Stochastically Bounded Burstiness for Communication Networks

Transcription

1 Stochastically Bounded Burstiness for Communication Networks David Starobinski and Moshe Sidi Department of Electrical Engineering Technion - Israel Institute of Technology Haifa 32000, Israel Abstract-We develop a network calculus for processes whose burstiness is stochastically bounded by general decreasing functions. This calculus enables us to prove the stability of feedforward networks and obtain statistical upper bounds on interesting performance measures such as delay, at each buffer in the network. Our bounding methodology is useful for a large class of input processes, including important processes exhibiting subexponentially bounded burstiness such as fractional Brownian motion. Moreover, it generalizes previous approaches and provides much better bounds for common models of real-time traffic, like Markov modulated processes and other multiple time-scale processes. We expect that this new calculus will be of particular interest in the implementation of services providing statistical guarantees. Keywords- Network Calculus, Statistical Bounds, Real-Time Traffic, Subexponential Distributions, Sum of Exponentials, Multiple Time-Scale Processes. I. INTRODUCTION The new broadband high speed networks [I] are designed to integrate a wide range of audio, video and data traffic within the same network. In these networks, the stringent Quality of Service (QoS) requirements of real-time multimedia applications are met by the mean of special services providing guarantees. One distinguishes between two main kinds of such services: deterministic and statistical. The deterministic service, also called Guaranteed Service within the Internet framework [2], is aimed to provide strict guarantees on the QoS requirements while the statistical service is intended to provide only statistical (probabilistic) guarantees. The main advantage of the statistical service over the deterministic service is that it can achieve a higher network utilization, at the expense of some minor quality degradation. The mathematical basis of the deterministic service is a network calculus first introduced in [3] and further developed in [4] which allows to obtain deterministic bounds on delay and buffering requirements in a communication network. This model provides bounds only on the maximal delay that traffic can undergo through the different elements in the network. For services offering statistical guarantees, one is rather interested in bounding the mean delay or the probability that the delay exceeds a certain value. For this reason, the concept of Exponentially Bounded Burstiness (EBB) has been introduced in [5](see also the related works [6], [7]). A continuous traffic stream with instantaneous rate R(t), is EBB with upper rate p and bounding function Ae-au if for all cr 20 and t > s 2 0 Pr { [R(u)du >_ p(t - s) + cr 5 Ae-au, (1) where p,a,a > 0 (notice that the EBB approach applies to discrete-time processes, as well). The EBB calculus is based on 1 an appropriate EBB characterization of the external processes feeding the network. It is shown in [5] that if all input processes to a feedforward communication network (and also some special non feedforward networks) are EBB, then, subject to throughput conditions, so are all traffic streams within that network and the network is said to be stable. This model enables to compute upper bounds on various performance measures such as delay survivor probabilities and mean delay, at each buffer in the net- work. For instance, it is shown in [5] that the delay W(t) sensed by a packet, entering at time t a general multiplexer fed by an EBB arrival stream, satisfies the following equation for some positive parameters B and,b Pr{W(t) 2 U> 5 ~e-fl., (2) provided that the service capacity of the multiplexer is larger than the upper rate of the EBB traffic stream. A process W(t) is said to be an Exponentially Bounded (EB) process with bounding function if it satisfies (2) for all t, CT > 0. The EBB model can he, therefore, considered as one of the helpful ways to implement statistical services. Nevertheless, the EBB model suffers from some major deficiencies. The contribution of this work is to reveal these deficiencies, find remedies for them, and, as a consequence, introduce an even more general and powerful bounding methodology. According to (2), the EBB model can be expected to provide relatively tight bounds when the delay probability distribution curve is linear on the logarithmic scale (the reasonable quality of the bounds, in such a case, has been validated in [5], [SI). However, for a variety of traffic models, such as Markov modulated processes and other multiple time-scale processes, it has been observed that the delay distribution in a multiplexer is better described by two linear regions (see [l], Section 3.8, and references therein). These regions are commonly termed cell region and burst region in the literature. As shown schematically in Fig. 1, the EB bound for such kinds of traffic may be rather loose, since the rate of decrease of the EB bound can not exceed the rate of decrease of the delay curve in the burst region. A call admission scheme based on EB bounds may therefore lead to significant network underutilization. But, it is not the only drawback of the EBB model. It turns out that a large class of traffic processes do not satisfy at all the characterization of exponentially bounded burstiness. This class includes all the processes exhibiting subexponentially bounded burstiness, i.e., burstiness stochastically bounded by functions decaying slower than any exponential. Many of the new models suggested in the literature for characterizing network traffic processes have /99/$ IEEE. 36

2 L g(pr(w>=x)) 7 I. Burst Region Fig. 1. CelVburst regions, typical delay curve distribution and EB bound subexponentially bounded burstiness. One of these models is the self-similar fractional Brownian motion (FBM) which has been shown to have good statistical fit with LAN data [9], [ 101. An FBM process has a subexponentially bounded burstiness since the delay distribution in a buffer fed by such a process is bounded by a function behaving like a Weibull distribution [ 111, i.e., Pr(W(t) 2 (T) 5 Ae-aab (3) where A, a > 0 and 1 > b > 0. In this work, we develop a new calculus in order to deal with processes which do not satisfy the EBB characterization. We consider processes whose burstiness is stochastically bounded by general functions f(a). We refer to these processes as "Stochastically Bounded Burstiness" (SBB) processes with bounding function f(a). We demonstrate the existence of a network calculus for SBB processes. This calculus allows to prove the stability of feedforward communication networks fed by SBB processes (subject to some conditions on the bounding functions) and obtain upper bounds on the interesting performance measures. It turns out that the SBB calculus is also a powerful tool for obtaining much better bounds for multiple time-scale processes. Since the function f(o) is general, one can choose to bound the traffic burstiness with a sum of exponentials instead of a single exponential, as is the case in the EBB model. We develop a network calculus for sum of exponentials and show that it leads to tighter bounds than those provided by the EBB calculus. This paper is organized as follows. In the next section, we present our model and introduce definitions used throughout the paper. In Section 111, we state and prove the basic theorems demonstrating the existence of a network calculus with general bounding functions. In Section IV, we illustrate the potential use of the SBB calculus for subexponentially bounded burstiness processes by proving the stability of feedforward networks fed by fractional Brownian motions, In Section V, we show how a judicious choice of the bounding function, i.e., sum of exponentials, can lead to much tighter bounds than those provided by the EBB approach. This result may be especially useful for the efficient provision of statistical guarantees in networks fed by multiple time-scale processes. The last section is devoted to discussion and open problems. 11. MODEL AND DEFINITIONS We consider the same network model as introduced in [5]. We assume that the system is started at time t = 0 and that all the network queues are empty at this time. The network queues are assumed to be of infinite size. Throughout this work we consider the continuous-time case (of stochastic processes) as well as the discrete-time case. In the latter case the concept of traffic stream {R(t)}tE~+ reduces to a sequence {R(t)}tE~ of random variables. For the sake of brevity, we present theorems and proofs only for the discrete model although similar results hold for the continuous model. We introduce the notation (sam du)" f(u) for denoting the n-fold integration of the function f(u). Note that the result of the operation (sam du)" f(u) is a function of the variable U. As an example, taking f(u) = e--au leads to ([ay d ~ f(u) ) = ~ (l/a)"e-"". We present, now, the new definitions: Definition I: Let.F represent the set of all the functions f(a) such that for any order n, the multiple integral (sam du)" f(u) is bounded for any a 2 0. Definition 2: A stochastic process, W(t), is Stochastically Bounded (SB) with bounding function f(a) if: (9 f(.) E 3. (ii) Pr(W(t) 2 a) 5 f(a) for all a 2 0 and all t 2 0. Definition 3: The rate of a continuous traffic stream, R(t), has a Stochastically Bounded Burstiness (SBB) with upper rate p and bounding function f(a) if: (i> f(0) E F. (ii) Pr { s," R(u)du 2 p(t - s) + a 5 f(a) for all (T 20 and 1 all 0 5 s 5 t. In this case we say that R(t) is SBB, and refer to it as an SBB process. Definition 4: A discrete-time process, R(t), has a Stochastically Bounded Burstiness (SBB) with upper rate p and bounding function f(a) if 0) f(q) E 2=. (ii) Pr { c:=~+~ ~ ( u ) 2 p(t - s) + o} 5 f(a) for all a 2 o and all 0 5 s < t. In this case we say that R(t) is SBB, and refer to it as an SBB process. The requirement that the function f(o) belongs to.f comes from a relation, proven in the sequel, between the bounding function of the output process from a network element and the bounding function of the input process to that network element. This relation involves the integral sum f(u)du which is required to be bounded for avoiding triviality. Since, we do not make, a priori, any assumption on the size of the network, it is required that for any order n the multiple integral (sam d ~ " f ) (~) is bounded. However, if it is known that the longest route in the network traverses no more than M network elements, it is enough to require that for all n 5 M the multiple integral (1," du) f(u) is bounded. Finally, we can assume, without any loss of the generality, that f(a) is not increasing with o since Pr(. 2 a) can not increase with D. 37

3 --r-rd BASIC RULES OF THE SBB CALCULUS C A. Sum of SB/SBB Processes Rin(t)- Wt) The following theorem states that the sum of two SBB processes is itself an SBB process. The case of any finite number W(t> of summed processes is easily dealt by applying iteratively the Fig. 2. An isolated network element: &(t) denotes the input traffic stream, theorem. This statement regard1ess Of the de- R,(t) denotes the output traffic stream, W(t) denotes the workload and C pendencies between the summed processes. denotes the element service capacity. Theorem 1: Let RI (t) be SBB with upper rate pl and bounding function fl(a) and R2(t) be SBB with upper rate p2 and function f(a) then the input process to the system is SBB with bounding function f2(a). Then Rl(t) + &(t) is SBB with upper rate upper rate p1 + p2 and bounding function g(a) = fl(p~) + p and bounding function f(a). Proof: By the assumptions of the theorem, there exists a ((1- p)a). The value of p is any real number such that bounding function f(a) such that O<p<l. Proof: Set U 2 0, 0 5 s < t and let 0 < p < 1 (in Pr(W(t> 2 U) 5 f(a) Section V-B, we present one of the possible criteria for setting the value of p). Then, we have for all U 2 0 and all t 2 0. Let 0 5 s < t, then L Pr ({ 2 Rl(U) U { 2 R2(U) zl=s+l u=s+1 2 Pl(t-s) +PO L p2(t - s) 4- (1 -PI0 1) theorem. It remains to prove that g(a) belongs to the set of functions F. We have (~mdu)nflbu) =(Up)" (S,,dU)nfl(U). (4) The rhs of (4) is bounded for any order n since f1 E 3 and p > 0. Thus, fl(pa) belongs to F. In the same way, one can show that f2 (( 1 - p)a) E F. Therefore g(a) = fl (pa) + f2 (( 1 - p)a) belongs to F. A similar proof will show that the sum of two SB processes is again an SB process. B. Characterization of SBB Processes The following theorem provides a very useful tool for characterizing SBB processes. It says that if the amount of unfinished work (workload) in a work-conserving system, transmitting at rate p, is SB with bounding function f(a) then the input process to the system is SBB with upper rate p and bounding function f (0). Theorem 2: Consider a system that transmits at rate p, and assume it is work-conserving. Suppose that it is fed with a single stream of traffic rate R(t), and let W(t) be the amount of data stored in the system at time t. If W(t) is SB with bounding since the system can transmit no more than p(t - s) information units during the time from s to t. Hence, Pr { c:=,+, R(u) 2 p(t - s) + a} 5 f(a) which asserts the C. Calculus for an Isolated Network Element The following result represents one of the main contribution of this work. We show that the output process from any workconserving network element (buffer) fed by an SBB input process is SBB and find a very simple relation between the bounding functions of these processes. We also show that in a such a case the workload in the buffer is SB. In the following theorem, we consider a single input process. Note that if there are many input processes, they can be aggregated to a single one by employing Theorem 1. Nevertheless, the theorem can be easily extended such that an SBB characterization is derived for each input-output stream (session) apart as we show in [12]. An illustration of a network element is shown in Fig. 2. Theorem 3: Let Ri,(t) be the aggregate input traffic stream to a work-conserving element, transmitting at rate C. Let W(t) be the amount of data stored in the system and let R,(t) be the output traffic stream of the system. If Rin(t) is SBB with upper rate p < C and bounding function f(a), then (i) R,(t) is SBB with upper rate p and bounding function do> = f(o) + (C - 4-l. J," f('ll)du. (ii) W(t) is SB with bounding function g(o) = f(a) + (C- p1-l. s," f(u)d'2l. Proof: (i) Let d(s) be the random variable defined by d(s) = min {U : W(s - U ) = 0) The quantity d(s) equals the time (number of slots) that has passed since the last time the queue was empty prior to s. As we assume the whole system is empty at time t = 0, we know R(u), we have that 0 5 d(s) 5 s. Letting RS>t Pr {R:J > p(t - S) + a} = cf=,+, 38

4 = 3 Pr ({ 2 p(t - s) + a} r l {d(s) = i>). (5) Notice that {d(s) = i} implies that in each one of the i time slots s - i + 1, s - i + 2,..., s C units of information have been transmitted, and {Rzit 2 p(t - s) + o} implies that at least additional p(t - s) + a information units have been transmitted from s to t. Moreover, {d(s) = i} also implies that the queue of the work-conserving element was empty at time s - i, and then all the above information units have entered it from s - i to t. Therefore, ({RZJ >_ p(t - s) + o} n {d(s) = i}) C Substituting (6) into (5), we get Pr { R:J 2 p(t - s) + a} {R,4;i>t 2 p(t - s) fa + Cz}. (6) 5 S Pr { 2 p(t - s) + a + ci} By the assumption of the theorem Pr {R.:: 2 p(t - s) + o} _< f(o) (8) holds for all integer 0 5 s < t and a 2 0. Thus, substituting (8) into (7), we get Pr { R:>' 2 p(t - s) + o} m Reminding that f(a) is not increasing with U, the following inequality, then, holds (see Fig. 3) based on the assumption that C - p > 0. Inserting (IO) into (9), we obtain Pr {R:J 2 p(t - s) + U } o o+a o+2a o+3a --._-- Ci Fig. 3. A pictorial explanation of the inequality f(o + az)a 5 s," f(u)du for any a > 0 and any non increasing function f(u). D. Calculus for a Feedforward Network and Stability When operating a communication network, one of the main concerns is to ensure that it is stable. By stability, we mean that at each network element i, limu+.cu Pr(Wi 2 a) = 0, where W, corresponds to the steady-state workload in the network element i. By now, we know that if the upper rate of an SBB input process is smaller than the service capacity of an isolated network element then the output process from that element is SBB. Furthermore, we know that the workload in that element is SB with bounding function g(a). Since g(a) E F, we know that sum g(u)du is bounded for any a 2 0. The finiteness of the integral implies that limu+m g(m) = 0, which, in turn, means that the network element is stable. The extension for any feedforward (acyclic) network fed by external SBB processes is immediate: We can choose a peripheral node (which accepts only outside world inputs), deduce that its output is SBB and drop it. Continuing inductively, we prove that all the traffic streams in the network are SBB implying that the workload in each network element is SB (assuming that the upper rate of the aggregate input process is smaller than the service capacity of the element). We conclude that any feedforward network fed by SBB processes is stable. -U The function g(o) belongs to F because that f(a) belongs to 3. (ii) Using the notations of the previous proof, we can write, t Pr {~(t) 2 o} = and by similar arguments, we get Pr ({~(t) 2 a> n {d(t) = i}), ({w(t> 2 0) n {d(t) = i}> c { R ~;~J 2 + ci}. The rest of the proof being identical to the proof of part (i), we obtain Pr {W(t) _> a} 5 g(a), with g(a) E F. Iv. STABILITY OF FEEDFORWARD NETWORKS FED BY FRACTIONAL BROWNIAN TRAFFICS As explained in the Introduction, a non-negligible number of recent traffic models for communication networks are characterized by subexponentially bounded burstiness. So far, several studies related to these traffic models have focused on the single node queueing performance. One of the more interesting applications of the SBB calculus is to allow derivation of results in a multiple node environment. As an example of application, we prove the stability of feedforward networks fed by fractional Brownian traffics. The fractional Brownian traffic, which is a special case of 39

5 long range dependent process, is defined as follows: lo R(u)du = mt + JamZ(t) fort 2 0, (1 1) where R(t) is the traffic rate and Z(t) is a normalized fractional Brownian motion (see [ 131, p.55, for the definition of fractional Brownian motion). This model has three parameters m, a and H with the following interpretations: m > 0 is the mean input rate, a > 0 is the index of dispersion at t = 1, and H E [1/2,1) is the Hurst parameter (parameter of self-similarity) of Z(t). The workload distribution in a buffer with service capacity p > m fed by a such a process is bounded by a function which behaves like a Weibull distribution [l 11, i.e., where Q = (p- m) /(2amH (l- H)2-2H), p < 2-2H (any positive number smaller than 2-2H), and where A is a positive number (which seems, unfortunately, difficult to compute). Therefore, according to Theorem 2, a fractional Brownian traffic is SBB with upper rate p and bounding function f(a) = Ae-Qub. The stability of a feedforward network fed by fractional Brownian traffics follows from Section 111-D, if one can show that f(a) E F. In [12], we show that for any E > 0 and any A/2=, e-axp dx < Ce-au (13) sxz, where C is a positive constant and y =,O - 6. We identify f(x)dx with the lhs of (13) which is bounded since we can choose E < p. In order to bound (s, =, dx)2f(z), we integrate both sides of (13) and obtain We bound the rhs of (14) by invoking (13) and obtain ( liu dx) f (x) < C e--aa, where C is a positive constant and y =,B Therefore, (~x~udx)2f(x) is bounded if we choose E < p/2. Using a similar procedure, it can be shown that for any order n the n- fold integral (sum dx)n f(x) is bounded. We come, thus, to the conclusion that f(o) belongs to F which implies that a feedforward network fed by fractional Brownian traffic processes is stable. v. NETWORK CALCULUS FOR SUM OF EXPONENTIALS A. Motivation: Tighter Bounds for Multiple Time-Scale TrafJic Markov modulated processes have been extensively used for representing real-time traffic like packetized voice traffic and video traffic (see, e.g., [14], [15]). For these processes, it has been observed that, on the logarithmic scale, the delay distribution in a multiplexer can be roughly broken into two linear regions (cell and burst regions). This behavior is a typical characteristic of multiple time-scale traffic [ 161, [ 171. Consider, for example, a stationary Markov modulated arrival process R(t), t E N, feeding a buffer with service rate C = 1. The modulating chain is assumed to have two states (say 1,2) with transition probabilities p12 = and p2] = 1/100. When in state 1, the source is producing i.i.d. arrivals A1 distributed according to Pr(A1 = 0) = 7/8 and Pr(A1 = 2) = 1/8. When in state 2, the source is producing i.i.d. arrivals A2 distributed according to Pr(A2 = 0) = 11/20 and Pr(A2 = 2) = 9/20. The parameters of this Markov chain are representative of multiple timescale traffic [18]. Following [18], we observe that the arrival process has a multiple time-scale structure: Transitions within state 1 or state 2 occur every few slots, transitions from state 2 to state 1 occurs every 100 slots on average and transitions from state 1 to state 2 occurs every IO6 slots on average. Using standard z-transform techniques we can find the steady-state buffer occupancy distribution, Pr(W = a) = lop5. (15) where U represents the buffer size. For this example, the bound on the survivor probability Pr(W 2 a) computed following the EBB approach of [5] is Pr(W 2 a) 5 for all a 2 0. This bound is loose. Using only two exponentials, we can provide a much tighter bound (corresponding, even, to the true probabilities in this case) Pr(W 2 a) 5 ( e-0.273~ 1 (17) for all a 2 0. The comparison of the two bounds with the true probabilities is shown in Figure 4. The question is whether there exists a network algebra for sum of exponentials. The answer is, obviously, in the affirmative if we resort to our general definition of stochastically bounded burstiness and choose the bounding function as a sum (mixture) of exponentials. Back to the example, one can see that W(t) is SB with bounding function f(a) = e-1.946~ e-0.273u, for all a 2 0, (18) and, by Theorem 2, R(t) is SBB with upper rate p = 1 and bounding function as defined in (18) (the factor - appearing before the first exponential in (17) has been omitted in (1 s), since it is meaningless). Note that the random variable W(t) is stochastically smaller than the random variable W, corresponding to the steady-state workload, since the buffer is empty at time 0 and the arrival process is stationary [ 191. B. Basic Rules of the Calculus for Sum of Exponentials The basic rules of the network calculus for processes with burstiness stochastically bounded by sum of exponentials are easily derived from the theorems presented in Section 111. The first rule is related to the addition of processes, i.e., if Rl(t) is SBB with upper rate p1 and bounding function f(a) = Cy a= 1 Aie-OLiU and &(t) is SBB with upper rate p2 and bound- M ing function g(a) = Bie-Piu then, according to The- orem 1, Rl(t) + R2(t) is SBB with upper rate p1 + p2 and bounding function h(a) = Cz Cie-Yi where 40

6 Prob[wi=xl Prob[wz=xl Fig. 4. Bounding a distribution (thin solid line) with a single exponential (thick solid line) and with a sum of two exponentials (thick dashed line). Fig. 5. Bounds on the workload distribution of a multiplexer fed by two dependent processes: EB bound (thin solid line) and SB bound (thick solid line) summed process Rin(t) = Rl,i,(t) + Rz,in(t) is EBB with upper rate 2 and bounding function 2. e-o.182u and that the workload distribution W(t) in the multiplexer is EB with bounding and where the value of p is any real number such that 1 > p > 0. function Using the network calculus for sum The usual criterion for choosing p is to maximize the asymp- of exponentials, one deduces that Ri,(t) is SBB with upper rate totic decay rate of h(a). Assuming that QN = miniai and 2 and bounding function + 2. PM = mini pi, then one should choose p = PM/(CYN + p ~ and ) W(t) is SB with bounding function e-1.295u e-0.735u in order to maximize the smallest yi. The second rule deals As one can see, from Fig. 5, with the calculus for an isolated network element. If a general work-conserving multiplexer with service rate C is fed by an SBB input process Rin(t) with upper rate p < C and bounding the SB bound is up to four orders of magnitude tighter than the EB bound. Finally, we obtain from Theorem 3 that the aggregate output stream from the multiplexer is SBB with upper rate function f(a) = Aie-Oiu then, according to Theorem 3, 2 and the same bounding function as for W(t). the output process R,(t) is SBB with upper rate p and bounding function g(a) = Ai(1 + (C- p)-'a,l)e-"tu, and the VI. DISCUSSION AND OPEN PROBLEMS amount of data stored in the multiplexer W(t) is SB with the same bounding function g(c7). In [ 121, we prove that the calculus for sum of exponentials offers tighter bounds than the EBB calculus, at each node of a feedforward network, provided that the SBB bounding function of each exogenous arrival process is upper bounded by the EBB bounding function. C. An Example As an application of these results, consider a general multiplexer with service capacity C = 3. The multiplexer is fed by the sum of two input streams with unknown degree of dependency. The first stream Rl,i,(t) corresponds to the Markov modulated process described in section V-A. The second stream, Rz,i,(t), is a Markov modulated process with two states and transition probabilities p12 = and pzl = 1/100. When staying in state 1, this stream is producing i.i.d. arrivals B1 distributed according to Pr(B1 = 0) = 9/10 and Pr(B1 = 2) = 1/10. When staying in state 2, it is producing i.i.d. arrivals Bz distributed according to Pr(B2 = 0) = 5/8 and Pr(B2: = 2) = 3/8. Using a similar procedure as described above for Rl,i,(t), that is by resorting to Theorem 2, it can be shown that Rz,in(t) is EBB with upper rate 1 and bounding function and SBB with upper rate 1 and bounding function e ,e-0.543u. We remark that although the exact dependency between the two incoming streams is unknown, and exact workload distribution probabilities can not be computed, the EBB and SBB calculus can be employed for obtaining upper bounds. Using the technique of [5], one finds that the 41 In this paper, we introduced a new network calculus, termed SBB calculus, which provides statistical upper bounds on various performance metrics, at each node of a feedforward network. This calculus is based on an appropriate characterization of the processes feeding the network. This characterization is achieved by probabilistically bounding the "burstiness", which in our context means an ensemble of linear envelop processes, by general decreasing functions. We showed that the SBB methodology is superior to previous approaches from several point of views. It handles a larger class of exogenous arrival processes, including important processes like fractional Brownian motion. It can provide a much more accurate and realistic characterization of real-time traffic, when the bounding function is chosen as a sum of exponentials. This enhanced characterization can substantially improve the utilization of networks implementing services based on statistical guarantees. The basic theorems presented in Section I11 and the techniques used for proving them should only be regarded as the building blocks of the SBB calculus. We show in [ 121 that these theorems can be further generalized and extended. For instance, besides the bounds on workload (buffer occupancy) distributions presented in this paper, it is not difficult to calculate bounds on various other performance measures like (virtual) waiting time, mean waiting time and end-to-end delay. We also argue in [12], that the concept of "exponentially bounded fluctuation'', developed in [8] for characterizing the fluctuation of variable rate links, can be extended to more general bounding functions. Note that the analysis we presented here assumes

7 nothing about the service discipline of the work-conserving element. For a specific element, a finer analysis may result in tighter bounds. This is, for example, the case for network elements implementing the WFQ (PGPS) service discipline [4], [201, v11. This work leaves several issues open for further research. For instance, the problem of handling processes with a bounding function f(o) = l/oa for some a > 0. In such a case, the calculus developed here can t be used for feedforward networks of arbitrary size since f.f (see Definition 1). One may also be interested in refining the calculus for sum of exponentials. One of the limitation of this calculus is that the number of exponentials needed, in order to stochastically bound the burstiness of the processes within the network, is growing each time we sum processes. For this reason, we propose in [ 121 an ad hoc procedure which limits the exponentials to a fixed number and still provides much better bounds than the EBB approach. [ 181 P.R. JelenkoviC, The Effect of Multiple Time Scales and Subexponentiality on the Behavior of a Broadband Network Multiplexer, Ph.D. thesis, Columbia University, [ 191 A. Borovkov, Stochastic P roce.wes in Queueing Theory, Springer-Verlag, [20] 0. Yaron and M. Sidi, Generalized Processor Sharing Networks with Exponentially Bounded Burstiness Arrivals, Journal qf High Speed Nerworks, Vo1.3, pp , [21] Z. Zhang, D. Towsley, and J. Kurose, Statistical Analysis of the Generalized Processor Sharing Scheduling Discipline, IEEE JSAC, Vol. 13, No.6, pp , August Acknowledgement The authors would like to thank Prof. Ofer Zeitouni for fruitful discussions. REFERENCES M. Schwartz, Broadband Inregrared Nehvorks, Prentice-Hall, S. Shenker, C. Partridge, and R. GuCrin, Specification of Guaranteed Quality of Service, RFC 2212, IETF, September R. Cruz, A Calculus for Network Delay, Part I: Network Elements in Isolation, IEEE Trans. on Information Theory, Vo1.37, No. I, pp. I I, January I99 I. A. Parekh and R. Gallager, A Generalized Processor Sharing Approach to Flow Control in Integrated Services Networks: The Multiple Node Case, IEEE/ACM TON, V01.2, No.2, pp , April Yaron and M. Sidi, Performance and Stability of Communication Networks via Robust Exponential Bounds, IEEE/ACM TUN, Vol.1, No.3, pp , June J. Kurose, On Computing Per-Session Performance Bounds in High- Speed Multi-Hop Computer Networks, Per:forni. 92, Newport, CA, June C.S. Chang, Stability, Queue Length and Delay of Deterministic and Stochastic Queueing Networks, IEEE Trans. on Auronzatic Cunrrol, Vo1.39, No.5, pp , May K. Lee, Performance Bounds in Communication Networks with Variable- Rate Links, Sigcoinm 95, pp Norros, On the Use of Fractional Brownian Motion in the Theory of Connectionless Networks, IEEE JSAC, Vol. 13, No.6, pp , August A. Erramilli, 0. Narayan and W. Willinger, Experimental Queueing Analysis with Long-Range Dependent Packet Traffic, IEEEIACM TuN, Vo1.4, No.2, pp , April N.G. Duffield and N. O Connell, Large Deviations and Overflow Probabilities for the General Single Server Queue, with Applications, Mathemutical Proceedings of the Cambridge Philosophictrl Sociefy, No. 1 18, pp , D. Starobinski and M. Sidi, Stochastically Bounded Burstiness for Communication Networks, CC Pub #233, Department of Electrical Engineering, Technion, February J. Beran, Srari.~rics,fur Long-Memory Processes, Chapman & Hall, H. Heffes and D.M. Lucantoni, Markov Modulated Characterization of Packetized Voice and Data Traffic and Related Statistical Multiplexer Performance, IEEE JSAC, Vo1.4, pp , B. Maglaris et al., Performance Models of Statistical Multiplexing in Packet Video Communication, IEEE Trans. on Cr)r,inzunic~rrions, Vo1.36, No.7, pp , July D.N. Tse, R.G. Gallager, and J.H. Tsitsiklis, Statistical Multiplexing of Multiple Time-Scale Markov Streams, IEEE JSAC, Vol. 13, No.6, pp , August P.R. JelenkoviC, and A.A. Lazar, The Asymptotic Behavior of a Network Multiplexer with Multiple Time Scale and Subexponential Arrivals, in Srochasric Networkr: Stability and Rare Evenrs, pp , Springer Verlag, editors: P. Glasserman, K. Sigman, and D.D. Yao,