slight detour to explore the correlationstructure of such input processes

9.4.1 Correlation Properties

In view of Proposition 9.2.1, the stationary version fb*; t ˆ 0; 1; g has a well-tde®ned (auto)covariance function G: R ! R, say,

G…h†  Cov‰b*; bt t‡h* Š; t; h ˆ 0; 1; : …9:15†Proposition 9.4.1 We have

G…h† ˆ lE‰…s h†‡Š ˆ lE‰sŠP‰^s > hŠ; h ˆ 0; 1; : …9:16†The ®rst equality in Eq (9.16) is established in Cox and Isham [7] and the secondequality follows readily from the de®nition (9.9) From Eq (9.16) we ®nd theautocorrelation function g: R ! R of the M=G=1 process …l; s† tobe given by

g…h† G…h†

G…0†ˆ P‰^s > hŠ; h ˆ 0; 1; : …9:17†Note that g…0† ˆ 1 as we recall that P‰s > 0Š ˆ 1

9.4.2 Inverting g

Proposition 9.4.1 shows that the correlation structure of the stationary M=G=1input process …l; s† is completely determined by the pmf of ^s (thus of s) It turns out

that the inverse is true as well Indeed, Eqs (9.9) and (9.17) together imply

g…h† g…h ‡ 1† ˆ P‰^s > hŠ P‰^s > h ‡ 1Š

ˆE‰sŠ1 P‰s > hŠ; h ˆ 0; 1; ; …9:18†sothat the mapping h ! g…h† is necessarily decreasing and integer-convex Takingintoaccount the facts g…0† ˆ 1 and P‰s > 0Š ˆ 1, we conclude from Eq (9.18) (with

E‰sŠ ˆ P1

hˆ0P‰s > hŠ ˆ1 lim1 g…1†h!1g…h† …9:21†and Eq (9.19) imposes limh!1g…h† ˆ 0 A moment of re¯ection readily leads to thefollowing invertibility result

Proposition 9.4.2 An R‡-valued sequence fg…h†; h ˆ 0; 1; g is the relation function of the M=G=1 process …l; s† with integrable s if and only ifthe corresponding mapping h ! g…h† is decreasing and integer-convex with

autocor-g…0† ˆ 1 > g…1† and limh!1g…h† ˆ 0, in which case the pmf G of s is given by

Eq (9.20)

9.5 LONG-RANGE DEPENDENCE

The existence of positive correlations in the sequence fbt*; t ˆ 0; 1; g is clearlyapparent from Eq (9.16) The strength of such positive correlations can beformalized in several ways, which we now describe; additional material is available

in Cox [8] and we refer the reader to Tsybakov and Georganas [32] for a discussion

of alternative de®nitions

The sequence fb*; t ˆ 0; 1; g is said tobe short-range dependent (SRD) ift

P1

Otherwise, the sequence fbt*; t ˆ 0; 1; g is said tobe long-range dependent(LRD) Easy calculations using Eq (9.16) readily lead to the following simplecharacterization.

Proposition 9.5.1 We have

P1 hˆ0G…h† ˆl2E‰s…s ‡ 1†Š …9:23†

so that the process is SRD if and only if E‰s2Š is ®nite

Interesting subclasses of LRD processes can further be identi®ed through thenotion of second-order self-similarity To do so, we introduce the rvs

b…m†t m1m 1P

kˆ0b*mt‡k; m ˆ 1; 2; ; t ˆ 0; 1; : …9:24†For each m ˆ 1; , the rvs fb…m†t ; t ˆ 0; 1; g form a (wide-sense) stationarysequence with correlation structure de®ned by

G…m†…h†  Cov‰b…m†t ; b…m†t‡hŠ and g…m†…h† G…m†…h†

G…m†…0†; h ˆ 0; 1; : …9:25†For each H > 0 consider the mapping gH: R ! R‡ given by

gH…h† 1

2…jh ‡ 1j2H 2jhj2H‡ jh 1j2H†; h ˆ 0; 1; : …9:26†

We say that the sequence fbt*; t ˆ 0; 1; g is exactly (second-order) self-similar if

Var‰b…m†t Š ˆ d2m b; m ˆ 1; 2; …9:27†for some constants d2> 0 and 0 < b < 1, a requirement equivalent to

G…h† ˆ d2gH…h†; h ˆ 0; 1; ; …9:28†where H  1 b=2 is known as the Hurst parameter of the process The parameter

H being in the range (0.5, 1), the mapping gH is strictly decreasing and convex, with gH…0† ˆ 1, and behaves asymptotically as

integer-gH…h†  H…2H 1†h2H 2 …h ! 1†: …9:29†

By Proposition 9.4.2 we can interpret gH as the autocorrelation function of theM=G=1 input process …l; sH† with

P‰sH > rŠ ˆjr ‡ 2j2H 3jr ‡ 1j4…1 22H2H 2‡ 3jrj†2H jr 1j2H; r ˆ 1; 2; ;sothat the M=G=1 input process …l; sH† is exactly second-order self-similar withHurst parameter H

In applications, the notion of exact self-similarity is often too restrictive and isweakened as follows The sequence fbt*; t ˆ 0; 1; g is said tobe asymptotically(second-order) self-similar if

lim

m!1 g…m†…h† ˆ gH…h†; h ˆ 1; 2; : …9:30†This will happen for the M=G=1 input process …l; s† if

with 1 < a < 2, for some slowly varying function L: R‡! R‡, in which case

H ˆ …3 a†=2

9.6 GENERAL BUFFER ASYMPTOTICS

Several authors [10, 16, 19] have derived asymptotics such as (9.1) by means oflarge deviations estimates associated with the sequence ft 1…Sb

t ct†; t ˆ 0; 1; g.These results (and their necessary extensions) are summarized below as they apply tothe present context

9.6.1 A General Setup

With a given R-valued sequence fxt‡1; t ˆ 0; 1; g, we associate the R‡-valued rv

q1 given by

q1 sup…St; t ˆ 0; 1; †; …9:32†where

S0ˆ 0; Stˆ x1‡


‡ xt; t ˆ 1; 2; : …9:33†

If the sequence fxt‡1; t ˆ 0; 1; g is assumed stationary and ergodic withE‰x1Š < 0, then q1is a.s ®nite We are interested in characterizing the asymptoticbehavior of the tail probability P‰q1> bŠ for large b

To ®x the terminology, a scaling sequence is any monotone increasing R-valuedsequence fvt; t ˆ 0; 1; g such that limt!1vtˆ 1 The sequence ft 1St,

t ˆ 1; 2; g is said tosatisfy the large deviations principle under scaling vt ifthere exists a lower-semicontinuous function I : R ! ‰0; 1Š such that for everyopen set G,

 P Sdb=yedb=ye> y

…9:38†

of the lower bound As the best lower bound is the largest, we can immediatelysharpen (9.40) intothe lower bound (9.4) with g given by (9.37) jThe existence of (9.34) (and for that matter, of (9.35)) is typically validatedthrough the GaÈrtner±Ellis theorem [9, Theorem 2.3.6, p 45] In that context, for each

exists (possibly as an extended real number) Under broad conditions, the process

ft 1St; t ˆ 1; 2; g then satis®es the large deviations principle under scaling vtwith good rate function L*: R ! ‰0; 1Š, where L* is the Legendre±Fencheltransform of the mapping L: R ! … 1; 1Š de®ned through Eq (9.42), namely,

L*…z†  sup

Expression (9.37) simpli®es when the large deviations principle for the process

ft 1St; t ˆ 1; 2; g holds with a good rate function I : R ! ‰0; 1Š, which isconvex Indeed, the relation

inf

follows readily from the goodness of I and the fact that limt!1t 1Stˆ E‰x1Š < 0a.s under the ergodic assumption However, by convexity we have I increasing(resp decreasing) on …E‰x1Š; 1† (resp on … 1; E‰x1Š†, and the conclusioninfx>yI…x† ˆ I…y‡† holds for all y > 0 The interior of the effective domain of I

is an interval of the form …y; y*† with 1  y E‰x1Š  y*  1, and Eq (9.37)becomes

g1ˆ inf

y>0g…y†I…y‡† ˆ inf

0<y<y g…y†I…y†: …9:45†The nondegeneracy condition y* > 0 holds in most applications

9.6.3 An Upper Bound

In Duf®eld and O'Connell [10], the companion upper bound (9.5) was derived under

a set of conditions that, unfortunately, do not cover some instances of the M=G=1process considered here Upon re®ning the arguments of Duf®eld and O'Connell[10], we have established the following asymptotic upper bound; details are available

in Section 9.14 and in Parulekar [27] An alternative approach was given by Duf®eld[11] but more explicit expressions are given here for the upper bound

Proposition 9.6.2 Assume the following conditions:

(i) For each y in R, the limit (9.42) exists (possibly as an extended real number)with

inf

y>0L…y† 2 R: …9:46†(ii) For some ®nite K  0, we have

The case K ˆ 0 is equivalent toHypothesis 2.2(iv) in Duf®eld and O'Connell[10] Moreover, the upper bound is trivial in cases where L*…0†  K As the leastupper bound is the sharpest, under the assumptions of Proposition 9.6.2 weimmediately get Eq (9.5) with

g* ˆ sup

y>0…min…a…y†; b…y†††: …9:51†

9.7 EVALUATION OF L…u† …u 2 R†

An important step in applying the results of the previous section consists in ®nding ascaling sequence fvt; t ˆ 0; 1; g such that for each y in R, the limit (9.42) exists(possibly as an extended real number) In Parulekar and Makowski [28±30], weshow that the selection of this scaling is governed by the behavior of the sequence

fv*; t ˆ 0; 1; g given byt

v* ˆ ln P‰^s > tŠ; t ˆ 0; 1; :t …9:52†This is done under the assumption that the limit

R ˆ 1 (Case I), 0 < R < 1 (Case II), or R ˆ 0 (Case III)

To state the results more conveniently, we set

exists (possibly in®nite), so does (9.42) with

and it suf®ces to concentrate on ®nding the limit (9.55) The main facts along theselines are developed in the next two theorems; proofs are available in Parulekar [27]and Parulekar and Makowski [30] Cases I and II are covered ®rst

Theorem 9.7.1 Assume R > 0, possibly in®nite, and take the linear scaling

vtˆ t; t ˆ 1; 2; : …9:57†Then, for each y in R, the limit Lb…y†  limt!1Lb;t…y† exists and is given by

with E‰eysŠ ®nite (resp in®nite) if y < R (resp R < y)

We now turn to Case III

Theorem 9.7.2 Assume R ˆ 0 with fv*=t; t ˆ 1; 2; g eventually monotonetdecreasing Then, with the scaling

provided there exists a mapping G: R ! R such that (i) G…t† < t for large

t ˆ 1; 2; , (ii) limt!1vt*…G…t†=t† ˆ 1, and (iii) limt!1…v*=t†…G…t†=v*t G…t†† ˆ 0.The assumptions of Theorem 9.7.2 are satis®ed in all cases known to the authorsand are easy to check for broad classes of distributions If v*  tt b…0 < b < 1†, wecan take G…t† ˆ tg with 1 b < g < 1 If vt*  …ln t†b…b > 0†, then the choice

G…t† ˆ t…ln t† gwith 0 < g < b will do

9.8 EXPONENTIAL TAILS

By exponential tails we refer to the situation where

E‰eysŠ < 1 for some y > 0: …9:61†

It is easy to see [30, Section 6] that for each y in R, the quantities E‰eysŠ and E‰ey^sŠare simultaneously ®nite (resp in®nite) Moreover, with R > 0, possibly in®nite,these quantities are ®nite (resp in®nite) whenever y < R (resp y > R) Thus, underlimit (9.53), exponential tails correspond to Cases I and II

9.8.1 The Asymptotics Under Exponential Tails

When R > 0, Theorem 9.7.1 suggests vtˆ t, sothat h…b† ˆ b, g…y† ˆ y 1, and

K ˆ 0 Therefore,

and the GaÈrtner±Ellis theorem [9, Theorem 2.3.6, p 45] guarantees the full largedeviations principle (under scaling t) fo r ft 1…Sb

t ct†; t ˆ 1; 2; g with ratefunction L* From Eq (9.45) we readily conclude

Proposition 9.8.1 Assume R > 0 Then, the asymptotics

lim

b!1

1

bln P‰qb1> bŠ ˆ g …9:66†holds with gˆ g*  g

Proposition 9.8.1 covers the situations where the tail of G decays at leastexponentially fast, for example, the Rayleigh, Gamma, and geometric cases Thisresult is of course not new; it has been obtained earlier by several authors [10, 16,19] and paves the way to the notion of effective bandwidth However, the argumentsleading toit, which are detailed in Parulekar [27], already show that the upper bound

of Proposition 9.6.2 is good enough to recover this ``classical'' case

9.8.2 Comparison with Instantaneous Inputs

Another input process closely related to the M=G=1 input process …l; s† is the inputprocess according to which the work associated with a session is offered instanta-neously tothe buffer, rather than gradually as was the case for M=G=1 inputprocesses

Such an instantaneous input process, say, fat‡1; t ˆ 0; 1; g, is composed ofi.i.d R-valued rvs given by

at‡1 :ˆbPt‡1

iˆ1st‡1;i; t ˆ 0; 1; ; …9:67†where the twofamilies of i.i.d rvs fbt‡1; t ˆ 0; 1; g and fst‡1;i; t ˆ0; 1; ; i ˆ 1; 2; g are as in Section 9.2.1 Let a denote a generic rv of thei.i.d sequence fat‡1; t ˆ 0; 1; g When offered to the multiplexer described by

Eq (9.11), the instantaneous arrival process generates a sequence of buffer contents

a D=GI=1 queue Hence, the system will be stable if E‰aŠ < c, a condition equivalentto(9.14), in which case qa

t )t qa

1for some R‡-valued rv qa

1 Here as well, we areinterested in the tail probabilities P‰qa

1> bŠ for b large, and in comparing theseasymptotics with those of P‰q1> bŠ

To apply Propositions 9.6.1 and 9.6.2, it is natural to consider the partial sumsequence fSa

t; t ˆ 0; 1; g associated with the instantaneous input process

fat‡1; t ˆ 0; 1; g, namely,

Saˆ 0; Sa

t ˆ a1‡


‡ at; t ˆ 1; 2; ; …9:69†sothat

qa

1ˆstsup…Sa

t ct; t ˆ 0; 1; †: …9:70†Under the enforced independence assumptions, for each t ˆ 1; 2; we have

E‰eyS a

tŠ ˆQt

sˆ1E‰E‰eysŠbsŠ ˆ e lt…1 E‰e ys Š†; y 2 R; …9:71†sothat