6 THE SINGLE SERVER QUEUE: HEAVY TAILS AND HEAVY TRAFFIC O. J. B OXMA 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 J. W. C OHEN CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands 6.1 INTRODUCTION Recently, there has been much interest in the behavior of queues with heavy-tailed service time distributions. This interest has been triggered by a large number of traf®c measurements on modern communication network traf®c (e.g., see Willinger et al. [38] for Ethernet LAN traf®c, Paxson and Floyd [30] for WAN traf®c, and Beran 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-range dependence in an input traf®c process is to take a ¯uid queue and to assume that at least one of the input quantities I (on or off periods in the ¯uid queue fed by on=off sources) has the following ``heavy-tail'' behavior: PI > t$ t3I h n t Àn ; 6:1 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 with h n a positive constant and 1 < n < 2 (here and later, f t$ t3I gt stands for ft=gt31 with t 3I; and many-valued functions like t Àn are de®ned by their principal value, so t Àn is real for t positive). In this context, regularly varying and subexponential distributions [5] have received special attention. We refer to Boxma and 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 with regularly varying service and=or interarrival time distribution. The ¯uid queue is closely related to this ordinary queue (cf. Remarks 6.2.6 and 6.6.3), so that several of the 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 À GxPG > x is said to be regularly varying at in®nity of index Àn when 1 À Gxx Àn Lx; x ! 0; 6:2 with Lx a slowly varying function at in®nity, that is, lim x3I Ltx Lx  1; Vt > 0: LÁ could, for instance, be a constant or a logarithmic function. If Eq. (6.2) holds, then we write PG > Á P Àn. Regular variation is an important asymptotic concept in probabilistic analysis. The main reference text is the book by Bingham et al. [5]. An early result concerning regular variation in queueing theory is due to Cohen [13]. He proved the following result for the GI=G=1 queue under the ®rst-come-®rst- served (FCFS) discipline: the tail of the waiting time distribution is regularly varying of index 1 À n if and only if the tail of the service time distribution is regularly varying of index Àn; n > 1. This result has turned out to be very useful in relating the regularly varying tail behavior of on periods of on=off sources in ¯uid queues to the 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 waiting time 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 the waiting time distribution is in®nite. The ®rst issue that we consider in this chapter is whether other service disciplines besides FCFS may lead to a less detrimental waiting time behavior. We consider the M=G=1 queue with the processor sharing (PS) discipline and the M=G=1 queue with the last-come-®rst-served preemptive resume (LCFS-PR) discipline. Note that PS and LCFS-PR are well-known and important disciplines, that both play a key role in product-form networks. For both disciplines, the waiting time tail behavior turns out to be regularly varying of index Àn iff the service time tail behavior is regularly varying of index Àn. We refer to Anantharam [2] for a related investigation of the in¯uence of the service discipline on the tail behavior of a single server queue. 144 THE SINGLE SERVER QUEUE: HEAVY TAILS AND HEAVY TRAFFIC The 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 3 1. This issue is of theoretical interest, since in the traditional heavy traf®c limit theorems it is assumed that the second moments 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 in situations with a reasonably light traf®c load. We ®rst focus on the M=G=1 queue, again with the three service disciplines FCFS, PS, and LCFS-PR. Subse- quently, we also allow interarrival times to be generally distributed and even heavy tailed. New heavy traf®c limit results are presented for the waiting time in the GI=G=1 queue with heavy-tailed interarrival and=or service time distribution. The case in which both tails are ``just as heavy'' is particularly interesting from a mathematical 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 4 1. The third issue under consideration is the convergence of the workload process fv t ; t ! 0g in the GI =G=1 queue with heavy-tailed interarrival and=or service time distribution. It is shown that Drv t=1ÀrDr converges in distribution for r 4 1, for all t ! 0. The thus scaled and contracted workload process converges weakly to the workload process of a queueing model of which the input process is described by n- stable Le  vy motion. This chapter is organized in the following way. In Section 6.2 we discuss the relation between the tail behavior of service times and waiting times, for the M=G=1 queue with service disciplines FCFS, PS, and LCFS-PR. In Section 6.3, for the same three M =G=1 variants, we present heavy traf®c limit theorems for waiting times, in the case of a regularly varying service time distribution with an in®nite variance. In the LCFS-PR case, the sojourn time distribution coincides with the M=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 in Section 6.4. Heavy traf®c limit theorems for the waiting time in the GI=G=1queue with the FCFS service discipline are presented in Section 6.5. A distinction is made between the cases in which the interarrival time distribution has a heavier tail, the service time distribution has a heavier tail, and both tails are ``just as heavy.'' In Section 6.6 the input process and the workload process are studied for the GI=G=1 queue with heavy-tailed interarrival and=or service time distribution. The heavy traf®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 this chapter have appeared in recent reports of the authors and their colleagues; in those cases, proofs are generally omitted. Also, several chapters in this volume are related to the present work. We mention in particular the contributions of Jelenkovic (Chapter 10), of Likhanov (Chapter 8), of Makowski and Parulekar (Chapter 9), of Norros (Chapter 4), of Brichet et al. (Chapter 5), and of Resnick and Samorodnitsky (Chapter 7).) 6.1 INTRODUCTION 145 6.2 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 queue with the FCFS discipline; it shows that the waiting time is ``one degree heavier'' than the 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. A similar 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 processor sharing, 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 the LCFS-PR discipline. In the present section we ®rst introduce some notation, and we present a very useful lemma that relates the tail behavior of a regularly varying probability 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 process with rate l; their service times B 1 ; B 2 ; . are i.i.d. (independent, identically distributed) random variables with ®nite mean b and LST bfsg. A generic service time is denoted by B.ByB* we denote a random variable of which the distribution is that of a residual service time: PB* > x 1 b  I x PB > u du; x ! 0: Its LST is given by b*fsg :1 À bfsg=bs. The traf®c load r : lb of the M=G=1 queue is assumed to be less than one, so that the steady-state waiting time distribution exists. A very useful property of probability distributions with regularly varying tails is a characterization of the behavior of its LST near the origin. Let FÁ be the distribution of a nonnegative random variable, with LST ffsg and ®nite ®rst n moments m 1 ; .; m n (and m 0  1). De®ne f n fsg :À1 n1 ffsgÀ P n j0 m j Às j j! "# : Lemma 6.2.1. Let n < n < n  1, C ! 0. The following statements are equiva- lent: f n fsgC  o1s n L1=s; s 5 0; s real; 6:3 1 À FtC  o1 À1 n G1 À n t Àn Lt; t 3I: 6:4 146 THE SINGLE SERVER QUEUE: HEAVY TAILS AND HEAVY TRAFFIC The case C > 0 is due to Bingham and Doney [4]. The case C  0 is treated in Boxma 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 FCFS discipline in full generality. There is no need to specify the interarrival time distribution; 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. Fo r r < 1 and n > 1, PB > x$ x3I n À 1 x b  Àn Lx@APW > x$ x3I r 1 À r x b  1Àn Lx: 6:5 Pakes [29] has extended this result to the larger class  of subexponential distributions (i.i.d. stochastic variables X 1 and X 2 have a subexponential tail if PX 1  X 2 > t=PX 1 > t$ t3I 2. His result states that PW > Á P  if and only if PB* > Á P , and if either is the case then PW > x$ x3I r 1 À r PB* > x: 6:6 R EMARK 6.2.3. Note that the interarrival time distribution has no in¯uence on the above-mentioned waiting time tail behavior. R EMARK 6.2.4. In the case of Poisson arrivals, the Pollaczek±Khintchine formula reads: Ee ÀsW  1 À r 1 À rb*fsg ; Re s ! 0; 6:7 and PW > x P I n1 1 À rr n PB 1 * ÁÁÁB n * > x; x ! 0: 6:8 Theorem 6.2.2 and Eq. (6.6) can be veri®ed very easily in the M =G=1 case. For example, the above-mentioned de®nition of subexponentiality implies that PB 1 * ÁÁÁB n * > x$ x3I nPB 1 * > x; and combination of Lemma 6.2.1 and Eq. (6.7) (that relates the LSTs of W and B*) readily yields the relation between the tail behavior of the distributions of W and B. 6.2 WAITING TIME TAIL BEHAVIOR 147 The possibility of having to wait at least a residual service time explains that the waiting time tail is one degree heavier than the service time tail: PB > Á P Àn implies that PB* > Á P 1 À n (cf. Bingham et al. [5]; obviously, integration increases the index of regular variation by one). In the next subsections we shall consider two service disciplines in which an arriving customer does not ®rst have to wait a residual service time, and where the waiting time tail is just as heavy as the service time tail. R EMARK 6.2.5. Denote the steady-state sojourn time by S; in the GI =G=1 FCFS queue, S 9 W  B, where W and B are independent and where 9 denotes equality in distribution. Hence PS > x$ x3I r 1 À r PB* > x: 6:9 Denote the steady-state workload by V ; in the GI =G=1 FCFS queue (cf. Cohen [15, p. 296]), Ee ÀsV 1 À r rb*fsgEe ÀsW ; Re s ! 0: 6:10 Hence, using Lemma 6.2.1, PB > Á P Àn@APV > Á P 1 À n. R EMARK 6.2.6. The goal of this remark is to indicate how results like Theorem 6.2.2 can readily be translated into results for ¯uid queues. Consider a ¯uid queue with an in®nite buffer and an output rate equal to one. This queue is fed by a source that alternates between silence periods s n , n ! 1, during which it generates no input, and activity periods a n , n ! 1, during which it generates ¯uid according to the rate process r n t t!0 , n ! 1 (this could be the aggregated input process due to several on=off sources). A crucial assumption is that Vn ! 1;Vt ! 0: 1 r n t R where R is some given constant: 6:11 In particular, the nth activity period results in a net input equal to ^ B n :  a n 0 r n tÀ1 dt: 6:12 Assume that s n  n!1 and, similarly, a n ; r n t; t ! 0 n!1 are i.i.d., and that these sequences are independent. The buffer content W b n at the beginning of the nth activity period satis®es the recursion W b n1  max0; W b n  ^ B n À s n ; n ! 1: 6:13 This is exactly the GI =G=1 Lindley waiting time recursion (identify W b n with W n , ^ B n with B n , and s n with the nth interarrival interval). Hence Theorem 6.2.2 also applies 148 THE SINGLE SERVER QUEUE: HEAVY TAILS AND HEAVY TRAFFIC to this ¯uid queue. Similar results for the tail behavior of the steady-state buffer content 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 queues fed 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 simulta- neously being served with rate 1=X , where X is the number of customers in the system. An extensive survey of processor sharing results is presented in Yashkov [39]. Let S PS denote the steady-state sojourn time in the M =G=1 PS queue. In Zwart and Boxma [42] the following result is proved. Theorem 6.2.7. Fo r n > 1 (n noninteger), PB > x$ x3I x Àn Lx@APS PS > x$ x3I 1 1 À r n x Àn Lx: 6:14 Both relations imply that, in the case of regular variation, PS PS > x$ x3I PB > 1 À rx: 6:15 Theorem 6.2.7 is proved by deriving a new expression for the LST of the sojourn time distribution, and applying Lemma 6.2.1. An interpretation of the factor 1 À r in Eq. (6.15) is the following. When a tagged customer is in the system for a long time, then the distribution of the total number of customers is approximately equal to the steady-state distribution of the number of customers in a PS queue with one permanent customer. The latter model is a special case of the M =G=1 queue with the generalized processor sharing discipline, as studied in Cohen [14]. It follows from the results in Cohen [14] that, in steady state, the mean fraction of service given to the permanent customer in that M =G=1 queue is 1 À r. Hence, if a tagged customer has been in the M =G=1 PS queue during a large time x, one would expect that the amount of service received is approximately equal to 1 À rx. 6.2.4 The M=G=1 LCFS-PR Queue In the LCFS preemptive resume discipline, an arriving customer K is immediately taken into service. However, this service is interrupted when another customer arrives, and it is only resumed when all customers who have arrived after K have left the system. Hence, the sojourn time of K has the same distribution as the busy period of this M =G=1 queue. The tail behavior of the busy period distribution in the M=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 thus immediately leads to the following result. Let S L denote the steady-state sojourn time in the M=G=1 LCFS-PR queue. Theorem 6.2.8. Fo r n > 1, PB > x$ x3I x Àn Lx@APS L > x$ x3I 1 1 À r n1 x Àn Lx: 6:16 Both relations imply that, in the case of regular variation, PS L > x$ x3I 1 1 À r PB > 1 À rx: 6:17 R EMARK 6.2.9. Theorems 6.2.7 and 6.2.8 show that in both PS and LCFS-PR, regular variation of the service time distribution gives rise to regular variation of the sojourn time distribution, of the same index. In LCFS±nonpreemptive, it is easily seen that regular variation of the service time distribution gives rise to regular variation of the waiting time distribution of one index higher ( just like FCFS). This result is obtained by applying Lemma 6.2.1 to the waiting time LST (for this LST, see, e.g., Cohen [15, (III.3.10)]). The increment of the index is not surprising, sinceÐas in FCFSÐa waiting time may include a residual service time. 6.3 HEAVY TRAFFIC LIMIT THEOREMS FOR THE M =G=1 QUEUE 6.3.1 Introduction When the variance s 2 of the service time distribution is ®nite, the standard heavy traf®c limit theorem for the stationary waiting time W in the M =G=1 queue holds, that is, lim r41 PDrW t1À e Àt ; t ! 0; 6:18 with Dr : l1 À r=1  l 2 s 2 =2. This exponential heavy traf®c theorem was obtained by Kingman in the early 1960s; see Kingman [25] for an early survey, and Whitt [36] for an extensive overview of heavy traf®c limit theorems for queues. In Boxma and Cohen [10] (see also Cohen [19]) a heavy traf®c limit theorem has been proved for the GI =G=1 FCFS queue in which the service time and=or interarrival time 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 the M=G=1 PS case is presented in Section 6.3.3. Section 6.3.4 contains a new heavy traf®c limit theorem for the waiting time distribution in the M=G=1 LCFS-PR queue with a heavy-tailed regularly varying service time distribution. 150 THE SINGLE SERVER QUEUE: HEAVY TAILS AND HEAVY TRAFFIC 6.3.2 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 service time distribution of index Àn, as speci®ed in Eq. (6.2), the ``contracted'' waiting time D W rW =b converges in distribution for r 4 1. The limiting distribution R nÀ1 t is speci®ed by its LST 1=1  r nÀ1 ,Rer ! 0, and the coef®cient of contraction D W r is the root of the equation r G2 À n n À 1 x nÀ1 Lx1 À r; x > 0; 0 < 1 À r ( 1; 6:19 with the property that D W r50 for r 4 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 we sketch a different proof. Using the Pollaczek±Khintchine formula (6.7) and Eq. (6.19) we can write, for Re s ! 0, Ee ÀsW  1 À r 1 À rb*fsg  1 1  r 1 À r 1 À b*fsg  1 1  1 À b*fsg G2 À n n À 1 D W r nÀ1 LD W r : 6:20 Now replace s by rD W r,Rer ! 0. Since D W r50 for r 4 1, it follows from Lemma 6.2.1 that 1 À b*frD W rg $ G2 À n n À 1 r nÀ1 D W r nÀ1 LrD W r for r 4 1: Hence, using the de®ning property of a slowly varying function, lim r41 Ee ÀrD W rW lim r41 1 1  1 À b*frD W rg G2 À n n À 1 D W r nÀ1 LD W r  1 1  r nÀ1 : 6:21 j R EMARK 6.3.2. In Boxma and Cohen [10] it is shown that Theorem 6.3.1 also holds for n  2. Furthermore, R nÀ1 t is discussed in detail in Boxma and Cohen 6.3 HEAVY TRAFFIC LIMIT THEOREMS FOR THE M =G=1 QUEUE 151 [10]. Its relation to the Mittag±Lef¯er function and (for n  3 2 ) to the complementary error function is pointed out. One can write, for t ! 0, 1 À R nÀ1 t P I n0 À1 n t nnÀ1 Gnn À 11 : R EMARK 6.3.3. Theorem 6.3.1 opens possibilities for approximating the waiting time distribution in the M =G=1 queue with heavy-tailed service time distribution; such possibilities are explored in Boxma and Cohen [9]. Replacing PW > x by 1 À R nÀ1 D W rx=b appears to yield remarkably accurate results, even if the traf®c load 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 ever waits, it is natural to concentrate on the sojourn time distribution rather than the waiting time distribution. In Zwart and Boxma [42] a novel expression has been derived for the LST vfs; tgEe ÀsS PS t  in the M=G=1 PS queue; here S PS t is the sojourn time of a customer with service request t. This expression leads to a new and easy proof [42] of the following M =G=1 PS heavy traf®c limit theorem of Sengupta [34] and of Yashkov [40]. Theorem 6.3.4. If b < I, then lim r41 vfs1 À r; tg 1 1  st ; Re s ! 0; t ! 0; 6:22 lim r41 P1 À rS PS t x1 À e Àx=t ; x ! 0; t ! 0: 6:23 Since vfs1 À r; tg 1, we have by dominated convergence and Theorem 6.3.4 the following heavy traf®c limit for the unconditional sojourn time distribution with LST vfÁg (the last statement of the theorem follows by observing that 1=1  st is the LST of the negative exponential distribution). Theorem 6.3.5. Fo r Re s ! 0, lim r41 vfs1 À rg   I 0 1 1  st dPB < t  I 0 e Àx bfsxg dx: 6:24 In Zwart and Boxma [42] the convergence of the moments of the sojourn time in heavy traf®c is also studied. The moments of the ``contracted'' sojourn time are shown to converge to the corresponding moments of the heavy traf®c limiting distribution. In particular, we have the following theorem. 152 THE SINGLE SERVER QUEUE: HEAVY TAILS AND HEAVY TRAFFIC