9 BUFFER ASYMPTOTICS FOR M=G= INPUT PROCESSES A RMAND M. M AKOWSKI AND M INOTHI P ARULEKAR 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®c models do not account for time dependencies observed at multiple time scales in a wide 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 the resulting temporal correlations are expected to have a signi®cant impact on buffer engineering practices, this ``failure of Poisson modeling'' has generated an increased interest 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 have exposed clearly the limitations of traditional traf®c models in predicting storage requirements and devising congestion controls. A discussion of these issues in the case of fractional Brownian motion is summarized in Chapter 4. In this chapter we focus instead on the class of M =G= input processes as potential traf®c models. An M=G= input process is understood as the busy server process of a discrete-time in®nite server system fed by a discrete-time Poisson process of rate l (customers=slot) and with generic service time s distributed according to G. As argued in Parulekar [27] and Parulekar and Makowski [29], these M=G= 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= input model to network traf®c modeling is perhaps best explained through its connection to an attractive model for aggregate packet streams proposed by Likhanov et al. [21]. They show that the combined traf®c generated by several independent, identically distributed (i.i.d.) on=off sources with Pareto distributed activity periods behaves in the limit, as the number of sources increases, like the M =G= input stream with a Paretodistributed s. This provides a rationale for the view that M=G= input processes could provide a natural alternative to existing traf®c models, at least for certain multiplexed applications. Second, the class of M =G= input processes is stable under multiplexing; that is, the superposition of several M =G= processes can be represented by an M =G= input process. Third, the M =G= model displays great ¯exibility in capturing positive dependencies over a wide range of time scales; this is achieved very simply through the tail behavior of s (Proposition 9.4.1). The degree of positive correlation can further be characterized by the sum of the autocovariances, or index of dispersion of counts (IDCs), with the process being short-range dependent (SRD) (i.e., IDC ®nite) if and only if Es 2  is ®nite (Proposition 9.5.1). Insights into how temporal correlations of M=G= input processes will affect queueing performance can be gained by analyzing the behavior of a multiplexer fed by an M =G= input process. For simplicity, we model the multiplexer as a discrete- time single server system consisting of an in®nite size buffer and a server with a constant release rate c (cells=slot). The number of customers in the input buffer at time t is denoted by q t . Our performance index is the steady-state buffer tail probability Pq  > b, as this quantity is indicative of the buffer over¯ow prob- ability in a corresponding ®nite buffer system with b positions. Computing these tail probabilities, either analytically or numerically, represents a challenging problem in the absence of any underlying Markov property for M =G= inputs. Instead, we focus on the simpler task of determining the asymptotic tail behavior of the queue-length distribution for large buffer size. More precisely, we seek results of the form lim b 1 hb ln Pq  > bg 9:1 for some positive constant g and mapping h : R   R  ; these quantities are characterized by l, G, and c and should be easily computable. Limits such as Eq. (9.1) suggest approximations of the form Pq  > be hbg b : 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 the multiplexer and could, in principle, be used to produce guidelines for sizing up its buffers. 216 BUFFER ASYMPTOTICS FOR M=G= INPUT PROCESSES In this chapter we provide an overview of some recent work on this issue. Drastically different behaviors emerge depending on whether v t *  Ot or v t *  ot (with t ), where v t * ln P ^ s > t; t  1; 2; ., and ^ s is the forward recurrence time (9.9) associated with s. The case v t *  Ot is associated with the service time s having exponential tails, while the case v t *  ot corresponds to heavy or subexponential tails for s. Our focus here is primarily (but not exclusively) on large deviations techniques in order to obtain Eq. (9.1). This approach has already been adopted by a number of authors [10, 16, 19]. Applying results by Duf®eld and O'Connell [10] (and some recent extensions thereof [11, 30]), we are able to compute hb and g under reasonably general conditions. In fact, for a large class of distributions, we can select hbv b * , and the asymptotics (9.1) and (9.2) then take the compact form Pq  > bP ^ s > b g b : 9:3 Hence, in many cases, including Weibull, lognormal, and Pareto service times, q  and ^ s (thus s) belong to the same distributional class as characterized by tail behavior. In many cases of interest, in lieu of Eq. (9.1), these large deviations techniques yield only the weaker asymptotic bounds g   lim inf b 1 hb ln Pq  > b9:4 and lim sup b 1 hb ln Pq  > by* 9:5 with g   g*. This situation typically occurs when s is heavy tailed (more generally, subexponential) with either ®nite or in®nite Es 2 , in which case large deviations excursions 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 be tight in the heavy-tail case and other approaches are needed. Of particular relevance are 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 that rely on the asymptotics of Pakes [26] for the GI =GI =1 queue under subexponential assumptions [12]. While Likhanov presents lower and upper bounds only when s is Pareto, these bounds are asymptotically tight. Results for the continuous-time model can 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 and Makowski [28] points already to the complex and subtle impact of (long-range) dependencies on the tail probability Pq  > b. Indeed, in Norros [25] the input stream to the multiplexer was modeled as a fractional Gaussian noise process (or rather its continuous-time analog) exhibiting long-range dependence (in fact, self- similarity), and the buffer asymptotics displayed Weibull-like characteristics. On the other hand, by the results described above, an M=G= input process with a Weibull 9.1 INTRODUCTION 217 service time also yields Weibull-like buffer asymptotics although the input process is now short-range dependent. Hence, the same asymptotic buffer behavior can be induced by two vastly different input streams, one long-range dependent and the other short-range dependent! To make matters worse, if the pmf G were Pareto instead of Weibull, the input process would now be long-range dependent, in fact asymptotically self-similar [28], but the buffer distribution would now exhibit Pareto-like asymptotics, in sharp contrast with the results of Norros [25]. To reiterate the main conclusion of Parulekar and Makowski [28], the value of the Hurst parameter as the sole indicator of long-range dependence (via asymptotic self- similarity) is at best questionable as it does not characterize buffer asymptotics by itself. Furthermore, buffer sizing cannot be determined adequately by appealing solely to the short- versus long-range dependence characterization of the input model used, be it of the M =G= type or otherwise. Of course, this is not too surprising since long-range dependence (and its close cousin, second-order self- similarity) is determined by second-order properties of the input process, while asymptotics of the form (9.1) invoke much ®ner probabilistic properties, which are embedded here in the sequence v t *; t  1; 2; .. The ®niteness of Es 2  (which characterizes the SRD nature of the M=G= input process) is obviously a poor marker for predicting the behavior of this sequence. To close, we note that the diverse queueing behavior demonstrated here is tied to the tail behavior of s, which determines the correlation structure of M=G= inputs. This clearly illustrates the tremendous impact that the correlation structure of an input stream can have on the corresponding queueing performance given that the M=G= inputs all have Poisson marginals! One more data point for the need of a cautious approach in modeling network traf®c when time dependencies are either observed or suspected. 9.2 THE M=G= INPUT PROCESS We summarize various facts concerning the busy server process of a discrete-time M=G= 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; b t1 new customers enter the system. Customer i; i  1; .; b t1 , is presented toits own server and begins service by the start of slot t  1; t  2; its service time has duration s t1;i (expressed in number of slots). Let b t denote the number of busy servers or, equivalently, of customers still present in the system, at the beginning of slot t; t  1. We assume that b servers are initially present in the system at t  0 (i.e., at the beginning of slot 0; 1) with customer i; i  1; .; b, requiring an amount of work of duration s 0;i from its own server. The busy server process b t ; t  0; 1; . is what we refer toas the M=G= input process. 218 BUFFER ASYMPTOTICS FOR M=G= INPUT PROCESSES The following assumptions are enforced on the R-valued random variables (rvs) b, b t1 ; t  0; 1; . and s t;i ; t  0; 1; .; i  1; 2; .: (1) The rvs are mutually independent; (2) The rvs b t1 ; t  0; 1; . are i.i.d. Poisson rvs with parameter l > 0; (3) The rvs s t;i ; t  1; 2; .; i  1; 2; . are i.i.d. with common pmf G on 1; 2; .. We denote by s a generic R-valued rv distributed according to the pmf G. Throughout we assume this pmf G tohave a ®nite ®rst moment, or equivalently, Es < . At this point, no additional assumptions are made on the rvs s 0;i ; i  1; 2; .. For each t  0; 1; ., we note the decomposition b t  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 in the system at the beginning of slot t; t  1 from those initially present (at t  0) and from the new arrivals in the interval [0, t], respectively. Under the enforced operational assumptions, we readily check that b a t  P t s1 P b s i1 1s s;i > t  s and b 0 t  P b i1 1s 0;i > t: 9:7 The rv b a t can alsobe interpreted as the number of busy servers in the system at the beginning of slot t; t  1 given that the system was initially empty (i.e., b  0). 9.2.2 The Stationary Version Although the busy server process b t ; t  0; 1; . is in general not a (strictly) stationary process, it does admit a stationary and ergodic version b t *; t  0; 1; .. 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  P b n1 1 ^ s n > t; t  0; 1; .; 9:8 where (1) the rvs b and  ^ s n ; n  1; 2; . are independent of the rvs b t1 ; t  0; 1; . and s t;i ; t  1; 2; .; i  1; 2; .; (2) the rvs  ^ s n ; n  1; 2; . are independent of the rv b, which is Poisson distributed with parameter lEs; and (3) the rvs  ^ s n ; n  1; 2; . are i.d.d. rvs distributed according to the forward recurrence time ^ s associated with s; the corresponding equilibrium pmf ^ G of ^ s is given by P ^ s  r Ps  r Es ; r  1; 2; .: 9:9 9.2 THE M=G= INPUT PROCESS 219 The following properties of b t *; t  0; 1; . follow readily from this repre- sentation [7, 18 (Theorem 3.11, p. 79), 27]. Proposition 9.2.1. The stationary and ergodic version b t *; t  0; 1; . of the busy server process has the following properties: (i) For each t  0; 1; ., the rv b t * is a Poisson rv with parameter lEs. (ii) The process is reversible in that b 0 *; b 1 *; .; b t * st b t *; b t1 * ; .; b 0 *; t  0; 1; .: 9:10 9.3 THE BUFFER SIZING PROBLEM As we shall see shortly in Sections 9.4 and 9.5, M =G= input processes display an extremely rich correlation structure. We expect these temporal correlations to have a signi®cant impact on queueing performance when such processes are offered to a multiplexer. Togain some insights intothis basic issue we map a multiplexer intoa discrete-time single server queue with in®nite capacity and constant release rate of c cells=slot under the ®rst-come-®rst-served discipline. The cell stream is modeled by an M=G= input process as de®ned above, with b t1 representing the number of new cells that arrive at the start of time slot t; t  1. Let q b t denote the number of cells remaining in the buffer by the end of slot t  1; t, sothat q b t  b t1 cells are ready for transmission during slot t; t  1. If the multiplexer output link can transmit c cells=slot, then the buffer content sequence q b t ; t  0; 1; . evolves according to the Lindley recursion q b 0  q; q b t1 q b t  b t1  c  ; t  0; 1; .; 9:11 for some initial condition q. Conditions under which the queueing system (9.11) admits a steady-state regime are well known and are given next. Proposition 9.3.1. If lEs < c, then there exists an R  -valued rv q b  such that q b t  t q b  for any choice of the initial conditions q, b, and s 0;i ; i  1; 2; .. The system is then said to be stable. This characterization of stability follows by extending Loynes's result [23] to Lindley recursions driven by sequences that couple with their stationary and ergodic versions [2]. Here, for any choice of the initial conditions b and s 0;i ; i  1; 2; ., the sequence b t1 ; t  0; 1; . indeed couples in ®nite time with the stationary and ergodic version introduced in Proposition 9.2.1. 220 BUFFER ASYMPTOTICS FOR M=G= INPUT PROCESSES Stationary M =G= processes being time-reversible, we have the representation q b   st supS b t  ct; t  0; 1; .9:12 for the steady-state buffer content q b  with S b 0  0; S b t  b 1 * b t *; t  1; 2; .: 9:13 Hereafter, by an M =G= input process we mean its stationary version b t *; t  0; 1; ., which is fully characterized by the pair l; G. Moreover, from now on, we always assume the stability condition r in  lEs < c: 9:14 9.4 SECOND-ORDER CORRELATIONS Before discussing the asymptotics associated with buffer over¯ow induced by M=G= input processes, we make a slight detour to explore the correlation structure of such input processes. 9.4.1 Correlation Properties In view of Proposition 9.2.1, the stationary version b t *; t  0; 1; . has a well- de®ned (auto)covariance function G : R  R,say, GhCovb t *; b th * ; t; h  0; 1; .: 9:15 Proposition 9.4.1. We have GhlEs  h  lEsP ^ s > h; h  0; 1; . : 9:16 The ®rst equality in Eq. (9.16) is established in Cox and Isham [7] and the second equality follows readily from the de®nition (9.9). From Eq. (9.16) we ®nd the autocorrelation function g : R  R of the M =G= process l; s tobe given by gh Gh G0  P ^ s > h; h  0; 1; .: 9:17 Note that g01 as we recall that Ps > 01. 9.4.2 Inverting g Proposition 9.4.1 shows that the correlation structure of the stationary M =G= input process l; s is completely determined by the pmf of ^ s (thus of s). It turns out 9.4 SECOND-ORDER CORRELATIONS 221 that the inverse is true as well. Indeed, Eqs. (9.9) and (9.17) together imply ghgh  1P ^ s > hP ^ s > h  1  1 Es Ps > h; h  0; 1; .; 9:18 sothat the mapping h  gh is necessarily decreasing and integer-convex. Taking intoaccount the facts g01 and Ps > 01, we conclude from Eq. (9.18) (with h  0) that Es 1  1  g19:19 with g1 < 1 necessarily by the ®niteness of Es. Combining Eqs. (9.18) and (9.19) we ®nd that Ps > h ghgh  1 1  g1 ; h  0; 1; .: 9:20 Note also from Eq. (9.20) that Es P  h0 Ps > h 1  lim h gh 1  g1 9:21 and Eq. (9.19) imposes lim h gh0. A moment of re¯ection readily leads to the following invertibility result. Proposition 9.4.2. An R  -valued sequence gh; h  0; 1; . is the autocor- relation function of the M=G= process l; s with integrable s if and only if the corresponding mapping h  gh is decreasing and integer-convex with g01 > g1 and lim h gh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 b t *; t  0; 1; . is clearly apparent from Eq. (9.16). The strength of such positive correlations can be formalized 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 b t *; t  0; 1; . is said tobe short-range dependent (SRD) if P  h0 Gh < : 9:22 222 BUFFER ASYMPTOTICS FOR M=G= INPUT PROCESSES Otherwise, the sequence b t *; t  0; 1; . is said tobe long-range dependent (LRD). Easy calculations using Eq. (9.16) readily lead to the following simple characterization. Proposition 9.5.1. We have P  h0 Gh l 2 Ess  1 9:23 so that the process is SRD if and only if Es 2  is ®nite. Interesting subclasses of LRD processes can further be identi®ed through the notion of second-order self-similarity. To do so, we introduce the rvs b m t  1 m P m1 k0 b* mtk ; m  1; 2; .;t  0; 1; .: 9:24 For each m  1; ., the rvs b m t ; t  0; 1; . form a (wide-sense) stationary sequence with correlation structure de®ned by G m hCovb m t ; b m th  and g m h G m h G m 0 ; h  0; 1; .: 9:25 For each H > 0 consider the mapping g H : R  R  given by g H h 1 2 h  1 2H  2h 2H h  1 2H ; h  0; 1; .: 9:26 We say that the sequence b t *; t  0; 1; . is exactly (second-order) self-similar if Varb m t d 2 m b ; m  1; 2; . 9:27 for some constants d 2 > 0 and 0 < b < 1, a requirement equivalent to Ghd 2 g H h; h  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 g H is strictly decreasing and integer- convex, with g H 01, and behaves asymptotically as g H hH2H  1h 2H2 h : 9:29 9.5 LONG-RANGE DEPENDENCE 223 By Proposition 9.4.2 we can interpret g H as the autocorrelation function of the M=G= input process l; s H  with Ps H > r r  2 2H  3r  1 2H  3r 2H r  1 2H 41  2 2H2  ; r  1; 2; .; sothat the M =G= input process l; s H  is exactly second-order self-similar with Hurst parameter H. In applications, the notion of exact self-similarity is often too restrictive and is weakened as follows. The sequence b t *; t  0; 1; . is said tobe asymptotically (second-order) self-similar if lim m g m hg H h; h  1; 2; .: 9:30 This will happen for the M =G= input process l; s if Ps > rr a Lr; 9:31 with 1 < a < 2, for some slowly varying function L : R   R  , in which case H 3  a=2. 9.6 GENERAL BUFFER ASYMPTOTICS Several authors [10, 16, 19] have derived asymptotics such as (9.1) by means of large deviations estimates associated with the sequence t 1 S b t  ct; t  0; 1; .. These results (and their necessary extensions) are summarized below as they apply to the present context. 9.6.1 A General Setup With a given R-valued sequence x t1 ; t  0; 1; ., we associate the R  -valued rv q  given by q   supS t ; t  0; 1; .; 9:32 where S 0  0; S t  x 1 x t ; t  1; 2; .: 9:33 If the sequence x t1 ; t  0; 1; . is assumed stationary and ergodic with Ex 1  < 0, then q  is a.s. ®nite. We are interested in characterizing the asymptotic behavior of the tail probability Pq  > b for large b. 224 BUFFER ASYMPTOTICS FOR M=G= INPUT PROCESSES