New heavy traf®c limit results are presented for the waiting time in theGI=G=1 queue with heavy-tailed interarrival and=or service time distribution.. In Section 6.3, for thesame three M
Trang 1THE SINGLE SERVER QUEUE:
HEAVY TAILS AND HEAVY TRAFFIC
O J BOXMA
Department of Mathematics and Computing Science,
Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands;
and CWI, P.O Box 94079, 1090 GB Amsterdam, The Netherlands
et al [38] for Ethernet LAN traf®c, Paxson and Floyd [30] for WAN traf®c, andBeran et al [3] for VBR video traf®c; see also various chapters in this volume).These measurements and their statistical analysis (e.g., see Leland et al [26])suggest that modern communication traf®c often possesses the properties of self-similarity and long-range dependence A natural possibility to introduce long-rangedependence in an input traf®c process is to take a ¯uid queue and to assume that atleast one of the input quantities I (on or off periods in the ¯uid queue fed by on=offsources) has the following ``heavy-tail'' behavior:
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.
143
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
Trang 2with hn a positive constant and 1 < n < 2 (here and later, f t t!1g t stands for
f t=g t ! 1 with t ! 1; and many-valued functions like t nare de®ned by theirprincipal value, so t n is real for t positive) In this context, regularly varying andsubexponential distributions [5] have received special attention We refer to Boxmaand Dumas [12] for a survey on ¯uid queues with heavy-tailed on-period distribu-tions In this chapter, we concentrate on the ordinary single server queue withregularly varying service and=or interarrival time distribution The ¯uid queue isclosely related to this ordinary queue (cf Remarks 6.2.6 and 6.6.3), so that several ofthe results of the present chapter will also be relevant for ¯uid queues
When G x is the probability distribution of a nonnegative random variable G,then 1 G x PG > x is said to be regularly varying at in®nity of index nwhen
of index 1 n if and only if the tail of the service time distribution is regularlyvarying of index n; n > 1 This result has turned out to be very useful in relatingthe regularly varying tail behavior of on periods of on=off sources in ¯uid queues tothe buffer content [7, 8]; see also Jelenkovic and Lazar [23] and Rolski et al [32].Cohen's result implies that, in the case of regular variation, the tail of the waitingtime distribution is one degree heavier than that of the service time distribution For
1 < n < 2, the mean of the service time distribution is ®nite, but the mean of thewaiting time distribution is in®nite
The ®rst issue that we consider in this chapter is whether other service disciplinesbesides FCFS may lead to a less detrimental waiting time behavior We consider theM=G=1 queue with the processor sharing (PS) discipline and the M=G=1 queue withthe last-come-®rst-served preemptive resume (LCFS-PR) discipline Note that PSand LCFS-PR are well-known and important disciplines, that both play a key role inproduct-form networks For both disciplines, the waiting time tail behavior turns out
to be regularly varying of index n iff the service time tail behavior is regularlyvarying of index n We refer to Anantharam [2] for a related investigation of thein¯uence of the service discipline on the tail behavior of a single server queue
Trang 3The second issue under consideration in this chapter is the heavy traf®c behavior
of a queue with heavy-tailed service time distribution A queueing system is said to
be in heavy traf®c when its traf®c load r ! 1 This issue is of theoretical interest,since in the traditional heavy traf®c limit theorems it is assumed that the secondmoments of service and interarrival times are ®nite, whereas Eq (6.1) with
1 < n < 2 leads to an in®nite second moment The issue is also of practical interest,since heavy traf®c limit theorems may give rise to useful approximations insituations with a reasonably light traf®c load We ®rst focus on the M=G=1queue, again with the three service disciplines FCFS, PS, and LCFS-PR Subse-quently, we also allow interarrival times to be generally distributed and even heavytailed New heavy traf®c limit results are presented for the waiting time in theGI=G=1 queue with heavy-tailed interarrival and=or service time distribution Thecase in which both tails are ``just as heavy'' is particularly interesting from amathematical point of view We identify coef®cients of contraction D r such that
D r times the waiting time has a proper limiting distribution for r " 1
The third issue under consideration is the convergence of the workload process
fvt; t 0g in the GI=G=1 queue with heavy-tailed interarrival and=or service timedistribution It is shown that D rvt= 1 rD rconverges in distribution for r " 1, forall t 0 The thus scaled and contracted workload process converges weakly to theworkload process of a queueing model of which the input process is described by n-stable LeÂvy motion
This chapter is organized in the following way In Section 6.2 we discuss therelation between the tail behavior of service times and waiting times, for the M=G=1queue with service disciplines FCFS, PS, and LCFS-PR In Section 6.3, for thesame three M=G=1 variants, we present heavy traf®c limit theorems for waitingtimes, in the case of a regularly varying service time distribution with an in®nitevariance In the LCFS-PR case, the sojourn time distribution coincides with theM=G=1 busy period distribution The heavy traf®c behavior of the latter distribution
is investigated in detail; the tail behavior of the limiting distribution is studied inSection 6.4 Heavy traf®c limit theorems for the waiting time in the GI=G=1 q ueuewith the FCFS service discipline are presented in Section 6.5 A distinction is madebetween the cases in which the interarrival time distribution has a heavier tail, theservice time distribution has a heavier tail, and both tails are ``just as heavy.'' InSection 6.6 the input process and the workload process are studied for the GI=G=1queue with heavy-tailed interarrival and=or service time distribution The heavytraf®c behavior of those processes is characterized Section 6.7 contains conclu-sions and some topics for further research (Note: Several of the results in thischapter have appeared in recent reports of the authors and their colleagues; in thosecases, proofs are generally omitted Also, several chapters in this volume are related
to the present work We mention in particular the contributions of JelenkovicÂ(Chapter 10), of Likhanov (Chapter 8), of Makowski and Parulekar (Chapter 9), ofNorros (Chapter 4), of Brichet et al (Chapter 5), and of Resnick and Samorodnitsky(Chapter 7).)
6.1 INTRODUCTION 145
Trang 46.2 WAITING TIME TAIL BEHAVIOR
6.2.1 Introduction
In Section 6.1 we have already mentioned a result of Cohen [13] that relates the(regularly varying) tail behavior of service and waiting times in the GI=G=1 queuewith the FCFS discipline; it shows that the waiting time is ``one degree heavier'' thanthe service time tail, in the case of regular variation In Section 6.2.2 this result, and
an extension, will be discussed in some more detail for the M=G=1 FCFS queue Asimilar result for the M=G=1 queue with the PS discipline (due to Zwart and Boxma[42]) will be discussed in Section 6.2.3 That result shows that, under processorsharing, the waiting time tail is just as heavy as the service time tail In Section 6.2.4
we prove that the latter phenomenon also occurs in the M=G=1 queue with theLCFS-PR discipline In the present section we ®rst introduce some notation, and wepresent a very useful lemma that relates the tail behavior of a regularly varyingprobability distribution and the behavior of its Laplace±Stieltjes transform (LST)near the origin
Consider the M=G=1 queue Customers arrive according to a Poisson processwith rate l; their service times B1; B2; are i.i.d (independent, identicallydistributed) random variables with ®nite mean b and LST bfsg A generic servicetime is denoted by B By B* we denote a random variable of which the distribution isthat of a residual service time:
A very useful property of probability distributions with regularly varying tails is acharacterization of the behavior of its LST near the origin Let F be thedistribution of a nonnegative random variable, with LST ffsg and ®nite ®rst nmoments m1; ; mn (and m0 1) De®ne
Trang 5The case C > 0 is due to Bingham and Doney [4] The case C 0 is treated inBoxma and Dumas [12, Lemma 2.2] The case of an integer n is more complicated;see Bingham et al [5, Theorem 8.1.6 and Chap 3].
6.2.2 The M=G=1 FCFS Queue
We ®rst formulate the main result of Cohen [13] for the GI=G=1 queue with FCFSdiscipline in full generality There is no need to specify the interarrival timedistribution; the mean interarrival time and traf®c load are denoted by 1=l and
r lb (as before) In what follows, W denotes the steady-state waiting time.Theorem 6.2.2 For r < 1 and n > 1,
6.2 WAITING TIME TAIL BEHAVIOR 147
Trang 6The possibility of having to wait at least a residual service time explains that thewaiting time tail is one degree heavier than the service time tail: PB > 2 R nimplies that PB* > 2 R 1 n (cf Bingham et al [5]; obviously, integrationincreases the index of regular variation by one).
In the next subsections we shall consider two service disciplines in which anarriving customer does not ®rst have to wait a residual service time, and where thewaiting time tail is just as heavy as the service time tail
REMARK6.2.5 Denote the steady-state sojourn time by S; in the GI=G=1 FCFSqueue, S ' W B, where W and B are independent and where ' denotes equality
in distribution Hence
PS > x x!1 1 rr PB* > x: 6:9Denote the steady-state workload by V; in the GI=G=1 FCFS queue (cf Cohen [15,
p 296]),
Ee sV 1 r rb*fsgEe sW; Re s 0: 6:10Hence, using Lemma 6.2.1, PB > 2 R n () PV > 2 R 1 n
REMARK6.2.6 The goal of this remark is to indicate how results like Theorem6.2.2 can readily be translated into results for ¯uid queues Consider a ¯uid queuewith an in®nite buffer and an output rate equal to one This queue is fed by a sourcethat alternates between silence periods sn, n 1, during which it generates no input,and activity periods an, n 1, during which it generates ¯uid according to the rateprocess rn tt0, n 1 (this could be the aggregated input process due to severalon=off sources) A crucial assumption is that
8n 1; 8t 0: 1 rn t R where R is some given constant: 6:11
In particular, the nth activity period results in a net input equal to
Wb n1 max0; Wb
This is exactly the GI=G=1 Lindley waiting time recursion (identify Wbwith Wn, ^Bnwith Bn, and snwith the nth interarrival interval) Hence Theorem 6.2.2 also applies
Trang 7to this ¯uid queue Similar results for the tail behavior of the steady-state buffercontent can be proved, using Kella and Whitt [24] The survey by Boxma and Dumas[12] contains an overview of the tail behavior of the buffer content in ¯uid queuesfed by on=off sources, in which at least one on-period distribution is heavy tailed.
6.2.3 The M=G=1 PS Queue
In the (egalitarian) processor sharing service discipline, every customer is neously being served with rate 1=X , where X is the number of customers in thesystem An extensive survey of processor sharing results is presented in Yashkov[39] Let SPSdenote the steady-state sojourn time in the M=G=1 PS queue In Zwartand Boxma [42] the following result is proved
simulta-Theorem 6.2.7 For n > 1 (n noninteger),
PB > x x!1x nL x () PSPS> x x!1 1
1 rnx nL x: 6:14Both relations imply that, in the case of regular variation,
PSPS> x x!1PB > 1 rx: 6:15Theorem 6.2.7 is proved by deriving a new expression for the LST of the sojourntime distribution, and applying Lemma 6.2.1 An interpretation of the factor 1 r
in Eq (6.15) is the following When a tagged customer is in the system for a longtime, then the distribution of the total number of customers is approximately equal tothe steady-state distribution of the number of customers in a PS queue with onepermanent customer The latter model is a special case of the M=G=1 queue with thegeneralized processor sharing discipline, as studied in Cohen [14] It follows fromthe results in Cohen [14] that, in steady state, the mean fraction of service given tothe permanent customer in that M=G=1 queue is 1 r Hence, if a tagged customerhas been in the M=G=1 PS queue during a large time x, one would expect that theamount of service received is approximately equal to 1 rx
6.2.4 The M=G=1 LCFS-PR Queue
In the LCFS preemptive resume discipline, an arriving customer K is immediatelytaken into service However, this service is interrupted when another customerarrives, and it is only resumed when all customers who have arrived after K have leftthe system Hence, the sojourn time of K has the same distribution as the busyperiod of this M=G=1 queue The tail behavior of the busy period distribution in theM=G=1 queue (which distribution obviously is independent of the service discipline,when the discipline is work conserving) has been studied by De Meyer and Teugels[27] for the case of a regularly varying service time distribution Their main theorem
6.2 WAITING TIME TAIL BEHAVIOR 149
Trang 8thus immediately leads to the following result Let SLdenote the steady-state sojourntime in the M=G=1 LCFS-PR queue.
6.3 HEAVY TRAFFIC LIMIT THEOREMS FOR THE M=G=1 QUEUE6.3.1 Introduction
When the variance s2 of the service time distribution is ®nite, the standard heavytraf®c limit theorem for the stationary waiting time W in the M=G=1 queue holds,that is,
lim
with D r : l 1 r= 1 l2s2=2 This exponential heavy traf®c theorem wasobtained by Kingman in the early 1960s; see Kingman [25] for an early survey, andWhitt [36] for an extensive overview of heavy traf®c limit theorems for queues InBoxma and Cohen [10] (see also Cohen [19]) a heavy traf®c limit theorem has beenproved for the GI=G=1 FCFS queue in which the service time and=or interarrivaltime distribution is regularly varying, with in®nite second moment In Section 6.3.2
we discuss that result for the case of the M=G=1 FCFS queue; in Section 6.5 the case
of the GI=G=1 FCFS queue will be treated The known heavy traf®c result for theM=G=1 PS case is presented in Section 6.3.3 Section 6.3.4 contains a new heavytraf®c limit theorem for the waiting time distribution in the M=G=1 LCFS-PR queuewith a heavy-tailed regularly varying service time distribution
Trang 96.3.2 The M=G=1 FCFS Queue
In Boxma and Cohen [10] the following result has been proved:
Theorem 6.3.1 For the stable M=G=1 FCFS queue with regularly varying servicetime distribution of index n, as speci®ed in Eq (6.2), the ``contracted'' waiting time
DW rW=b converges in distribution for r " 1 The limiting distribution Rn 1 t isspeci®ed by its LST 1= 1 rn 1, Re r 0, and the coef®cient of contraction DW r
is the root of the equation
r G 2 n
n 1 xn 1L x 1 r; x > 0; 0 < 1 r 1; 6:19with the property that DW r # 0 for r " 1
Proof We refer to Boxma and Cohen [10] for a detailed proof of the theorem,albeit under slightly weaker conditions on the service time distribution Here wesketch a different proof Using the Pollaczek±Khintchine formula (6.7) and Eq.(6.19) we can write, for Re s 0,
Trang 10[10] Its relation to the Mittag±Lef¯er function and (for n 3
2) to the complementaryerror function is pointed out One can write, for t 0,
1 Rn 1 DW rx=b appears to yield remarkably accurate results, even if the traf®cload r is much smaller than one (in particular when x is large)
6.3.3 The M=G=1 PS Queue
In view of the fact that, under the processor sharing discipline, no customer everwaits, it is natural to concentrate on the sojourn time distribution rather than thewaiting time distribution In Zwart and Boxma [42] a novel expression has beenderived for the LST vfs; tg Ee sS PS t in the M=G=1 PS queue; here SPS t is thesojourn time of a customer with service request t This expression leads to a new andeasy proof [42] of the following M=G=1 PS heavy traf®c limit theorem of Sengupta[34] and of Yashkov [40]
Trang 11v > 0 Let P1; P2; be i.i.d stochastic variables with distribution that of the busyperiod of the M=G=1 queue Let K v be the number of arrivals in an interval oflength v, as de®ned above Then P 0 0 and
Trang 12On the other hand, since P V can be viewed as the time to empty the system whenstarting in steady state, we can write
It may be shown [20] that these equations have one and only one solution for
Re s 0 with k 0 1; and k s is regular for Re s > 0, continuous for Re s 0,and decreasing in s for s real Using Theorem 6.3.1, the heavy traf®c limit theoremfor the waiting time W in the M=G=1 FCFS queue (or equivalently for the workloadV), one can now prove the following heavy traf®c limit theorem for P V, orequivalently, see Eq (6.29), for P*
Theorem 6.3.7 For the stable M=G=1 queue with regularly varying service timedistribution of index n, as speci®ed in Eq (6.2), the ``contracted'' busy period
DP rP V converges in distribution for r " 1 The limiting distribution Pn 1 t isspeci®ed by the fact that its LST p r is that zero of the equation
with the property that y 0 1; and the coef®cient of contraction DP r equals
1 rDW r The same results hold for DP rP*, the ``contracted'' residual busyperiod
REMARK6.3.8 In Cohen [20] it is shown that Theorem 6.3.7 also holds for n 2
REMARK6.3.9 It follows from Eqs (6.27) and (6.29), with s rDP r, Re r 0,and v > 0, that
Trang 13In view of the just established limiting behavior of DP rP V and the fact that
DW r ! 0 for r " 1, it follows that the left-hand side of Eq (6.33) tends to onewhen r " 1, for all ®nite v However, taking v u=DW r, apparently
lim
r"1 Ee rD P rP u=D W r e rp ru; Re r 0; u > 0: 6:34Consequently, DP rP u=DW r converges in distribution for r " 1, and the LST ofthe (nondegenerate) limiting distribution equals exp rp ru Note that its meanequals u Expression (6.34) also occurs in a study of Prabhu [31], as entrance timefrom u in 0 of an in®nitely divisible process with only positive jumps; its appearance
as a heavy traf®c limit result for M=G=1 is new to the best of the authors'sknowledge
REMARK6.3.10 Since the sojourn time SLin the M=G=1 LCFS-PR queue has thesame distribution as the busy period, Theorem 6.3.7 also holds for the residualsojourn time SL*
1
D P rtPP > x dx;one can, via differentiation, obtain a heavy traf®c limit theorem for P and SL Weshall not go into detail here See Ott [28] and Abate and Whitt [1] for such heavytraf®c results in the case of ®nite service time variance
6.4 LCFS-PR: ASYMPTOTICS FOR THE HEAVY TRAFFIC
LIMITING DISTRIBUTION
The main goal of this section is to derive an asymptotic series, for t ! 1, of thelimiting distribution Pn 1 t that occurs in Theorem 6.3.7 In the previous section wehave seen that p r, the LST of this limiting distribution, is the unique solution of Eq.(6.32) with y 0 1 A picture readily shows that, for 1 < n < 2,
has only one real root for s > 0, and that this root is positive for s > 0, equal to 1 for
s 0, and bounded by 1 Introduce Y s : y s=b1= n 1 To investigate thesingularities of this solution of Eq (6.35), note that