Traffic Equivalence and Substitution in a Multiplexer with ... - CiteSeerX

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Traffic Equivalence and Substitution in a Multiplexer with Applications to Dynamic Available Capacity Estimation Costas A. Courcoubetis, Antonis Dimakis, and George D. Stamoulis Abstract— For a multiplexer fed by a large number of sources, we derive conditions under which a given subset of the sources can be substituted for a single source while preserving the buffer overflow probability and the dominant timescales of buffer overflows. This notion of traffic equivalence is stronger than simple effective bandwidth equality and depends on the multiplexing context. We propose several applications of the above traffic substitution conditions. First, we show that fractional Brownian motion as a single source substitute can effectively model a large number of multiplexed sources using information obtained purely from traffic traces; this has direct application to simple but accurate traffic generation. Second, we focus on dynamic (i.e., on-line) estimation of available capacity and buffer overflow probability. This requires the solution of a double optimization problem expressed in terms of functions whose values are obtained from time averages of the traffic traces over a large range of timescales. We show how to solve this problem on-line by reducing it to the calculation of a fixed-point equation that can be solved iteratively by combining traffic substitution using fractional Brownian motion with dynamic measurements of the actual traffic. We have validated this approach by extensive experimentation with large numbers of real traffic sources that are fed to a high bandwidth link, and comparing our on-line estimation of available capacity and the resulting dynamic call admission control with other existing approaches. The superior accuracy of our approach also suggests that taking the buffer size into account, as does our on-line algorithm, may be vital for achieving approximations of practical interest.

I. I NTRODUCTION Modern broadband networks, with their large capacities, are able to carry simultaneously a very large number of bursty connections, with diverse traffic characteristics. In order to achieve large economies of scale and increase their competitive advantage, such networks must use statistical multiplexing effectively. Thus, it is important for the network manager to know what maximum load the network can handle without degrading its performance below some predetermined level. In practice, the available information about the load is in terms of the actual history of the input traffic. Using this information, one wishes to predict performance parameters that correspond to rare eve nts, such as the probability of buffer overflow. Typically, this must 5 for real-time traffic applications. Related isbe less than sues are the estimation of the amount of capacity that is available for servicing new connections, and the capability to run accurate simulations. Such simulations are useful to validate the end-toend performance of real-time multimedia applications that are multiplexed with the rest of the traffic carried by the network.

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This work is partly supported by the European Union under IST Project M3I (RTD No IST-1999-11429). C. A. Courcoubetis and G. D. Stamoulis are with the Department of Informatics, Athens University of Economics and Business, 47A Evelpidon Str & 33 Lefkados Str., Athens, GR 113 62, Greece, and the Institute of Computer Science (ICS), Foundation for Research and Technology Hellas (FORTH), E-mail: courcou,gstamoul @aueb.gr A. Dimakis is with the Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA 94720-1770, USA, E-mail: [email protected]

f

g

However, in this context, it is difficult to generate background traffic that has the same effects to the above multimedia traffic as the actual network traffic. This is mainly due to the fact that traffic models are inaccurate and it is hard to simulate such a large number of actual sources. In this paper, we propose a new approach for creating a simple source that can substitute for an arbitrarily large number of actual sources feeding a network link. Interestingly enough, not only does this traffic substitution technique simplify the traffic generation process, but it does also speed-up considerably the on-line estimation of available capacity thus being applicable to dynamic call admission control too. Let us first consider the problem of traffic generation. When based on simple traditional traffic models, the prediction of performance as well as dynamic available capacity estimation cannot be very efficient. Such models are not adequate to capture most essential aspects of the actual network traffic. For example, the encoding mechanisms used for multimedia traffic (e.g., MPEG) introduce important structure in the traffic stream, as well as burstiness at various discrete timescales, which depend on the structure of the frame sequence. This structure is believed to greatly influence important performance parameters, such as the buffer overflow probability and the available capacity, especially in the case of small buffers, as is common in networks carrying real-time traffic. A popular alternative to performance analysis is the simulation of reality. In this approach, traffic that looks as close to real as possible is generated and fed to the multiplexer, which is then monitored. As already mentioned, the key obstacle in this approach is the generation of a realistic equivalent for the traffic resulting from a large number of sources (hundreds or thousands). In the past, large traffic generators used dedicated hardware programmed to explicitly emulate the sources in real time. A popular approach uses Markov Modulated Poisson Process models (MMPPs) implemented as state machines. Besides the high cost of such dedicated hardware and doubts about the validity of MMPP models for multimedia traffic1 , state explosion soon becomes the bottleneck if multiple state models are used for improving accuracy ( N sources, each represented by a k state model will in general result in kN states). In this paper we propose a new approach for creating a simple single source equivalent to a large number of such sources. This source can substitute for the given set of sources at the multiplexer without perturbing the dominant phenomena that cause buffer overflows in the particular multiplexing situation. The equivalent source can be defined so that either the buffer overflow probability or the available capacity is preserved after sub1 due to heavy tails of state sojourn times instead of exponential, see [1].

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stitution. Our methodology uses actual traffic measurements and source information given in terms of a number of representative actual traffic traces; there is no need of assuming that the traffic multiplexed is generated according to known traffic models. This approach has several applications including performance prediction, estimation of available capacity, and dimensioning of the network. These can be carried out by means of computationally efficient experiments, employing, instead of many actual sources, the traffic that substitutes for them. The novelty in our approach for traffic substitution lies in the fact that we construct a “local” model for a set of sources, instead of a “universal” one. It is by now well understood that not all aspects of a source contribute equally to the phenomena that cause the overflows at the multiplexer [2], [3]. Depending on the buffer size and the overall traffic mix, burstiness at different timescales becomes the major contributor to the overflows. Our motivation now becomes clear: we approximate the bursty behavior of the sources for the timescales that are the most relevant. In doing this, we use the many sources asymptotic which provides us with the dominant term of the buffer overflow probability and focuses on the most important timescale in which source burstiness contributes to the overflows. This provides the formal framework for defining traffic equivalence: a single source can substitute for a set of sources if the substitution preserves certain important properties of the above large deviation coefficient. This equivalence is more subtle than simple effective bandwidth equality. An equivalence based solely on effective bandwidth equality will in general perturb the operating behaviour of the multiplexer, if applied to a significant fraction of the traffic. Which processes should be used as single source substitutes? Although we do not provide a complete answer to this question, we exploit the use of fractional Brownian motion (fBm) in constructing the equivalent source, due to its analytical effective bandwidth representation allowing simple parameter fitting. Using a single Gaussian process for traffic generation instead of hundreds of sources simplifies considerably the above task, and suggests that such “background” traffic can be generated in real time by software. This scenario is useful when one wants to conduct real experiments for assessing the performance of network control mechanisms, such as admission control, and resource allocation. An important issue is whether fBm can always be used to substitute for actual traffic. We have established by combining analytic properties of the large deviation coefficient with properties validated through experimentation that this is probably the case for MPEG traffic. In the second part of the paper, we extend the work on traffic substitution and develop a new methodology for the fast on-line estimation of the operating point (the value of the parameters s; t in the asymptotic) and of other important performance parameters such as the buffer overflow probability and available capacity. The latter is employed in dynamic call admission control (CAC). Estimation of available capacity is key to our dynamic CAC: a new call is accepted only if a conservative estimate of its effective bandwidth is less than the available capacity estimated over the time scale of the duration of the new call. In fact, the CAC algorithm is combined with the methodology of estimation of operating point and performance parameters, thus resulting

in a control method based exclusively on measurements taken on-line from the actual traffic. We show by means of extensive experiments the accuracy of our estimation methodology and the efficient exploitation of resources that results from the use of our CAC algorithm. Although CAC is traditionally related to the ATM technology, in our case we use the same term in a generic sense to denote any network control that may be used to protect the network from overloads and keep performance within predefined levels. Such a control must decide whether there is enough capacity to accept new connections specified by their traffic contracts. We now briefly discuss related work. Most of the basic results on traffic substitution and their experimental validation were presented in a previous conference paper [4] of ours. The importance of the notion of operating point for the analysis of multiplexers is also apparent in the work of [5]. In particular, the authors establish that if the traffic flows, the buffers and the link capacities of several heterogeneous multiplexers are aggregated, then the QoS level required by each flow is still guaranteed provided that the individual flows (prior to aggregation) are “wellmatched”, i.e., they result in the same operating point. (A counterexample is also provided for the opposite case.) Thus, the many sources asymptotic applies to both our work and [5], and traffic behavior is captured at the important timescales that are given by the operating point. However, our work is in another direction, since we construct a single new traffic flow that can be substituted for the aggregate traffic, regardless of whether this emerges from “well-matched” flows or not. We show that this substitution preserves QoS if the two flows (the single source and the aggregate) are “well-matched” and have equal effective bandwidths. Furthermore, it should be noted that the applicability of the many sources asymptotic formula to estimating buffer overflow probability, the accuracy of such estimations, as well as the usage of the critical timescales in analyzing and interpreting the performance of a multiplexer were analyzed in [2] and [6]. In this paper, we go considerably further, dealing with the notion of traffic substitution (which was first introduced in our previous work [4]), as well as with dynamic estimation of available capacity based on the aforementioned notions and tools. Call admission control based on traffic measurements has been proposed by several researchers as an attractive solution in supporting real-time services over a network offering statistical guarantees [7], [8], [9], [10]. By using traffic measurements instead of traffic descriptors, the levels of utilization achieved are higher than with static CAC, without compromising QoS. In [11] and [9], Chernoff bounds are employed in the estimation of available capacity using ON/OFF traffic approximation arguments. ON/OFF Markov as well as Normal approximations of available capacity are given in [12]. Also, in [13], measurement-based CAC is performed by employing estimates of buffer overflow probability, which are derived online by means of a method based on virtual systems. The effects of measurement errors and call dynamics in performance is considered in [7] and [14], and robust control mechanisms are proposed. Our work on dynamic CAC should be compared to that of [11] and [9]. Since, the estimation of available capacity by our algorithm relies entirely on measurements, it is expected that the utilization attained is in general higher. Moreover, since

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our algorithm employs the value of the buffer size, contrary to algorithms based on Normal approximation of the distribution of the input load, it is expected to outperform such algorithms as well. Finally we should note that the practical applicability of the many sources asymptotic formula to estimating buffer overflow probability is demonstrated in [2] and [6]. In this paper, we go considerably further, dealing with the notion of traffic substitution, as well as with dynamic available capacity estimation. The remainder of this paper is organized as follows: In Section II, we present some background material from the theory of multiplexing and introduce that theory’s notion of available capacity. In Section III, we present our approach for traffic substitution. (We refer to [4] for the experimental validation of traffic substitution.) Furthermore, we present and validate our approach for on-line calculation of the operating point and certain performance parameters including the available capacity (in Section IV), and for dynamic CAC (in Section V) by extending the approach in [7]. Finally, in Section VI we provide some concluding remarks.

(

(

=

=(

=

)

=

[0 ℄ [0 ℄

(0 ℄

h i 1 j (s; t) = log E esXj [0;t℄ ; st

(1)

where s, t are system parameters which are defined by the context of the source, i.e., the characteristics of the multiplexed traffic, their QoS requirements, and the link resources (capacity and buffer). Specifically the time parameter t (measured in, e.g., milliseconds) corresponds to the most probable duration of the busy period of the buffer prior to overflow [16] (i.e., the time-to-fill the buffer). The space parameter s (measured in, e.g., kbits 1 ) corresponds to the degree of multiplexing and depends, among others, on the size of the peak rate of the multiplexed sources relative to the link capacity. Effective bandwidths are increasing in s [15]. In particular, for link capacities much larger than the peak rate of the multiplexed sources, s tends to and j s; t approaches the mean rate of the source; for link capacities not much larger than the peak rate of the sources, s is large and j s; t approaches the maximum value that the random variable Xj ; t =t can attain.

0

( )

[0 ℄

)

=

=

)

=(

)

lim 1 log Q(N ; Nb; Nn) =

N !1 N

2

4s(b + t) inft sup s

st

J X j =1

3

nj j (s; t)5  I ;

(2)

where I is called the asymptotic rate function. The last equation is referred to as the many sources asymptotic and has been proved for discrete time in [16], for continuous time in [17] and for a more special case in [18]. Due to equation (2), the buffer overflow probability (BOP) can be written as P overflow e NI +o(N ), which leads to the following approximation when N is large:

(

log P(overflow)  [s(B + Ct) inft sup s

II. BACKGROUND: T HE M ANY S OURCES A SYMPTOTIC In this section we review some basic results from the theory of multiplexing. The model for the multiplexer is simple in order to illustrate the basic concepts; it can be extended to handle priorities as in [15]. The arrival process at a broadband link is the superposition of independent sources of J types. Let Nj Nnj be the number n1 ; : : : ; nJ ; note that the nj s of sources of type j , and let n are not necessarily integers. This system can be viewed as having N sources of the same type, where a single source consists of a proportion of the J source types and can be characterized by the vector n. The broadband link has a shared buffer of size B Nb and link capacity C N . Parameter N is the scaling parameter (size of the system), and parameters b, are the buffer and capacity per source, respectively. Furthermore, let Xj ; t be the total load produced by a source of type j in the time interval ; t , which feeds the above link. We assume that Xj ; t has stationary increments. The effective bandwidth of a source of type j is defined as [15]

)= (

Let Q N ; Nb; Nn P overflow be the probability that, in an infinite buffer that multiplexes Nn Nn1 ; : : : ; Nnj sources and is served at rate C N , the queue length is above the threshold B Nb. The following holds for Q N ; Nb; Nn :

stR (s; t)℄ 

)=

R ;

(3)

( )

where R s; t is the effective bandwidth function of the aggregate input traffic; since the input sources are independent, this is the sum of the individual effective bandwidths. R can be viewed as the scaled asymptotic rate function. If a QoS requirement of the type P overflow  e is imposed, then the acceptance region (i.e., the set of points n where the above requirement is satisfied) for large N is approximated by an affine in N boundary [2]



(

1

J X

Nj j (s ; t )  C +  B t j =1

(

)

  C ; s

(4)

)

where s ; t is an extremizing pair in equation (2), and C  is the “effective capacity” of the system at the above operating point s ; t . Equation (4) justifies the use of the effective bandwidth j s; t as a relative measure of resource usage for a particular operating point of the link, expressed through parameters s, t. For example, if a source of type j1 has twice as much effective bandwidth as a source of type j2 , then, for this particular operating point of the link, one source of the first type can be substituted for two sources of the second type, while satisfying the QoS constraint. Thus, effective bandwidths can be used as a basis for substitution for sources at a particular operating point, when small percentages of the traffic are to be substituted for2 . Clearly, the point s ; t constitutes a saddle point of the mapping s; t 7! s B Ct stR s; t . That is, the derivatives of this function with respect to s and t both vanish at s ; t , while the corresponding Hessian matrix is non-definite (i.e., neither positive semi-definite nor negative semi-definite). In our case, we have a  Hessian matrix; for such a matrix

(

)

( )

( ) ( ) [( + )

( ) (

( )℄

)

2 2

2 Substituting large percentages of traffic will change the value of (s ; t ) as computed from (2) or (3).

4

being non-definite is equivalent to its determinant being negative. Henceforth, when referring to saddle points, it will always be taken that these correspond to t s. In [19] the proof of the many sources asymptotic was extended to show that as N ! 1

inf sup

   1 NI 1 + O N1 ; (5) e P(overflow) = p 2 2  2N s where (s ; t ) is the operating point (obtained by (2)) and  2 is

given by



h  i 2 M 00 (s ) 2 = 2 log E esXj [0;t ℄ = s M (s ) s=s

( )=

( t + b)2 ;

  E esXj [0;t℄ is the moment generating function

where M s of the traffic process. Based on (5)), we obtain the following approximation:

(

)  p 1 2 2 e NI 2N s = e NI 12 log(2N2 s2 ) :

P overflow

(6)

The above equation is referred to as the many sources asymptotic with the Bahadur-Rao improvement [6]. The term 1 2 s 2
can be approximated by 1 N NI [20]. 2 2 Hence, (6) does not require any additional computations compared with (2). Moreover, if a target buffer overflow probability of e is imposed, then, using the Bahadur-Rao improvement, the corresponding acceptance region is obtained by replacing

log 2

with

log (4

1 2 log (4 )




1 + 21



1

~

capacity can be calculated again by means of an infsup optimization problem, namely

~ =

The many sources asymptotic is very generally applicable. Illustrative examples of its applications discussed in [15] and [3] employ periodic sources, fractional Brownian motion, Markovian sources, policed and shaped sources. Moreover, it is not necessary to assume a particular traffic model; equation (3) can be solved based on trace information of the actual traffic sources [21]. Given a trace of a source of type j , one can construct off-line the log-moment generating function on a grid on the s; t plane; having done this for each j , one can then explicitly solve the optimization problem. A way to considerably speed-up the above procedure is described in Subsection IV-A.

#

P

sb st Jj=1 nj j (s; t) Æ : st

~

(7)

The equivalent capacity C sufficient to serve these sources when N . the buffer overflow probability is e NÆ , is given by C Thus C C can be viewed as the available capacity, which of course, depends on the buffer overflow probability. Equation (7) is essentially the “dual” of (2). This is justified rigorously below. Proposition 1: The following results hold: 1. The pair s ; t that solves the right-hand side of (2) also solves (7) for Æ I . Furthermore, . 2. The pair s; t that solves (7) also solves (2) for . Furthermore, I Æ . Proof: We first show that for t t the s of the term between the brackets in (7) is attained at s s , and is equal to

. Then, it suffices to show that for any t > , the s in (7) is greater than or equal to : P By (2), I  s b t st Jj=1 nj j s; t , for every s > with equality for s s . After rearranging terms we obtain

~= ~

~

( ) = (~ ~) =

~=



sb st

=~

=

( + )

0

)

in (4) (see [6]).

inft sup s

"

=

=

PJ

j =1 nj j

st

where again equality holds if s

(s; t)

I

sup 0 sup ( ) ; 8s > 0 ;

= s. Thus,

P s b s t Jj=1 nj j (s ; t ) I

=  s t P sb st Jj=1 nj j (s; t ) I ; st for every s > 0. On the other hand, I  sups [s(b + P

t) st Jj=1 nj j (s; t)℄, 8t > 0, so sups [s(b + t) P st Jj=1 nj j (s; t) I ℄  0, while the same holds if the term inside the brackets is divided by st. This implies, P

sb st Jj=1 nj j (s; t) I ; 8t > 0 :

 sup st s

( )

Combining the previous two inequalities, we obtain A. Available Capacity Equation (4) can also be used for the purposes of estimating available capacity and performing CAC. In particular, if a new call arrives at a system, this should only be admitted if the inequality in (4) is not violated after adding its effective bandwidth to that of the sources already admitted. (This is valid under the implicit assumption that the operating point of the system will not be significantly perturbed after admitting this call.) An alternative way of applying this idea is to compute the minimum capacity that should be dedicated per source in order for buffer overflow probability not to exceed a prespecified value e NÆ , and then compute the available capacity. Indeed, it follows  from (2) that the1 capacity sufficient per source equals

N Q N ; Nb; Nn  Æ . Optimal N !1

~ = inf : lim

log (

)

s b s t

PJ

j =1 nj j

s t

0

(s; t) sb sup s

I



PJ

st j=1 nj j (s; t) I ; st

for every t > . Hence, result 1 is established. Result 2 is proved similarly. We first introduce some notation to be used throughout the rest of the paper. The multiplexer considered has a buffer of size B and a service capacity C . It is originally fed by traffic R V , where R is the traffic to be replaced, and V is the rest of the traffic. After substitution, the multiplexer is fed by traffic M V , where M is the “model” of the real traffic R. If the

+ +

5

complete traffic is replaced, then by abusing notation we take V . s B Ct stX s; t We define the functions GX s; t and AX s; t s B Ct stX s; t Æ =st, where X s; t is the effective bandwidth of traffic mix X 2 fR; V; M; R V; M V g. Computing the buffer overflow probability leads to the infsup problem of the type t s GX s; t , where s; t is an operating point. On the other hand, computing the available capacity, requires solution of t s AX s; t . We will say that the function X s; t is differentiable at a given point if both its partial derivatives exist and are finite. Suppose that for given traffic R, target buffer overflow probability e NÆ and buffer size B , by solving (7), the equivalent capacity C is determined. Furthermore, assume that the traffic is multiplexed into a link with capacity C , and buffer B . The following result implies that the available capacity C C can indeed be utilized by some extra traffic V the effective bandwidth of which does not exceed it. This can be easily seen when the extra traffic is a constant rate source (e.g., CBR), because its addition is equivalent to reducing the initial capacity of the link accordingly. Below, we prove this for variable rate traffic (e.g., VBR). Proposition 2: If traffic V is also multiplexed with the given traffic R, and V satisfies V s; t < C C for every s, t > , then

=0

( )=[ ( + )

+

( )

( )= ( + ) ( ) ℄

inf sup inf sup

( ) ( ) + ( )

( ) ( )

~

~

~

( )

0

inft sup GR+V (s; t) > NÆ : s ~) Proof: By Proposition 1, we have inf t sups [s(B + Ct stR (s; t)℄ = NÆ. Fix some t > 0; then sups [s(B + Ct) ~ ) stR(s; t)℄, since st(R (s; t)+ V (s; t))℄ > sups [s(B + Ct ~ V (s; t) < C C for every s, t > 0. Since the above hold for every t > 0, then inf t sups [s(B + Ct) st(R (s; t) + ~ ) stR(s; t)℄ = NÆ. V (s; t))℄ > inf t sups [s(B + Ct

Clearly, the above result can be applied to dynamic CAC. Thus, on the basis of an estimate of the available capacity, the extra traffic V can be accepted if V s; t < C C for every s, t > , or equivalently if the peak rate of V is less that C C ; such a method is presented in detail in Section V. One might wonder whether there is any advantage in using equation (7) in conjunction with CAC rather than using equation (2). The answer is affirmative: source V can definitely be accepted C for every s, t > , or equivalently if if V s; t < C the peak rate of V is less that C C . On the other hand, if equation (2) is used, then condition t s GR+V s; t > NÆ must be verified. This is more demanding computationally, because it requires solution of an infsup problem in order to derive the operating point after acceptance of V . On the other hand, the condition V s; t < C C for every s, t > , involves the outcome of (7) under the existing traffic R. It should be noted that there is also another reason for performing CAC on the basis of equation (7) rather than equation (2). This reason is related to the convergence properties of the on-line estimation of buffer overflow probability, and is explained in Section V. Finally, although the condition V s; t < C C for every s, t > employs the peak rate of traffic V , it does not lead to peak rate allocation, because the notion of available capacity C C reflects the multiplexing gain achieved. After adding the new source, our on-line estimation procedure will compute the new

~

( )

( )

0

~

( )

0

~

0 ~ inf sup

( )

~

0

( )

~

~

~

value of C , and hence adjust to the actual value of available capacity that results after the new admission. III. T RAFFIC S UBSTITUTION As already explained in Section II, effective bandwidths can be used as a basis for substitution for sources at a particular operating point. Unfortunately this argument fails if one wants to replace a larger fraction of the total traffic, since this will in general lead to a different operating point s; t . Indeed, it is reasonable to assume that the operating point s; t is dictated by the interaction of the overall traffic mix, and substituting for a single source will have a negligible effect. On the other hand, changing the traffic composition can introduce new dominant timescales of buffer overflows, see [2]. The key for providing the right equivalence condition with the appropriate substitution properties is in equation (2). The conditions we introduce ensure that the term inside the brackets remains approximately unchanged in a neighborhood around the original operating point. Unless a new local optimum is introduced with a smaller value, this will guarantee that the buffer overflow probability will be the same and overflows will occur with the same mode (with respect to the dominant overflow timescales). One can easily extend the approach to include conditions on the first two moments of the processes in order to match other properties of interest such as the average queue length; this extension falls outside the scope of this paper.

( ) ( )

A. The Equivalence Condition In this subsection, we define the condition to be satisfied by a traffic model in order to be substituted for the actual one, and then we prove some important properties of the substitution. The next definition defines traffic equivalence at some arbitrary point s0 ; t0 . Definition 1: Let M and R be traffic with effective bandwidth functions M s; t and R s; t respectively that are differentiable at some point s0 ; t0 . Traffic M is said to be equivalent with traffic R at the point s0 ; t0 if and only if

(

)

( )

( ) ) 8 > R (s0 ; t0 ) = M (s0 ; t0 ) > > > > M < R ( s; t0 ) ( s; t0 ) = s s > > R > > > :

t

(s0; t)

(

s=s0

) (

=

s=s0 M ( s0 ; t) t t=t0

:

(8)

t=t0 It is easy to see that the above conditions define a reflexive, symmetric, and transitive relation, i.e., an equivalence relation defined locally with respect to the point s0 ; t0 . An important observation is that in order to simultaneously satisfy the above conditions between a real traffic R and the traffic generated by a model M , any such candidate model must in general have at least three free parameters whose value will be defined by the above system of three equations. (Note that in some trivial cases less free parameters may be enough; e.g., when R is a CBR source and M is a CBR model, one free parameter is sufficient.) The important property of traffic equivalence is that when defined on a saddle point it preserves the above saddle point after substitution. The problem is that substitution might introduce more saddle points which might in turn produce a new operating point. The above property is formally stated as follows.

(

)

6

Theorem 1: Assume that GR+V (s; t) has a saddle point at (s0 ; t0), in a neighborhood D of which R; M ; V are twice differentiable. If at (s0 ; t0 ) M is equivalent with R, and GM +V has a non-definite Hessian matrix, then GM +V has also a saddle point at (s0 ; t0 ), and furthermore GM +V (s0 ; t0 ) = GR+V (s0 ; t0 ).

Proof: See [4]. The main consequence of Theorem 1 is as follows: If traffic R V feeds the link and has s ; t as the operating point, then, if M is substituted for R, s ; t will still be a saddle point of GM +V s; t and thus a potential operating point; it will not be the new operating point if more saddle points are introduced that achieve a smaller value for GM +V . An intuitive interpretation for this is that, after model M is substituted for R, new more probable timescale(s) of overflow may be introduced. Thus, eventually the two systems would have different buffer overflow probabilities, due to different modes of buffer overflow. Summarizing the above, an essential requirement for our approach of traffic substitution to be effective is that traffic model M preserves the operating point at which is substituted for the actual traffic R. If traffic M satisfies this requirement, then it preserves both the type and the frequency of the dominant phenomena that determine overflows. Below we show that the above always holds in the case of substitution with fractional Brownian motion, in the absence of background traffic V . Regarding the assumption of a non-definite Hessian matrix, since this involves real traffic, it can be checked only through experiments; it can be validated analytically in special cases where all the traffic is generated by models. We have conducted extensive experiments involving combinations of modeled (M ) and MPEG traffic (V ) (see also Section IV in [4]); in all cases, the Hessian matrix was indeed non-definite. In [4] we studied experimentally the accuracy of substitution by comparing the BOP of the actual traffic against the BOP achieved after substituting fBm and ON/OFF Markov modulated fluid models for the actual traffic. In all cases, substitution was highly accurate and outperformed common traffic modeling approaches based on matching the first and second order statistics of the actual traffic. Moreover, substitution using effective bandwidth equality only, i.e., without taking into account the first order conditions, can lead to considerable mismatch in terms of BOP, as a particular example in [4] demonstrates. Thus, Definition 1 presents a concise traffic description that captures the relevant multiplexing information. So far, we have only discussed traffic substitution at the operating point s ; t , by means of which the buffer overflow probability is preserved. Another interesting application of the property of traffic equivalence has to do with traffic substitution at the point s; t which is the pair solving t s AR+V s; t in the calculation of the available capacity. This point is still a saddle point after substitution, and if no more saddle points are introduced, then the available capacity is also preserved. These results are summarized in the following proposition, which can be proved similarly as Theorem 1. Proposition 3: Assume that AR+V s; t has a saddle point at s0 ; t0 , in a neighborhood D of which R ; M ; V are twice differentiable. If M is equivalent with R at s0 ; t0 , and

+

(

( )

(

(~ ~)

(

)

(

)

)

)

inf sup

( )

( )

(

)

AM +V

has a non-definite Hessian matrix, then AM +V has also a saddle point at s0 ; t0 , and furthermore AM +V s0 ; t0 AR+V s0 ; t0 .

(

)

(

)

(

)=

Deriving substitution conditions from traffic traces: an example As already explained in the Introduction, network designers would like to be able to assess performance under real load scenarios. Using our method, this can be done by replacing large traffic aggregations with an equivalent source having a simple model (with at least three free parameters as we discussed previously). In order to do this, one should evaluate numerically for the actual system the value of the operating point s ; t and the values of the left-hand sides of the equations (8). We illustrate the necessary steps involved by means of an example. Suppose that we want to assess the perceived QoS of a particular video (say James Bond) and experiment in a real network link with high degree of multiplexing, by using our method for substituting for the background traffic. To make things concrete, consider a link with capacity 155 Mbps and buffer size of 2000 cells3 , which is fed by the aggregation of 40 independent James Bond movies and 400 independent Star Wars movies (both encoded in MPEG), where we are interested to observe visually the performance of the James Bond movies only. In this case we need a single modeled source to be substituted for all Star Wars movies. Traffic in accordance to this source should then be generated in the experiments, rather than employing 400 different generators of Star Wars. To apply the proposed approach of traffic substitution, we first calculate the effective bandwidth functions SW s; t , JB s; t of Star Wars and James Bond movies respectively using only a single trace for each4 , and then calculate the operating point of the system. The latter is computed by solving numerically st JB s; t SW s; t t s s B Ct involving the effective bandwidth function of both movies. It should also be noted that if we add say one (or only a few) more video movies (e.g., Simpsons cartoon), then the new operating point will not be far away from that already calculated, and that for most practical purposes can be taken to be the same. This remark can be particularly helpful in cases where the application traffic is small relative to the overall traffic mix and is not known prior to substitution; e.g., a videoconferencing application.

(

)

( )

( )

inf sup [ ( + )

(40 ( ) + 400

( ))℄

B. Substitution by Fractional Brownian Motion Models Fractional Brownian motion (fBm) is a Gaussian source with mean rate , and variance of increments over time t given by 2 t2H , where  and H 2 ; are constants. The effective bandwidth of fBm is given as (see [15])

(0 1)

s M (s; t) =  + 2 t2H 1 :

2

(9)

is the Hurst parameter; depending on whether H > 12 or  12 , the source is long- or short-range dependent respectively. Long-range dependent processes have non-summable

H H

3 Note that although our approach is not particular to ATM technology, as explained in the Introduction, buffer sizes and link capacities are given valuesrepresentative of ATM, in order to make the presentation more concrete. 4 the functions are encoded by their values as estimated from the trace, on a finite grid over the (s; t) positive orthant

7

autocorellation functions, while short range dependent sources have summable ones. A nice feature of fBm is that it is completely characterized by three parameters; that is, as many parameters as equations in (8). Solving the system (8) in this case, we have

8 > >  > > > > >
0 s 0 s=s0 > > > >  > R > ( s; t0 ) : 2 = 2t01 2H s s=s0



:

(10)

12

The fact that the number of parameters equals the number of equations makes one wonder whether there is enough freedom to guarantee that the fBm with parameters derived from the above equations is always a valid one, i.e., that  < C , H 2 ; and 2 > . Note that the condition on 2 is always guaranteed, since effective bandwidths are increasing in s [15]. On the other hand, there is no algebraic proof that , as given by the first of the above equations, satisfies  < C for an arbitrary point s0 ; t0 . However, there is important evidence that this holds if s0 ; t0 is the operating point s ; t , as discussed at the end of this subsection. Henceforth, we focus on studying equivalence of real traffic R and an fBm source at the operating point, and we assume that the condition guaranteeing  < C does hold, namely that

(0 1)

0

(

(

)

)

(

R (s ; t ) s

)



R  (s; t )  < C : s s=s

(11)

The following theorem establishes that if condition (11) ap, then H indeed plies to a case of traffic equivalence with V lies in ; , and thus the fBm model derived is valid. Moreover, the theorem states that the operating point is always preserved after substitution. Theorem 2: Consider real traffic R with operating point s ; t . If (11) holds, then there exists a valid fBm source M that is equivalent to R at s ; t . Furthermore, if M is substituted for R, then the same operating point is preserved. The following remark establishes a condition equivalent to (11), expressed in terms of the scaled asymptotic rate function R defined in (3).  R Remark 1: The condition R s ; t s  s s; t s=s < C is equivalent to  R (12) > R: B B Proof: See proof of Lemma 1 in [4]. Regarding the validity of (12), there are interesting indications that in many cases a stronger condition holds, namely

=0

(0 1)

(

)

(



)

(

2

)

(

B

)



R >  R :

(13)

B

This is equivalent with the concavity of the scaled asymptotic rate function R w.r.t. B . Indeed, for the superposition of ON/OFF Markov fluids p when B is small it was shown in [26] that R  C1 C2 B , where C2 > . Also, for large B , it is shown in [16] that R converges to a linear function in B ;



 +



0

similar indications can be found in [24]. On the other hand, concavity is not always the case, as for the sub-bursty Markovian sources considered in [17]. Since we could not hope for a general result validating (13) and/or (12), we had to resort to experimentation. Thus, we have done extensive experiments to validate (12) and (13) for actual traffic of interest (MPEG). In [4] R is plotted as a function of the buffer size B for several cases where many independent yet identical video movies feed a link; In all cases, both conditions were verified. An interesting observation is that if (13) holds, then (3) and (10) imply that H > = , and the corresponding fBm is long-range dependent. It is somewhat surprising that in certain cases a long-range dependent fBm source can be equivalent to a short-range dependent traffic. IV. O N -L INE C ALCULATION OF O PERATING P OINT, P ERFORMANCE PARAMETERS AND AVAILABLE C APACITY A. Iterative Methods for the Calculation of Operating Point The application of our substitution approach relies on knowledge of the operating point s ; t . We have already mentioned that this can be computed off-line by solving the infsup formula in equation (2). This solution is a saddle point of GR s; t . There is no general method for solving such problems especially when R s; t is computed from real traffic traces. A straightforward approach, described in [27], to solve this problem is by taking the maximum w.r.t. s for each t, and then taking the minimum over t. This method is computationally demanding, because two optimization problems need to be solved sequentially5 . Having in mind that a single evaluation (on a particular point s; t ) of the effective bandwidth function R for a real trace may take a few seconds, this method becomes prohibitively slow to be used for the on-line evaluation of the above operating point in a serial implementation. On the other hand, a fully parallel implementation of this approach is non-trivial; moreover, it is not clear whether the potential performance gains will warrant its high implementation cost. Since we are interested in performing on-line estimation ultimately, we are looking for an approach that it is both simple and fast (accurate) enough for practical purposes. We have devised an iterative method for calculating the operating point (and the buffer overflow probability) using the idea of traffic substitution. The key idea is to construct a mapping for which the value of the operating point is a fixed point. The method starts with an initial guess of the operating point, say s0 ; t0 . Then we assume that an fBm model is substituted for the actual input traffic at this point, and using system (10) we compute the parameters of the fBm model. (The details of the computations are given in the next section.) In particular, let fBm0 denote the fBm model obtained, and let 0 , 0 , and H0 denote its parameters. Next, we compute the new value s1 ; t1 to be the operating point of the link fed by fBm0 . This can be computed explicitly (see [15])

(

)

( )

( )

( )

(

(

)

)

5 We can exploit the concavity of s(B + Ct) stR (s; t) in terms of the parameter s in order to speed-up the solutions of the sups problems (cf. proof of Theorem 1 in [4]).

8

giving,

H0 B

t1 =

(1 H0)(C 0 ) ; B + (C 0 )t1 : s1 = 2 t2H0 01

(

(14) (15)

)

The method then iterates by using s1 ; t1 as the current guess of the operating point, at which a new fBm model is substituted for the actual traffic. Since we are iterating by using the substitution equations (10), it follows from Theorem 1 that the operating point s ; t is a fixed point in the above algorithm; i.e., having s ; t as the initial guess, will produce an fBm model preserving the same operating point (see Subsection III-B). We have not yet been able to prove convergence properties for the above procedure, although extensive experimentation has provided extremely positive results: mostly less than six steps were needed for achieving a relative error of less than 10%, between i) the buffer overflow probability (BOP) estimate as given by equation (3) combined with the Bahadur-Rao approximation, evaluated at the operating point estimate obtained at each step of our algorithm, and ii) the corresponding BOP estimate evaluated at the correct fixed point; no divergent case was observed. Figure 1 depicts the series of steps for two starting points. For the starting : ; t0 , the relative errors in BOP in steps 1 point s0 through 5 were 34895%, 170%, 34%, 6%, 1% respectively.

( (

) )

of actual traffic, must be relatively large. One could construct cases where more than one timescales for buffer overflows are likely, in which case the initial point for starting the algorithm could lead to different values for the operating point. Although theoretically possible, this occured rarely in practice under reaB=C , i.e., the sonable choices of t0 . In particular, using t0 minimum time taken to empty the buffer, as a rule of thumb, always led to convergence to the right point for all MPEG traffic we have considered. Since the function s 7! GR s; t is concave, convergence to the maximizing s for a fixed t by iterating using only equation (15) is always achieved. Hence, the choice of s0 is not crucial in determining convergence to the operating t0 kept fixed for a small point, provided one iterates with t number of steps. (In the numerical example above, no initial iteration on s with t0 kept fixed was used though.) The above method was based on the iteration of equation (14) and (15), which stem from the infsup problem for calculating BOP. A similar iterative method can be developed for the infsup problem giving the available capacity (see Subsection II-A). The two iterative methods differ only in the expressions giving the operating point estimates. Here, the relevant equations for updating the estimates of the operating point are:

=

( )

=

ti+1 =

= 0 01 = 300



i (1

 H1 i B p and Hi ) 2Æ 2Æ(1 Hi ) ; si+1 = B

inf sup

0.0121 3+

0.0081 s (1/kbits) 0.0061

+ 1 +2

0.0041 0.0021

50

O 100

0.0001 150

200

250

300

(17)

( )

which are obtained by solving explicitly t s AX s; t for X fBm. Again, in this case by invoking Proposition 3, it follows that the optimal point is a fixed point of the above iteration. Regarding the speed of convergence, again a few steps were sufficient in all cases encountered in the experiments. The fast calculation method described above assumes the availability of the effective bandwidth function (and its derivatives) of the various sources. This can indeed be the case when transmitting traffic that is either known in advance (e.g., stored video) or has a typical behavior (e.g., uncompressed audio). In the next subsection, we combine this method with on-line estimation of the effective bandwidth and its derivatives.

=

O 0.0101

(16)

t (msec)

B. On-Line Calculation Fig. 1. The steps taken to converge by the method for fast calculation of the operating point. 510 Star Wars sources are multiplexed in a 155 Mbps link with 2000 cells buffer size. The contour of GR (s; t) is depicted for the aggregate actual traffic. Two starting points are considered, one at s = 0:01, t = 300 and the other at s = 0:0001, t = 110. It can be seen that convergence is fast for both starting points. The operating point of the system is s = 0:006, t = 168 (marked as ‘1’ at the figure), while there exist two more fixed (saddle) points at s = 0:0042, t = 248 and at s = 0:0097, t = 82 (marked as ‘2’ and ‘3’ respectively). For both starting points, the method converged to the desired fixed point.

We must add that the actual traffic (MPEG) used in the experiments produced a number of saddle points, but there was a substantial difference in the value of the optimum between the dominant one and the rest of the points, which were also rather flat compared with the dominant one. This behavior indicates that the global optimum’s region of attraction, in the case

In order to perform the substitution step involved in the method described above, we must solve the system of equations (10) on-line. This can be done by estimating the effective bandwidth and its derivatives for the incoming traffic. Each pair of (14), (15) and (16), (17) gives rise to a different estimator of the operating point, while the estimators for the effective bandwidth and its derivatives are the same in both cases. Assume tn 1 ; sn 1 is the operating point estimate at the end of the n -th step of the iterative procedure described in Sub; ; : : : For notational convenience, we section IV-A, for n take that the n-th step starts at time 0. Assuming that the input process has stationary increments, we estimate the effective bandwidth during step n, at timescales tn 1 and tn 1 t, by replacing the expectation in (1) with the sample mean of K measurements of load in neighboring time intervals. In particular,

^ ^ ( 1)

=12

^

^ +


9

( ^ ) ^

for each source j , the estimation of j s; tn 1 requires measurements of the load in intervals of duration tn 1 , while the est , requires measurements of the load timation of  s; tn 1 in intervals of duration tn 1 t. Thus, let Xj ta ; tb denote the load produced by source j in the time interval ta ; tb , where  ta < tb . For the estimation of j sn 1 ; tn 1 , we measure Xj at the time intervals ; tn 1 ; tn 1 t; tn 1 t ; : : :, that is

( ^ +
) ^ +


[ ℄ ( ℄ (^ ^ ) ^ ^ (0 ℄ ( +
2^ +
℄

0

1)(t^n 1 +
t); k(t^n 1 +
t)
t℄ ; for k = 1; 2; : : : ; K . In order to estimate j (^ sn 1 ; t^n 1 +
t) we measure Xj at the K consecutive intervals of duration t^n 1 +
t, namely we measure Xj [(k 1)(t^n 1 +
t); k(t^n 1 +
t)℄ ; for k = 1; 2; : : : ; K . Note that the load samples concerning timescale t^n 1 are not taken in consecutive intervals; there is a
t gap between samples. This was done in order for the initiations of the measurements regarding intervals t^n 1 and t^n 1 +
t to be synchronized in each iteration step. Xj [(k

Thus, we obtain the following estimates of the effective bandwidth, which should be maintained on-line:

1

^nj (^sn 1 ; t^n 1 ) = s^n 1 t^n 1 ! K X 1 ^ ^ s ^ X [( k 1)( t +
t ) ;k ( t +
t )
t ℄ n 1 n 1 log K e n 1 j : k=1

(18)

Note that, as revealed in the experiments, the above estimates of the derivatives are not sensitive to the values of s and t. Having estimated the effective bandwidth of the total input traffic and its derivatives, by solving the system of equations (10), we obtain the parameters of the equivalent fBm traffic source. Plugging these parameters in (14), (15) or (16), (17) we obtain the operating point estimate sn ; tn to be used in the next iteration [(n )-th step], and then we continue similarly as above. Based on the two methods for deriving sn ; tn , we derive in the sequel two on-line estimators; namely, one for the buffer overflow probability [based on (14), (15)] and the other for the available capacity [based on (16), (17)]. Finally, it should be noted that the above estimators of the effective bandwidth are derived on a per-flow basis. This approach, however, was selected only in order to facilitate numerical stability in the computations of the estimators. Estimation of the effective bandwidth of aggregations of flows could have also been applied.







^ ^

+1

^ ^

B.1 Buffer Overflow Probability estimation The estimate of the scaled asymptotic rate function [see (3)], at (the end of) step n, is



^ n = s^n 1(B + C t^n 1 ) s^n 1t^n 1 ^nR(^sn 1 ; t^n 1) ; (20) where  ^nR (^sn 1; t^n 1) was defined in (19), and depends on traffic measurements taken during the n-th step only. Using (6), we obtain the Bahadur-Rao approximation of , namely,   ^ n 21 log 4^ n : (21)

^ (^ +
^ ) In Figure 2 we plot the estimates obtained by this on-line it^ (^ ^ +
) erative process for a 34 Mbps link with 500 cells buffer, where s^0 = 0:01 kbits 1 and t^0 = 20 msec. The results were taken [( 1)(^ +
) (^ +
)℄ from 250 replicated experiments, where the input process con= 12 sisted of 90 multiplexed Star Wars sources. The left-hand side ^ (^ ^ ) (^ +
) figures depict the sample means (over this set of 250 experiments) for the quantities of interest, while the right-hand side ^ +
figures depict the corresponding empirical distributions. The mean values of the estimates of the operating point and (^ +
)

Estimates of nj sn 1 s; tn 1 are taken similarly, while t are taken by employing the those of nj sn 1 ; tn 1 tn 1 t ; k tn 1 t , for measurements of Xj k k ; ; : : : ; K , as explained above. Both of these estimates are needed for estimating the two partial derivatives of nj sn 1 ; tn 1 . Iteration step n lasts for time K tn 1 t, since we take K samples of load in consecutive intervals of dut. Thus, parameter K essentially filters out traffic ration tn 1 fluctuation in timescales smaller than K tn 1 t. Since traffic sources are independent, we can add the estimates for each source and estimate the effective bandwidth for the total input process R during iteration step n:

^nR (^sn 1 ; t^n 1 ) =

J X j =1

^nj (^sn 1 ; t^n 1 ) ;

(19)

where J is the number of sources. In the same way, we estit and nR sn 1 s; tn 1 where mate nR sn 1 ; tn 1 s > . Furthermore, the derivatives in (10) are approximated as follows:




0

(^

^

+
)

(^

+
^ )

 ^nR ^ ^nR (^sn 1 +
s; t^n 1 ) ^nR (^sn 1 ; t^n 1 ) ( s; tn 1 )  s
s s=^sn 1

and

 ^nR ^nR (^sn 1 ; t^n 1 +
t) ^nR (^sn 1 ; t^n 1 ) (^ sn 1 ; t) :  t
t t=t^n 1

the effective bandwidth are accurate and the corresponding empirical distributions have relatively small variance. On the other hand, this is not the case with the BOP estimates as seen from the empirical distribution in Figure 2(b). This is due to the fact that estimator (20) depends only on the traffic observed during each iteration and ignores past traffic behavior, i.e., depends only on the last K samples. K captures the trade-off between the speed of convergence and the variance of estimates: for small K , the algorithm converges faster but the variance of the value obtained in different experiments is large; on the other hand, for large K , the variance is considerably smaller, at the expense of much larger estimation intervals, thus resulting in large convergence time. Since such estimates are to be used for the purpose of dynamic CAC, large convergence times are clearly not acceptable. Figure 3(b) depicts estimates of taken by n for one run with K , i.e., about one sample is taken every 20 seconds. Clearly, this highly varying behavior in time cannot provide a robust estimator of BOP for the purpose of dynamic CAC where the traffic mix may considerably vary as time

= 500



^

10

8

0.06

Φ (on-line) Φ with Bahadur-Rao (on-line) Φ (exact) actual BOP

7



single run. The estimates of obtained using (22) are depicted in Figure 4; these should be compared to Figure 2(a),(b).

distribution of Φ estimates

0.05

6 0.04 5

8

0.03

0.02

3

12

actual BOP Ψn (run 1) Ψn (run 2)

4

Φn Ψn

11

actual BOP

7

0

20

40

60

80 100 time (sec)

120

140

160

0

1

2

3

4

(a)

5

6

7

8

-log10(BOP)

1

0.01

9

-log10(BOP)

10 2

6

5

(b) empirical distribution of eff. band.

3

0.08

0

500

1000

0.07

35

0.05

Mbps

1500 2000 time (sec)

2500

3000

4

3500

0

500

1000

(a)

0.06

1500 2000 time (sec)

2500

3000

3500

(b)

^ n for two Fig. 3. (a) Samples taken by (the Bahadur-Rao approximation of) ^ n and ^ n . independent runs. (b) Samples taken by

30 0.04 0.03 25

7

5

0.09 eff. band. (on-line) eff. band. (actual)

8

6

4 40

9

0.02 0.01

20

0

20

40

60

80 100 time (sec)

120

140

0 10

160

15

20

(c)

25 Mbps

30

35

40

8

0.08

Ψ with Bahadur-Rao (on-line) Φ with Bahadur-Rao (exact) actual BOP

(d)

distribution of Ψ (w. Bahadur-Rao) estimates

0.07

7 100

0.06

0.06 t* (on-line) t* (actual)

90

empirical distribution of t*

0.05

6 0.05

0.04

80

msec

5

0.04

70 60

0.03

50

0.02 0.01

0.02

3

40

0

20

40

60

0.01

30 20

0.03

4

0

20

40

60

80 100 time (sec)

120

140

0 10

160

20

30

40

50

60

70

80

(e)

(f)

0.09

140

160

180

0

1

2

3

4

5

6

7

8

9

(b)

Fig. 4. Estimates of the BOP produced by means of the Bahadur-Rao approxi^ n . (a) Sample mean of the estimates and (b) empirical distribumation of tion of estimates at t = 100 sec.

0.35 s* (on-line) s* (actual)

120

(a)

msec

0.1

80 100 time (sec)

empirical distribution of s* 0.3

0.08 0.25

1/kbits

0.07 0.06

0.2

0.05

0.15

^

0.04 0.1 0.03 0.05

0.02 0.01

0

20

40

60

80 100 time (sec)

120

140

0 0.01

160

(g)

The accuracy of the estimator given by (22) could be further improved by discounting the weight of older values of i . We do not deal with this issue any further, because we shall not study dynamic CAC based on the estimate of BOP, as explained in Section V.

0.02

0.03

0.04

0.05 0.06 1/kbits

0.07

0.08

0.09

0.1

(h)

B.2 Available Capacity estimation Fig. 2. On-line estimation of the BOP (a)-(b), effective bandwidth (c)-(d), time parameter (e)-(f), and space parameter (g)-(h) for MPEG-1 traffic. All empirical distributions are taken at t = 100 sec.

Using (7), we obtain the available capacity estimate at the nth iteration step:

bust estimator of BOP in a static scenario where the traffic mix does not vary in time. In fact, in order to limit the variations in estimating , we use the following estimator instead of n itself: n X e ^ i : (22) n n i=1

^



^ = log 1

Similarly to (30), the Bahadur-Rao approximation in this case is 1  n . n 2

^

log 4 ^

Using (20) and the strong law of large numbers it is easy to see that n is consistent if sn and tn are fixed (as n changes), i.e., , with probability 1. Though the operating n!1 n point estimates are not fixed, they do not change significantly, as depicted in Figure 2(e)-(h), and n is still consistent, with good approximation, as depicted in Figure 3(a), where the samples taken in two independent runs are depicted. Figure 3(b) illustrates the estimates n and n for the same input traffic over a

lim

^

^

^

^ = 

^

^

^

s^n 1 B s^n 1 t^n 1 ^nR (^sn 1 ; t^n 1 ) ; s^n 1 t^n 1

A^n = C

^ n as a basis to provide a ropasses. In what follows, we use 

(23)

where e is the specified buffer overflow probability. When 
the Bahadur-Rao approximation is employed, 21 

1 + 21



log(4 )

1

is used instead of in (23). We aim at using the above estimator in dynamic call admission control. Thus, when a connection request arrives for a session lasting for time T , we should use the available capacity estimate over this sessions’ duration. If all sessions last for the same time T , then K (i.e., the number of samples at each itert  T . [Recall ation) should be chosen such that K tn 1 that iteration step n requires time K tn 1 t .] For example, take a session that lasts for 10 sec, and another one that lasts for 2 hours. Then, in order to decide whether the first session should be admitted or not, it suffices to check that enough bandwidth is available for the next 10 sec. As for the second session, enough available capacity should be ensured for the 2 hour duration. Thus, available capacity makes sense

(^ +
) (^ +
)

11 6

0.84 actual BOP specified BOP

actual utilization

0.835 0.83

5 link utilization

0.825 -log10(BOP)

to estimate over a given time-horizon rather than for indefinite time. Therefore, for a given link, available capacity should be estimated for a number of different values of K , thus giving rise to different estimates each corresponding to a different timehorizon.

4

3

0.82 0.815 0.81 0.805 0.8

2

0.795 0.79

We have estimated available capacity using the same link scenario as in Subsection IV-B.1, for five different values of K . Thus, five different algorithms implementing the fast calculation method described in Subsection IV-A, based on the same observed traffic stream were run concurrently and independently for different values of K . In particular K ; ; ; ; ; . Figure 5 depicts how the available capacity estimates evolve in time for K . We determined the accuracy of the available ; ; capacity estimates obtained during each iteration period as follows: First, we recorded these estimates, and then, we ran the same experiment again but we multiplexed during each time period an additional constant rate source that consumed an amount of bandwidth precisely equal to the estimate recorded at the previous experiment for the specific time period. The above pair of experiments was repeated for an adequate number of times, so that 95% confidence intervals were obtained for the BOP and the achieved link utilization measured in the second series of experiments.

= 100 500 1000 2000 10000 18000 500 1000 2000

=

10 K=500 K=1000 K=2000

available capacity (Mbps)

8 6

1

0.785 0

5000

10000 K

15000

(a)

20000

0

5000

10000 K

Fig. 6. Accuracy of the available capacity estimator for different time-horizons, i.e., values of K . (a) achieved BOP, using 95% confidence intervals, (b) achieved link utilization.

V. DYNAMIC C ALL A DMISSION C ONTROL A. CAC Algorithm We propose a dynamic CAC algorithm based on the on-line method for calculation of available capacity developed in Subsection IV-B.2. We have opted not to study dynamic CAC based on the on-line estimate of BOP, because verification of the corresponding acceptance condition is more computationally demanding (see end of Subsection II-A), while with a realistically varying mix of sources such an estimate does not converge as fast as necessary (see Subsection IV-B.1). At the moment of a call arrival, a decision is being made depending on whether the estimated available capacity A can accommodate the call or not, assuming that its traffic has the worst possible behavior subject to the constraints imposed by the traffic contract. Since ON/OFF traffic in known to exhibit this worst case behavior, a call is admitted if and only if

^





2 0 -2 -4 0

200000

400000 600000 time (msec)

800000

1e+06

Fig. 5. Available capacity estimates for different time-horizons, i.e., values of K . The link is 34 Mbps, with 500 cells buffer and 90 Star Wars sources are multiplexed together. Note that, for different call durations, different amounts of capacity are estimated as being available.

Figure 6 depicts the experimental results for the BOP and the utilization. In Figure 6(a) the actual measured BOP is plotted for each value of K . It is very close to the specified BOP for all values of K used, thus implying that the estimator of available capacity is very accurate. Figure 6(b) depicts the measured link utilization for the different values that K assumes. As expected, there is a gain in utilization when considering available capacity in short time-horizons (small K ); using the long-term available capacity estimate (large K ) is conservative, however, only resulting in 0.05 less utilization. Thus, if the duration of a session is known, then an increase in utilization is possible. In general, however, the duration of a session is not known. We can then employ the long-term available capacity estimate, which provides a lower bound to the available resources, without significant reduction in the utilization attained.

20000

(b)

m exp s^nt^np
1
A^  log 1 + p

4

15000

:

(24)

The right-hand quantity is the ON/OFF bound of the effective bandwidth of a source with known mean rate m and peak rate p; the bound is tight when the source is indeed ON/OFF (see, e.g. [28]). In general these rates are not known in advance but it is common to provide upper bounds through leaky bucket descriptions which are used to police that source. In the context of ATM the relevant parameters are the sustainable and peak cell rates (SCR, PCR) [29]. (While the interpretation of PCR is straightforward, it should be noted that SCR is the maximum permissible average rate over any time window of a prespecified large duration.) In what follows we assume that m SCR PCR. sn and tn (although, in general m  SCR) and p are the current operating point estimates obtained in the recentmost iteration, while A is estimated as given in (23) (with the Bahadur-Rao approximation), by taking into account only traffic measurements since the beginning of the current iteration. A backoff strategy is employed, similar to the one described in [7], whereby at a call rejection all subsequent call requests are rejected until a call leaves the system. For the sake of comparison, we compare the performance of our algorithm with that of the optimal one, where the exact statistics of all traffic sources are known in advance, and hence the exact acceptance region can be computed. It should also be noted that although the above algorithm is based on Proposition 2, the acceptance condition (24) that was

^

=

^

=

^

12

employed is different (and less conservative) than that implied by this result, namely A  PCR. This is motivated by the fact that we are experimenting with small sources, which are not expected to alter considerably the operating point upon admission. Thus, it is sufficient to check the admission condition of Proposition 2 for the new call V (namely, that V s; t < C C A) only for the present estimate of the operating point; this check is performed by employing the ON/OFF bound of the effective bandwidth of V .

^

~= ^

( )

B. Experimental Results

0.035 0.03 0.025 0.02 0.015 380 360 340 320 300 280 260 # of voice calls 240 220 200 180 160

0.01 0.005 0 65 70

We simulated a 34 Mbps ATM link with 500 cells buffer, carrying video and voice calls. Voice calls are modeled as ON/OFF Markov fluids, where the sending rate while the source is in the ON state is 64 kbps and the mean sojourn times are 352 msec and 650 msec for the ON and the OFF states respectively. Video calls carry Star Wars MPEG-1 traffic, and each of them starts at a random instant within the entire duration of the trace; when the end of the trace is reached, traffic generation loops back at the beginning. Calls of each type arrive according to independent Poisson processes with rate 5 calls/sec. Call durations are exponentially distributed, with mean values 5 min and 3 min, for the video and the voice calls respectively. These are considered as representative values for the respective durations. Calls are admitted according to the CAC algorithm described in the 4. The estimates previous subsection. The target BOP was . of the operating point were obtained for K We ran one experiment for about 135 hours of simulation time, and measured the achieved BOP periodically in 10 minute intervals. This interval length was selected so that the collected measurements correspond to a time-horizon of practical interest. The warm-up period was 1000 seconds. With the target BOP 4, the value achieved in the simulation was 4:3437, being 4:2430. Link while the upper 95% confidence boundary was utilization was 79.11%0.07% at a confidence level of 95%. In order to compare these results with that of the ideal CAC algorithm, we derived the acceptance region of the latter (for tar4) by brute force simulations. As already get BOP equal to explained, such a CAC algorithm corresponds to the case where traffic statistics are known in advance. The maximum utilization in this case, for points of the acceptance region close to the points that our CAC algorithm operated, is 82.2%, i.e., our algorithm achieved about 96.2% of the ideal performance in terms of utilization. Figure 7 depicts the distribution of calls for the two traffic types, as well as part of the ideal acceptance region. The fraction of time spent in non-permissible states is 24.2%. However, as apparent from Figure 7, the largest portion of this time is spent in non-permissible states that are very close to the boundary of the acceptance region. (Thus, the deterioration of the BOP associated with these states is minor.) Indeed, if the boundary were shifted so that 2, 4, 6, 8, 10 more video calls were included in the acceptance region, then the above fraction would be 9.1%, 2.5%, 0.54%, 0.1%, 0.006% respectively. To investigate the effect of the time-horizon parameter K in the performance of our algorithm, we conducted experiments for different values of K using the same simulation parameters as above. In Table I, we present the achieved BOP and utiliza-

10 = 1000

10

10

10

10

75 80 85 90

# of video calls

Fig. 7. Empirical distribution of calls in progress. The curve depicted is the boundary of the acceptance region.

tion. (BOPhi denotes the upper 95% confidence boundary of the BOP.) Observe that the BOP achieved was always below the 4 and relatively insensitive to K . target of

10

K

BOP 4:3437 4:4104 4:3732 4:5247

10 10 10 10

1000 2000 3000 5000

BOPhi 4:2430 4:3253 4:2202 4:3793

10 10 10 10

util.(%) 79.11 79.04 79.03 78.90

TABLE I

To further motivate our approach, which takes buffer size into account, we compare our CAC algorithm to another one that does not take buffer size into account. The following CAC approach, which is based on the Central Limit Theorem, is widely employed in practical situations (e.g., see [10], [14]): the mean and variance of the input traffic are first estimated, and then the BOP is estimated using a Normal distribution of the input load. Such an approach has the advantage that it does not rely on any specific traffic model. (Recall that our approach does not rely on such models either.) Its wide use mainly stems from the fact that it requires relatively simple measurements. On the other hand, such approximations are based on a bufferless model and it is not clear how to incorporate buffer sizes. In what follows, we describe a simple algorithm based on a Normal approximation and compare its performance with our algorithm. Let the mean and variance estimates at time t of the input traffic be  t and  2 t respectively, where

^( )

^(t) = ^ 2 (t) =

Z t

0 Z t

0

0 

0 

^ ()

n X j =1

1

X_ j [0; t  ℄A h( )d ;

n  X j =1

X_ j [0; t

1

1 ^(t)2A h( )d ; ℄ n ) _ [0 ℄

for each t  , where h  T 1 =T . Xj ; t denotes the instantaneous load produced by source j at time t, i.e.,

0

()=

exp(

13

[0 ℄

the time derivative 6 of Xj ; t . [The index j varies over the set of calls present at each time instance t.] Here T is a “memory” time-constant which controls the impact of old measurements on current values, i.e., the memory of the estimators. Thus, T is analogous to the parameter K introduced in the CAC algorithm based on the available capacity. By the Central Limit Theorem, the marginal distribution of the aggregate input traffic will be approximately Normal, and its mean and variance can be estimated by  t and  2 t respectively. Hence, we are led to the following CAC rule: When a new call arrives, it is accepted if and only if

^( )

T

(sec) 10 50 100 200

^ ()

Q



C







 BOPtarget ;

= ^( ) +

= ^ ( )+

(

(

)

= 10

10

= 200 = 5000 65 6 78 9 13%

6 Since we are considering a fluid flow simulation model for the input traffic,

time derivatives of Xj [0; t℄ are well-defined.

BOP 6:1605 5:5901 5:2730 4:8130

10 10 10 10

BOPhi

10 6:000 10 5:5529 10 5:2218 10 4:7696

util.(%) 68.84 67.31 66.54 65.61

TABLE II

(25)

where Q is the upper tail of the standard Normal probability denSCR and  2  2 t SCR PCR sity function,   t SCR . Thus, besides the contribution of the traffic load of the calls in progress,  (resp.  ) above also incorporates the mean (resp. a bound on the variance) of the incoming call, in order to sharpen the BOP estimate. Note that SCR PCR SCR bounds from above the variance of the new call, and corresponds to the case where this call is an ON/OFF source of peak PCR and mean SCR. Observe that the approximation of a new call as an ON/OFF source was used in our algorithm too [cf. inequality (24)]. For insights regarding the performance of such CAC schemes the reader is referred to [10] and [14]. In Table II, we present the BOP and the utilization achieved by simulating the algorithm described above with different values of the “mem4 . In all simulations, ory” time-constant T for BOPtarget the duration of calls, call arrival rates, simulation duration and the way estimates of the BOP and utilization were taken was the same as in the simulations of our CAC algorithm based on the available capacity. Note that since the CAC algorithm based on the Normal distribution does not employ the buffer size at all, it is expected that its BOP is lower than BOPtarget . On the other hand, as T increases, the BOP comes closer to the target value and it is within the range of the BOP that our algorithm seconds, achieves as shown in Table I. Indeed, for T 4:8130, which is roughly half of the BOP of the the BOP is . At the CAC based on the available capacity for K same time, their achieved utilization is : % and : % for the “Normal-distribution”-based and for the available capacitybased CAC respectively; i.e., the latter achieves more utilization. Observe that as the utilization increases, the estimate of the BOP decreases. This apparent paradox can be explained as follows: For different values of the “memory” time-constant T , a different traffic mix is admitted; the temporal characteristics of this mix [which influence BOP, as evident by equation (2)] vary with T considerably more than its marginal distribution, which for all values of T satisfies condition (25); on the other hand, utilization depends only on the marginal distribution of the input traffic. Thus, a simple CAC algorithm that does not take buffer into account may fail to capture the trade-off between loss and utilization; this makes system tuning (e.g., selecting an “optimal” T ) an intricate task. Observe that one could “tamper” with C in (25) so as to ac-

)

count for ignoring the buffer size, thereby achieving higher utilization levels. However, the problem mentioned in the last paragraph lies deeper and is not avoided by merely changing the value of some parameters.

VI. C ONCLUDING R EMARKS In this paper we have proposed a new notion of traffic equivalence based on preserving important properties of the large deviation coefficient of the buffer overflow probability. It can be used for constructing simple traffic generators in software, which can emulate an arbitrarily large number of real sources from their actual traces, possibly in real time. We have also developed a new methodology for on-line estimation of the operating point, and important performance parameters (such as the buffer overflow probability, and available capacity), which was then employed in dynamic CAC. Extensive experiments showed that this estimation methodology is very accurate, while the dynamic CAC algorithm results in very efficient exploitation of resources. The nice properties of our algorithms are also owed to the fact that it employs the value of the buffer size, contrary to most of the commonly used CAC algorithms. Our work applies to any technology offering statistical QoS guarantees, such as ATM and Integrated Services. It can also be applied to technologies dealing with aggregations of sources, such as to Differentiated Services, for dimensioning purposes. Although the use of fBm for constructing traffic substitutes is promising, using processes with more parameters should be further investigated. This is important for extending the approach to match other measures of the queueing process, such as the mean queue length. Since this particular direction is well understood by now (e.g., see [23], [30]), combining these approaches should be feasible. Another direction for future research is the application of our work on a network-wide basis, possibly in conjunction with routing that is based on link metrics. Acknowledgement: The authors are grateful to Frank Kelly and Vasilios Siris for useful discussions on the subject of the paper. R EFERENCES [1]

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Costas A. Courcoubetis was born in Athens, Greece. He received his Diploma (1977) from the National Technical University of Athens, Greece, in Electrical and Mechanical Engineering, his M.S.(1980) and Ph.D (1982) from the University of California, Berkeley, in Electrical Engineering and Computer Science. From 1982 until 1990 he was Member of the Technical Staff (MTS) in the Mathematical Sciences Research Center, Bell Laboratories, Murray Hill, NJ, and from 1990 until 1999 he was Professor in the Computer Science Department at the University of Crete in Heraklion, Greece, and headed the Telecommunications and Networks Group at the Institute of Computer Science, FORTH. Since Fall 1999 he is a Professor in the Computer Science Department at the Athens University of Economics and Business. His current research interests are economics of networks with emphasis in the development of pricing schemes that reduce congestion and enhance stability and robustness, quality of service and management of integrated services, performance and traffic analysis of large systems, applied probability models. Other interests include the combination of e-commerce technologies with telecommunications, the design of auctions for the allocation of scarce resources such as spectrum and power distribution capacity, and formal methods for software verification. He is a member of the National Research Consul, and has consulted for the Greek Telecommunications and Energy Regulatory Commissions.

Antonis Dimakis received the B.Sc. and M.Sc. degrees in computer science from the University of Crete, Greece, in 1996 and 1999 respectively. From 1994 to 2000 he has been a member of the Telecommunications and Networks Division of the Institute of Computer Science (ICS), FORTH. Since 2000 he is pursuing a Ph.D. degree in the department of Electrical Engineering and Computer Sciences at the University of California, Berkeley. His research interests include resource allocation, network control, pricing, stability and performance analysis.

George D. Stamoulis received the Diploma in Electrical Engineering (1987, with highest honors) from the National Technical University of Athens, Greece, and the M.S. (1988) and Ph.D. (1991) degrees in Electrical Engineering from the Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. Since 1995 he has been a member of the Telecommunications and Networks Division of the Institute of Computer Science, Foundation for Research and Technology Hellas (ICS-FORTH). From 1995 to 2000 he was also an Assistant Professor at the Computer Science Department of the University of Crete, Greece. Since the end of 2000 he is an Associate Professor in the Department of Informatics of Athens University of Economics and Business (AUEB). His main research interests are in network performance analysis, network economics and charging for telecommunications services, and auction mechanisms for bandwidth.