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7 FLUID QUEUES, ON=OFF PROCESSES, AND TELETRAFFIC MODELING WITH HIGHLY VARIABLE AND CORRELATED INPUTS S IDNEY R ESNICK AND G ENNADY S AMORODNITSKY Cornell University, School of Operations Research and Industrial Engineering, Ithaca, NY 14853 7.1 INTRODUCTION Large teletraf®c data sets exhibiting nonstandard features incompatible with classical assumptions of short-range dependence and exponentially decreasing tails can now be explored, for instance, at the ITA Web site www.acm.org=sigcomm=ITA=. These data sets exhibit the phenomena of heavy-tailed marginal distributions and long-range dependence. Tails can be so heavy that only in®nite variance models are possible (e.g., see Willinger et al. [49]), and sometimes, as in ®le size data, even ®rst moments are in®nite [1]. See also Beran et al. [3], Crovella and Bestavros [12±14], Leland et al. [33], Resnick [38], Taqqu et al. [48], and Willinger et al. [49]. Other areas where heavy tails and long-range dependence are crucial properties are ®nance, insurance, and hydrology [4±7, 16, 17, 24±26, 35, 37]. New features in the teletraf®c data discussed in recent studies suggest several issues for study and discussion. Statistical. How can statistical models be ®t to such data? Finite variance black box time series modeling has traditionally been dominated by ARMA or Box± Jenkins models. These models can be adapted to heavy-tailed data and work very well on simulated data. However, for real nonsimulated data exhibiting 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. 171 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 dependencies, such ARMA models provide unacceptable ®ts and do not capture the correct dependence structure. For discussion see Davis and Resnick [15], Resnick [38, 39], Resnick et al. [42], and Resnick and van den Berg [43]. Probabilistic. What probability models explain observed features in the data such as long-range dependence and heavy tails. Consequences. Dothe new features revealed by current teletraf®c data studies mean we have to give up Poisson derived models and exponentially bounded tails and the highly linear models of time series? Various bits of evidence emphasize the de®ciencies of classical modeling. There are simulation studies [34] and the experimental queueing analysis of Erramilli, Narayan, and Willinger [18]. An analytic example [40] shows that for a simple G=M =1 queue, a stationary input with long-range dependence can induce heavy tails for the waiting time distribution and for the distribution of the number in the system. Connections between long-range dependence and heavy tails need to be more systematically explored but it is clear that in certain circumstances, long-range dependent (LRD) inputs can cause heavy-tailed outputs and (as we discuss here) heavy tails can cause long-range dependence. We discuss three models where heavy tails induce long-range dependence: 1. A single channel on=off source feeding a single server working at constant rate r > 0. Transmission or on periods have heavy-tailed distributions. 2. A multisource system where a single server working at constant rate r > 0isfed by J > 1 on=off sources. Transmission periods have heavy-tailed distributions. 3. An in®nite source model feeding a single server working at constant rate r > 0. At Poisson time points, nodes or sources commence transmitting. Transmission times have heavy-tailed distributions. In each of the three cases, our basic descriptor of system performance is the time for buffer content to reach a critical level. Such a measure of performance is path based and makes sense without regard to stability of the model, existence of moments of input variables, or properties of steady-state quantities. 7.2 A SINGLE CHANNEL ON/OFF COMMUNICATION MODEL 7.2.1 BasicSetup We consider ®rst communication between a single source and a single destination server. The source transmits for random on periods alternating with random off periods when the source is silent. During the on periods, transmission is at unit rate. Let fX on ; X n ; n ! 1g be i.i.d. nonnegative random variables representing on periods. The common distribution is F on . Similarly, fY off ; Y n ; n ! 1g are i.i.d. 172 FLUID QUEUES, ON=OFF PROCESSES, AND TELETRAFFIC MODELING nonnegative random variables independent of fX on ; X n ; n ! 1g representing off periods and these have common distribution F off . The means are m on I 0 F on s ds; m off I 0 F off s ds; which are assumed ®nite and the sum of the means is m : m on m off . Using these random variables we generate an alternating renewal sequence characterized as follows. 1. The interarrival distribution is F on à F off and the mean interarrival time is m m on m off . 2. The renewal times are 0; P n i1 X i Y i ; n ! 1 : Because of the ®niteness of the means, the renewal process has a stationary version: D; D P n i1 X i Y i ; n ! 1 : where D is a delay random variable satisfying PD > x I x PX on Y off > s m ds I x 1 À F on à F off s m ds: However, making the process stationary in this manner has the disadvantage that the initial delay period D does not decompose into an on and an off period the way subsequent inter-renewal periods do and the following procedure is preferable for generating the stationary alternating renewal process. De®ne independent random variables B; X 0 on ; Y 0 off , which are assumed independent of fX on ; X n ; n ! 1g and fY off ; Y n ; n ! 1g,by PB 1 m on m 1 À PB 0; PX 0 on > x I x 1 À F on s m on ds : 1 À F 0 on x; PY 0 off > x I x 1 À F off s m off ds : 1 À F 0 off x: 7.2 A SINGLE CHANNEL ON/OFF COMMUNICATION MODEL 173 The delay random variable D 0 is de®ned by D 0 BX 0 on Y off 1 À BY 0 off : This delayed renewal sequence fS n ; n ! 0g : D 0 ; D 0 P n i1 X i Y i ; n ! 1 is a stationary renewal process. 7.2.2 High Variability Induces Long-Range Dependence Consider the indicator process fZ t g, which is 1 iff t is in an on period. Thus, for t ! D 0 , Z t 1; if S n t < S n X n1 ; some n 0; if S n X n1 t < S n1 ; some n and if 0 t < D 0 we de®ne Z t 1; if B 1 and 0 t < X 0 on ; 0; otherwise: ( A standard renewal argument gives the following result [22]. Proposition 7.2.1. fZ t ; t ! 0g is strictly stationary and PZ t 1 m on m : Conditional on Z t 1, the subsequent sequence of on=off periods is the same as seen from time 0 in the stationary process with B 1. It is easiest to express long-range dependence in terms of slow decay of covariance functions so we consider the second-order properties of the stationary process fZ t g (See Heath et al. [22].) The basis for the next result is a renewal theory argument. Theorem 7.2.2. The covariance function gsCovZ t ; Z ts 174 FLUID QUEUES, ON=OFF PROCESSES, AND TELETRAFFIC MODELING of the stationary process fZt; t ! 0g is gs m on m m off m À s 0 F off s À uF 0 on à Udu m on m m off m À F 0 on à U Ã1 À F off s m on m off m 2 À 1 m s 0 zs À oU dw; where U P I n0 F on à F off nà and zt t 0 F off x F on t À x dx m on F 0 on Ã1 À F off t m off F 0 off Ã1 À F on t: How do we analyze the asymptotic behavior of gÁ as a function of s? Note gs is of the form gsconst lim v3I z à U vÀz à Us hi so we need rates of convergence in the key renewal theorem. This can be based on a theorem of Frenk [19] and is given in Heath et al. [22]. Theorem 7.2.3. Assume that there is an n ! 1 such that F on à F off nà is non- singular. Suppose F on tt Àa Lt; t 3I; where 1 < a < 2 and L is slowly varying at in®nity and assume F off to F on t; t 3I: Then gt$ m 2 off a À 1m 3 t ÀaÀ1 Lt; t 3I: 7.2 A SINGLE CHANNEL ON/OFF COMMUNICATION MODEL 175 So gt decreases like a constant times t F on t. Such a slow decay of gt at an algebraic rate is characteristic of long-range dependence. One way to think about this result is that, with heavy-tailed on periods, there is a signi®cant probability that a very long on period can cover both the time points s and t s, thereby inducing strong correlation between these two time points. Taqqu et al. [48] use the long-range dependence of the on=off process and superimpose many such processes. This superposition is approximately a fractional Brownian motion, giving one explanation of the observed self-similarity of Ethernet traf®c. See alsoLeland et al. [33]. Other limiting procedures leading toLe  vy motion with heavy tails are possible and are also brie¯y discussed in Leland et al. [33]. See also Konstantopoulos and Lin [32]. 7.2.3 Single Channel Fluid Queues with Constant Service Rates Suppose work enters a communication system according to the on=off process. The server works off the load at constant rate r assuming there is load to work on. Here are the formal model ingredients for the single source model. 1. The Input Process At : t 0 Z v dv: Since At$tm on =m, the long-term input rate is m on =m. 2. The Output Process. There is a release rate functionÐthe release rate from the system when contents are at level x is rx : r; if x > 0; 0; if x 0: 3. The Stability Condition. Input does not overwhelm the release rate. This necessitates 1 > r > m on m : 7:1 The restriction that r < 1 results from the normalization that work arrives at rate 1 during on periods and prevents the contents process from having 0 as an absorbing state. 4. The contents process fX tg satis®es the storage equation dX tdAtÀrX t dt: Note that during an on period, the net input rate is 1 À r since work is inputted at unit rate but the server works at rate r. During an off period, the release rate is r 176 FLUID QUEUES, ON=OFF PROCESSES, AND TELETRAFFIC MODELING (provided there is liquid to release). This means the paths of X Á are sawtoothed shaped. 7.2.3.1 Regeneration Times Recall that the stationary alternating renewal pro- cess is S n D 0 P n i1 X i Y i ; n ! 0 : Since the contents process is stable, we can de®ne regeneration times fC n g :fS n : X S n À 0g; which are times when a dry period ends and input commences. So the standard limit theorems due to Smith for regenerative processes [45] guarantee limit distributions exist in discrete and continuous time: PX S n > x31 À W x; PX t > x31 À V x: There are also connections with standard random walk theory and Lindley's equation holds. If we compare X S n with X S n1 we get X S n1 X S n 1 À rX n1 À rY n1 X S n x n1 ; where x n1 1 À rX n1 À rY n1 and fx j g i.i.d. It is important to distinguish between the random walk with steps fx n g and the random walk with steps fX n Y n g. Assume 1 À F on xx Àa Lx; a > 1; x 3I: From standard random walk theory [11, 36] we get 1 À W x$ r 1 À r 1 À r aÀ1 a À 1m on x ÀaÀ1 Lx : bx ÀaÀ1 Lx; x 3I; 7.2 A SINGLE CHANNEL ON/OFF COMMUNICATION MODEL 177 sothat the tail of W is heavy and comparable to the integral of the tail of F on . The de®nition of r is r m on m off 1 À r r < 1: As expected from the sawtooth shape of the paths, the tail of V x is heavier because of a bigger multiplicative factor 1 À V x$ b 1 À r aÀ1 ma À 1 ! x ÀaÀ1 Lx: See Boxma and Dumas [9] and Heath et al. [22] and the references therein and also the discussion in Chapter 10. The tails of both W x and V x are of the form const  x F on x. So the equilibrium content of the system, in either continuous or discrete time, can get quite large with nontrivial probability. This point will be reinforced in the discussion of the time it takes for buffer content to reach a critical level. 7.2.4 Extremes, Level Crossings, and Buffer Over¯ow The distributions W x and V x are standard queueing quantities and convey some performance information. Another performance measure, one that is less dependent on notions of stability and existence of moments, is the time until buffer over¯ow. We formulate the time to buffer over¯ow as the hitting time of a high level L: tL : inf ft ! 0 : X t!Lg: De®ne M t W t s0 X smaxfX s : 0 s tg, and the reason for interest in the maximum content up to t is that as a process it is the inverse of the tÁ process since tLinf fs > 0 : X s!Lginf fs > 0 : M s!Lg : M 2 L; where M 2 Á is the right continuous inverse of the monotone function MÁ.Ifwe understand the asymptotic behavior of M Á, then we will understand the asymptotic behavior of tÁ. To understand the behavior of MÁ we ®rst study the extremes of fX S n g and then ®ll in the behavior between the discrete points fS n g. We study the maxima of the random walk generated by fx n g over cycles and then knit cycles together. Recall x n1 1 À rX n1 À rY n1 : 178 FLUID QUEUES, ON=OFF PROCESSES, AND TELETRAFFIC MODELING The ®rst downgoing ladder epoch of f P n i0 x i ; n ! 0g is N inf n > 0 : P n i0 x i 0 : 7:2 The tail behavior of the maximum in a (discrete) cycle is described next. See Asmussen [2] and Heath et al. [21]. Proposition 7.2.4. For the stable queueing process fX S n g satisfying 1 À F on xx Àa Lx; a > 1; x 3I; the maximum over a cycle has a distribution tail asymptotic to the tail of the on distribution and, in particular, P W N n0 X S n > x "# $ Px 1 > xE N $ P1 À rX 1 > xE N $1 À r a F on xE N: Note that this tail is lighter than the tail of W or V from the previous section. Interestingly, the off distribution in¯uences this expression only through the factor E N. From maxima over discrete cycles, it is relatively simple to derive tail behavior over a continuous time cycle: de®ne C 1 S N . Corollary 7.2.5. Assume the contents process fX tg is stable and 1 À F on xx Àa Lx; a > 1; x 3I: The distribution tail of the maximum of the contents process over one cycle is asymptotic to the tail of the on distribution; P W C 1 s0 X s > x $1 À r a F on xE N: Again note the minimal effect of F off , which only affects the answer through the multiplicative factor E N. Now the behavior up to an arbitrary time t is derived from the random number of cycles squeezed into 0; t and we have the following result from Heath et al. [21]. (Let D r 0; I be the right continuous functions on 0; I with ®nite left limits; D l 0; I are the left continuous functions with ®nite right limits.) 7.2 A SINGLE CHANNEL ON/OFF COMMUNICATION MODEL 179 Theorem 7.2.6. Assume fX tg stable and 1 À F on xx Àa Lx; a > 1; x 3I: De®ne the quantile function bs 1 1 À F on 2 s: Let fY a t; t > 0g be the extremal process [44] generated by F a xexpfÀx Àa g; x > 0 so that PY a t xF t a x: De®ne the rescaled extremal process S a t 1 À r m 1=a Y a t: Then in D r 0; I  D l 0; I, as u 3I, MuÁ bu ; MuÁ bu 2 AS a Á; S 2 a Á: In particular, we get for the ®rst passage process, as u 3I, 1 À F on utuÁ A Y 2 a m 1=a 1 À r : and lim L3I P 1 À r a m 1 À F on LtL x PE1 x1 À e Àx ; x > 0; where E1 is a unit exponential random variable. Also, as x 3I, 1 À F on xEtx 3 m 1 À r a ; 180 FLUID QUEUES, ON=OFF PROCESSES, AND TELETRAFFIC MODELING
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