EXPERIMENTAL VALIDATION OF SELECTED RESULTS ON ATM STATISTICAL MULTIPLEXING IN THE EXPLOIT PROJECT

Transcription

1 EXPERIMENTAL VALIDATION OF SELECTED RESULTS ON ATM STATISTICAL MULTIPLEXING IN THE EXPLOIT PROJECT N. Mitrou +, K. Kontovasilis ++, N. Lykouropoulos + and V. Nellas + + National Technical University of Athens, Division of Computer Science, Heroon Polytechneiou 9, Zografou GR , Athens, GREECE mitrou@softlab.ntua.gr ++ NCSR Democritos Institute for Information & Telecommunications GR , Aghia Paraskevi, Athens, GREECE kimon@cyclades.nrps.ariadne-t.gr Abstract A number of experiments were designed and performed on the EXPLOIT ATM testbed, aiming at validating selected results on statistical multiplexing of bursty traffic. The influence of the burst-size and silence-duration distributions on the multiplexer's performance, the performance experienced by the individual classes in a heterogeneous traffic environment and the performance in the case of heterogeneous traffic featuring different time scales are the main issues investigated.. Introduction Exploitation of the statistical multiplexing gain is of prime importance for the successful introduction of ATM networks, since it is the only way to obtain efficient operation under significant and diverse traffic demands. Understanding the multiplexer's behaviour, especially for heterogeneous input traffic, is a basic prerequisite for designing efficient traffic control procedures with the above objective. During the last five years, a great deal of work has been carried out on this subject, mainly theoretical or through simulation due to the lack of real ATM networks. Experimental validation of the obtained results under realistic conditions is a necessary complement of this work, before using it in real networks. Towards this end, a number of important results on statistical multiplexing have been selected out of recent publications and appropriate experiments were designed and performed in the EXPLOIT Testbed (a small scale ATM network developed by RACE projects) as part of the work within the EXPLOIT project. Following this introduction, section outlines the equipment set-up used for the reported experiments. It also states some general observations and comments regarding the derived results. Section 3 addresses an important topic in bursty traffic multiplexing, namely the influence of the distribution of burst and silence duration on the multiplexing behaviour. This work has been partially funded by the European Union. ++ Work done while this author was with NTUA

2 Using simple 3- or 4-state Markovian ON/OFF models, it is demonstrated that hyperexponential distributions result in a worse performance than their exponential or hypoexponential counterparts [8]. Section 4 presents results from heterogeneous traffic multiplexing, with two different ON/OFF classes and a CBR component multiplexed together. A Little-like formula relating the complementary buffer occupancy distribution seen at random instances, G(x), with the one seen by the arriving cells, H(x), is experimentally validated, according to the respective theoretical developments of reference [5]. It is also demonstrated that CBR streams always see G(x), being therefore useful as test streams. An important result regarding heterogeneous traffic multiplexing of streams, some of which feature much larger burst sizes (class ) than the rest (class ), is investigated in section 5. Conventional wisdom dictated that no significant multiplexing gain would be achieved in such a case, because the large bursts could not be absorbed by a buffer of reasonable size. However, according to results stemming from Near-Complete Decomposability developed for fluid models [6], this is not always so. An important relationship between the rates of the two classes that ensures stable operation, no matter how large the bursts of class are, is fully validated. Furthermore, a closed form approximate solution for the long-term buffer occupancy is verified, under the stated stability conditions.. Equipment set-up, traffic generation and measurements. EXPLOIT Testbed (ETB) configuration for experiments on statistical multiplexing All experiments described in this paper were conducted on the EXPLOIT Testbed (ETB), a small-scale experimental ATM network, built by RACE projects for experimentation with ATM traffic. Details about the ETB are given in reference [] within this issue, while some information on the architecture, the modelling and the performance of the main units can be found in ref. []. Since this paper focuses on burst-level statistical multiplexing, all experiments employed just a single ATM multiplexing element (LaTEX - see []) loaded by permanent ATM connections which were driven by sources of specific traffic profiles (see figure (a)). This scheme intended to isolate the output-port congestion phenomena under study from irrelevant details inherent in (and particular to) the more sophisticated switching structures. Figure (b) is the actual testbed configuration used in the experiments. Two Synthesized Traffic Generators (STG# and STG#) are connected in two different input ports of the LaTEX switch, while the Real Traffic Analyser (RTA) module is connected to an output port (through an ATM interface at 55 Mb/s speed). The STGs and the RTA are parts of the W&G ATM-00 Generator-Analyser [3]. Another input port of the switch is used to connect a PC Terminal Adapter [4].

3 3 PC-TA LaTEX ATM-00 Real Time Analyser ATM multiplexer. output line input lines STG # ATM-00 Generator STG # ATM at 55 Mb/s (a) (b) Figure : (a) ATM multiplexer, (b) Configuration of the ETB for the experiments. Traffic generation and analysis capabilities The W&G ATM-00 Synthesized Traffic Generators, used as the main traffic generators in the set of experiments presented, are capable of producing mixes of traffic profiles, each containing up to 047 sources. Each source can be either CBR or of the Markov-Modulated-Rate-Process (MMRP) type with up to 7 states each. The cellpattern sent by the sources in any of the active states can also be defined by the experimenter. A Test Cell Generator module was used to select part of the transmitted stream and put sequence numbers and time-stamps in the payload of the cells on-thefly. The PC Terminal adapter [4] is also capable of generating CBR or VBR traffic. As a CBR generator it can produce traffic with a user-selectable rate up to 78 Mbit/s. As a VBR traffic generator it can produce ON/OFF patterns with exponentially or hyperexponentially distributed ON and OFF phases [0]. Other traffic profiles can also be programmed. The ATM-00 Real Time Analyser has the ability of filtering the received cell stream according to specified VPI/VCI masks and of measuring cell losses, transmit- and receive- interarrival times, cell delay and Cell-Delay Variation (CDV) for the test cell streams on-line. The values obtained from these measurements are also available for further processing at the end of the experiment. In the experiments presented here, celldelay histograms were captured and processed further, as explained in paragraph.4. The traffic generators were carefully programmed so that the aggregate peak rate of each produced stream was kept below the link rate; without this precaution, the traffic profiles would have been distorted, due to saturation within the generators. This constraint has the consequence that the maximum achievable overloading factor equals the number of the input lines fed to the multiplexer (3 in the set-up of fig. ). This drawback can be overcome by artificially reducing the output link rate serving the multiplexer, through using a CBR component of appropriate rate, as explained in the next paragraph.

4 4.3 Traffic models used Traffic models of the ON/OFF type are quite popular for describing bursty traffic for many reasons, including their clear representation of the notion of burstiness and their usual amenability to tractable analysis. Although multirate models may capture more closely the behaviour of some VBR sources, these models need an increased number of parameters for their specification and they require greater complexity at the network side, both for the exploitation of the extra multiplexing gain (due to the finer rate granularity) and for the policing of the parameters specifying the model. Consequently, it is not uncommon to specify a simpler, conservative ON/OFF model that captures the bulk of the VBR properties of such sources and use this throughout. For these reasons we adopted the ON/OFF paradigm for the VBR traffic streams in our experiments, although the theoretical results under validation hold for arbitrary Markov-Modulated Rate models. Besides VBR traffic, CBR streams played an important role in our experimentation. Firstly they were used as "background" traffic, reducing the available bandwidth by a constant amount and creating congestion conditions for the statistically varying traffic component. Secondly they were used as "test" traffic, since as is explained in section 4, the buffer occupancy and the delay distributions experienced by the cells of the CBR component are the same with the respective distributions at arbitrary random time instants and related in a simple fashion to the respective performance measures experienced by the cells of the VBR traffic..4 Performance measures The performance metrics under validation were buffer occupancy and delay distributions under the assumption of an infinite buffering capability. Given any such Complementary Probability Distribution Function (CPDF), say G(x), the actual overflow probability for a real multiplexer with a finite buffer of size B is approximated by its upper bound G(B). The approximation error is very small if the overflow probability is small (so that the accumulated probability at finite buffer boundary is not significant). Figures (a-b) show a typical cell-delay histogram through the LaTEX switch, as measured by the Real Time Analyser. Function h(d) in fig. (a) depicts the actual sample frequencies, i.e. h(d) is the actual proportion of samples having experienced a delay of d slots. The function H(d) in fig. (b) is the sample CPDF, i.e. d H( d ) = h( k ) k = 0. Both functions are discrete (measured in integral ATM slots), although continuous interpolations are shown in the figure. The figures may be divided in four main regions; region R covers an approximately constant delay of 8-9 ATM slots (ending at the ordinate of the narrow peak at the beginning of the h( d ) curve), which is caused by the internal architecture of the LaTEX switch. Region R represents the cell-level congestion (due to the simultaneous cell arrivals from different input lines), being

5 5 usually small (a few cell-slots). Region R 3 (shaded in fig. (b)) is the burst-level congestion region, being of primary importance in statistical multiplexing. Finally, region R 4 reflects the boundary at buffer limit, with the characteristic overshoot of the histogram in fig. (a). log0(h( d ) ) Delay, d (ATM slots) (a) log0(h ( d ) ) Delay, d (ATM slots) Figure : (a) Histogram and (b) CPDF of cell delay through a switching element (b) As already stated, R 3 is the region of our interest. All of the theoretical results that we shall experimentally validate here, were derived through a fluid-flow analysis with an infinite-buffer boundary condition. As we will see, the approximation of the experimental curves within R 3 is quite satisfactory. We avoid to relate the results with any specific switch architecture, considering only the congestion at their output lines. Thus, the approximately constant delay of 8-9 ATM slots, observed through the LaTEX switch, was subtracted from the measurements to

6 6 obtain the final delay figures. This is equivalent to a shifting of the curves of fig. to the left by 8 or 9 positions. 3. Homogeneous traffic multiplexing: Influence of the distribution of burst and silence duration to the multiplexing behaviour Given the popularity of ON/OFF traffic models, it is important to determine the minimal number of parameters which is adequate to provide, at each case, a close representation of the source under modelling. The usual, and extensively used, assumption of exponentially distributed ON and OFF periods leads to a minimum order model with just three parameters (e.g. peak and mean rate and mean duration of the ON period). It has been demonstrated [7,8], however, that the details of the distribution for the ON and OFF period duration have an important effect on the multiplexing potential. As a means for more accurate description, simple 3- or 4-state Markovian models that match the first three moments of the ON and OFF periods have been devised [7,8]. The analytical results demonstrated a considerable variation of the multiplexer's performance, depending on the distributions of the ON and OFF periods. This is also in accordance with other studies [4,5]. Matching experiments were designed and performed on the ETB to validate the analytical results. Figure 3, borrowed from reference [8], shows seven different low-order Markov- Modulated-Rate-Process (MMRP) models for an ON/OFF stream, including the twostate exponential one - in the middle of the figure. The three diagrams on the left side (ii, iii & iv) correspond to hypo-exponential ON and/or OFF sojourn time distributions (each hypo-exponential state being composed by two exponential stages in series), while the three diagrams on the right-hand side (v, vi & vii) correspond to hyperexponential cases (with two exponential stages in parallel composing a hyperexponential state) with the same mean sojourn times. Each elementary state in the diagrams has an exponentially distributed sojourn time with a mean value shown next to the state circle. The composite ON (OFF) phases all have the same mean duration, 560 (840). All values refer to the same, unspecified in the figure, time unit. The transition probabilities, wherever different from unity, are indicated by the arrows. Experiments were performed on each of the models of fig. 3. In each experiment seven statistically independent but identical traffic streams of the respective model were generated and multiplexed on the same output link of the LaTEX. The values of the mean sojourn times, shown for each state in fig. 3, were considered in units of 0µsec, sizing the smallest mean sojourn time (00) at msec. The rate c in the ON state was adjusted to 5 Kcells/sec or. Mb/sec ( cell=44 bits) so that an average of 0 cells were generated during the smallest ON sojourn time (that of the msec). The bandwidth, 8 DC $ D;;, made available to the seven streams of the ON/OFF class was adjusted to yield a normalised peak rate c/8 DC $ D;; = 0., i.e. 8 DC D;; induce this effect a CBR component of rate equal to (55-0.6)Mb/sec was multiplexed together with the ON/OFF streams. $ =0.6 Mb/s. To

7 7 (iii) 840 (ii) 80 3 ON phase OFF phase (iv) 3 OFF phase OFF phase 3 OFF phase (i) ON phase ON phase (v) 840 transient state OFF phase ON phase ON phase (vii) (vi) Figure 3: Low-order MMRP models for ON/OFF traffic streams Fig. 4 shows experimental data (solid lines), along with the analytical results from the fluid-flow analysis (dotted lines). The latter were obtained by solving the fluid model of the system with an infinite-buffer boundary condition. As already commented, there is an expected deviation between the fluid-flow analytical curves and the experimental ones close to the two boundaries, namely zero and full buffer size (48 cells for the LaTEX). The experimental curves of fig. 4 were obtained by measuring the delay of the CBR stream multiplexed together with the ON/OFF traffic. As will be fully discussed in the next section, these measurements do yield the buffer occupancy at random instances. The calculated histograms were shifted to the left, to compensate for the approximately constant delay of 9 time slots caused by the internal switch architecture.

8 8 0 logcpdf(x) theoretical experimental (i) exp ON, exp OFF (ii) exp ON, hypo OFF (iii) hypo ON, exp OFF (iv) hypo ON, hypo OFF (v) exp ON, hyper OFF (vi) hyper ON, exp OFF (vii) hyper ON, hyper OFF (vii) (vi) -5 (iv) (v) buffer occupancy, x (cells) Figure 4: Buffer occupancy distributions for the low-order Markovian models of fig. The results clearly indicate that the -state exponential model is not always sufficient to describe the multiplexing behaviour of ON/OFF streams and that the distribution of the ON and OFF sojourn times is very important. Hyper-exponential distributions lead to a worse performance, contrary to hypo-exponential ones that yield lower delay and overflow probabilities, compared to their exponential counterparts. These results are in accordance with the theoretical predictions [7,8] and, in a different setting of [4,5]. (i) (ii) (iii) 4. Heterogeneous traffic multiplexing: per-class performance The second set of experimental investigations is concerned with heterogeneous traffic multiplexing and the performance seen by individual classes. Suppose that different traffic classes are multiplexed without any priority scheme and let G( x) denote the complementary buffer occupancy distribution seen at random instances, H( x) the buffer occupancy at instances of cell arrivals (of any class) and Hi ( x) the buffer occupancy at arrivals of cells belonging to class i. (Of course, in a fluid-flow setting, a cell is replaced by an infinitesimal amount of fluid entering the buffer.) Since H( x) polls the cell arrivals and since the service time per cell is constant, H( x) is essentially the global delay-distribution. In a similar fashion, the distributions Hi ( x) express the performance experienced by particular classes of traffic.

9 9 These class-dependent performance metrics are important, especially when particularly "difficult" traffic classes are multiplexed with other, less demanding ones; in such cases the performance enjoyed by the difficult classes may be quite different from the "average" performance and the computation of Hi ( x) becomes important. For arbitrary Markovian fluid models, H( x) and the class dependent distributions can be readily computed, without any computational cost in excess of the cost required for the "standard" computation of G( x) [5]. Furthermore, the performance metrics satisfy the following relations: G( x) = ρ H( x) () G( x) = ρ H ( x) () i i i If class j is CBR, H j ( x) = G( x) (3) where ρ i is the normalised load contributed by class-i and ρ = ρ i i is the total normalised load. Relations (-3) were proved in ref. [5] for arbitrary time-reversible Markovian fluid models of multiplexers with infinite buffering capabilities. The timereversibility assumption can be readily relaxed (see the method of proof in [5]). Furthermore, given () and (), relation (3) may be derived in general from first principles, but calling upon the server's work-conserving principle of operation and the fact that in a fluid regime, only the differences between input and output rates matter, not their absolute values (see [9]). The fact that the class of distributions representable by Markov models is dense promotes the conjecture that relations (-3) are valid for arbitrary fluid models of infinite buffer multiplexing. Interpretation-wise, relation () is a kind of "Little's Law" for distributions; this fact becomes apparent by recalling the association of G( x) to the buffer content and of H( x) to the delay experienced. Equation () formalises the intuitive fact that the global performance is a "weighted average" of the performance seen by individual classes. Finally, relation (3) states that the cells from any CBR components always see upon arrival a buffer distribution equal to G( x). This is in line with our intuition, since a CBR stream "samples" the buffer regularly. This property of CBR components is useful in using them as "test streams". In such a case their influence on the multiplexer's performance is restricted to just spending a constant amount of the available bandwidth. This was exploited already in the experiments of the previous section. To test the above mentioned theoretical results, two different ON/OFF classes, with exponentially distributed ON and OFF sojourn times, and a CBR component were multiplexed on the same output link of the LaTEX switch, featuring the characteristics shown in Table. The settings in Table result in the traffic parameters shown in Table, which refer to a multiplexer with reduced output-capacity and no CBR loading.

10 0 ON/OFF() mean ON sojourn time t = 4. 4 msec mean OFF sojourn time s = 99t peak rate c =5 Mb/s number of sources n = 30 ON/OFF() mean ON sojourn time t = msec mean OFF sojourn time s = 4t peak rate c =3 Mb/s number of sources n = 0 CBR Rate 40 Mb/s Table : Experiment-settings Bandwidth available to ON/OFF traffic C ON / OFF = 5 Mb/s mean burst sizes: V = V = 50 cells Normalised peak rates: c% c / CON / OFF = 0. 33, c% = 0. normalised mean rates: r% = c% / 00, r% = c% / 5 Normalised loads: ρ = n r% = 0. 05, ρ = n r% = 0. 4 ρ = ρ + ρ = Table : Equivalent traffic parameters For each of the three classes - ON/OFF(), ON/OFF() and CBR - the distribution of the delay through the switch was measured and the results (after subtracting the constant delay) are shown in fig. 5, denoted by H ( x), H ( x) and G ( x) respectively. The delay distribution, seen by the cells of either ON/OFF class, was also measured and is shown in the figure, denoted by H ( x). Fig. 5 also includes the theoretical curves G( x) and G( x) / ρ (dotted lines), as derived by applying the heterogeneous Markovian fluid-flow multiplexing analysis, presented in [5]. It can be evidenced that the curves G( x) and G ( x), on the one hand, and G( x) / ρ, H ( x) and ρ ρ H( x) + H ( x) on the ρ ρ other hand are very close together, apart from the regions close to the boundaries.

11 0 logcpdf(x) -0.5 V=50, c=0.33, r=c/00, n=4, V=50, c=0., r=c/5, n=0 - H (x) H (x) + H (x) -.5 H (x) G(x) G(x) H(x) G(x) buffer occupancy, x (cells) Figure 5: CBR + ON/OFF- + ON/OFF- traffic, G ( x) : measured on the CBR stream H ( x), H ( x) : measured on the respective classes 5. Nearly-Completely-Decomposable traffic mixes This section deals with traffic featuring excitations happening at more than one timescale i.e., some rate fluctuation patterns occur much more frequently than others. Such phenomena give rise to Nearly Completely Decomposable (NCD) Markovian fluid models [6]. Two generic reasons may be identified behind such structure. In the first case the different time-scales are internal to the behaviour of one or more of the traffic sources multiplexed together (as in e.g., moving video packetised traffic, featuring line frame and scene changes). In the second case, the NCD structure is due to the heterogeneous multiplexing of traffic with very different time-constants (the term timeconstant referring to the order of magnitude of the mean sojourn times that a traffic source spends at the states describing the various rate-levels of its activity). Thus, we refer to the case of mixing "slow" and "fast" sources. In this second case, the disparate time-scales phenomena are not built-in in any of the sources; instead they emerge due to source-superposition. Regardless of its origin, the NCD structure gives rise to analysis techniques, based on model decomposition, that, not only lead to efficient and tight approximations for the global system behaviour, but also clearly identify the important contributors to the

12 small- and large-buffer dynamics, thus enabling a better insight into the multiplexer's performance [6]. When the NCD structure originates from mixing slow and fast sources, the relevant results take a particularly simple form and we now briefly review them. Assume there are two traffic-classes, where class- consists of the slow and class- of the fast sources; in consequence, sources of class- change, on the average, their information flow-rate much more frequently than those of class-. Notice that each class may well be composed of many heterogeneous sources, as long as the order of magnitude of the state-sojourn times of each individual source conforms with the source's classification. Let the aggregate class- traffic span K different rate levels ( ) ( ) ( ) ( ) r, K, r K. These are reached in steady-state with probabilities π, K, πk, respectively. The class- peak rate is then $ ( r ) r ( = max ). We similarly denote the cumulative mean rate of class- by r ( ). The multiplexer's output capacity is C. j j Consider now a fictitious queueing system where the same multiplexer is loaded by the traffic of class- only and the output capacity is readjusted to C r ( ) i.e., the class- traffic is replaced by a CBR stream of rate equal to the mean rate of class-. By solving the fictitious queueing system one obtains the joint steady-state probabilities Fj( ) ( x), viz. the probabilities with which the buffer content is less than or equal to x ( and the input rate is r ) ( ) ( ) j. The CPDF of the buffer occupancy is G ( x) = F ( x). ( ) ( ) In a similar fashion, for each index j such that rj + r < C, consider a fictitious queueing model where the multiplexer is now loaded by the traffic of class- only and the output capacity of the multiplexer is readjusted to C ( ), i.e., the class- traffic is ( replaced by a CBR stream with rate equal to r ) j. Let the CPDF of the buffer occupancy of this fictitious model be denoted as G ( ) ( x ). Given these performance metrics, the CPDF of the buffer occupancy of the original multiplexer, being loaded by both classes, is approximated as j r j j j ( ) ( ) ( ) G( x) G ( x) + F ( 0) G ( x) j j: r ( ) r ( + ) < C j j (4) and the approximation becomes tighter, as the difference between the time-scales of the two classes becomes greater [6]. Equation (4) reflects the model-decomposition benefit obtained by the exploitation of the NCD structure; the global solution is achieved by "pasting" together partial solutions of fictitious lower-order systems that involve only one of the two traffic classes. Besides providing the means for efficient computation, equation (4) brings to light an important systemic property, albeit in an implicit fashion. To uncover it, observe that all ( CPDFs G ) ( x) and G ( ) j ( x ) are sums of negative exponential functions of the buffer occupancy x and the decay-rates of these exponentials are inversely proportional to the time-scales of the respective traffic classes. As a consequence, the presence of the ( "slow" class- incurs a slow overall decay rate for G( x), identifiable in G ) ( x).

13 3 Fortunately, this undesirable effect does not always happen. In case the loading is such that $ ( r ) r ( + ) < C (5) i.e., the cumulative peak rate of the "slow" class plus the cumulative mean rate of the "fast" class does not exceed the link-capacity, then the fictitious queueing system ( featuring class- only never overloads the multiplexer and G ) ( x) = 0, x 0. Equation (4) then reduces to K ( ) ( ) G( x) π j G j ( x) j= (6) and no slow decay-rates occur any more. In other words, equation (5) determines the maximal allowable loading, so that no large-scale buffering phenomena occur. Given that (5) is satisfied, much smaller buffer sizes are required for a target overflow probability. In a qualitative way of interpreting (5), it means that the "slow" class is allotted enough bandwidth, so that its large bursts need not be buffered; only the smaller bursts of the fast class are buffered and this reduces the buffer size required. This is an important implication for traffic engineering and (5) becomes a valuable tool for traffic control, on the presence of traffic with very large bursts (slow time-scales). We will return to these control issues in the discussion for the second experiment in this section. The first experiment was designed to demonstrate the issues behind condition 5. As in the experiment of section 4, two different ON/OFF traffic classes and a CBR component (to artificially reduce the link bandwidth) were multiplexed over the same output link of the LaTEX switch. The settings for the experiment are displayed in Table 3, while the transformed traffic parameters appear in Table 4. The time-constants of the sources in the two classes are quite different, bringing this configuration well into the NCD setting. This can be easily observed by the values in Table 3, where t / t = 750; a similar evidence is offered by the difference in the burst-sizes in Table 4. The experiments are arranged so that, while both the number of sources from each class changes (see Table 3), the total number of ON/OFF sources is kept constant to n + n = 5; this has the consequence of maintaining a constant traffic intensity over all combinations. Furthermore, condition (5) translates to 0. n ( 5 n ) <, equivalently to n < 7. 85, so that the buffer occupancy curves for the first three values of n in table 3 should exhibit a much faster decay rates than the curves for the other values. ON/OFF() mean ON sojourn time t = 696 msec mean OFF sojourn time s = ( 7 / 3) t peak rate c =.5 Mb/s number of sources n =, 3, 5, 8, 9, 0,, 4

14 4 ON/OFF() mean ON sojourn time t =. 633 msec mean OFF sojourn time s = 4t peak rate c =.5 Mb/s number of sources n = 5 n CBR Rate 40 Mb/s Table 3: Experiment settings Bandwidth available to ON/OFF traffic C ON / OFF = 5 Mb/s mean burst sizes: V = 6000 cells, V = cells Normalised peak rates: c% c / CON / OFF = 0., c% = 0. 5 normalised mean rates: r% = 0. 3c% = = r% = 0. c% Normalised load: ρ = n r% + n r% = 0. 03( n + n ) = Table 4: Equivalent traffic parameters The results are displayed on fig. 6, which includes the CPDF of the buffer occupancy, for the various combinations in the number of sources from each class; solid lines correspond to the experiment's data, while dotted lines are the theoretical results from fluid-flow analysis. The experiment's data were again obtained by measuring the delay of the CBR-stream's cells. It can be readily evidenced from fig. 6 that the last three curves (clustered at the bottom of the figure), which correspond to values of n (, 3, 5) that satisfy condition (5), exhibit a much higher decay rate than the rest. In fact, the decay rate of the slowly decaying curves does not vary greatly, even though the number of the slow sources varies from 4 down to 8 (notice in particular the two top-most curves corresponding to n = 4 and n =, respectively). In consequence, the theoretical issues addressed earlier in this section are fully justified.

15 5 0 log(cpdf(x)) - V=6000, c=0., r=0.03, V=, c=0.5, r= n=4, n+ n= theoretical (fluid-flow) experimental buffer occupancy, x (cells) Figure 6: Propagation of large-burst effects on the buffer occupancy depending on condition (5). The second experiment in this section focuses in the domain where condition (5) is valid; in this case the global performance is approximated by equation (6). An important application in this setting is the management of an ABR service environment, recently introduced by the ATM Forum and the ITU [,3]. Class would then correspond to the aggregate Guaranteed-QoS (G-QoS) traffic, while class to the collective ABR traffic. The G-QoS class is allotted peak-rate bandwidth, in contrast to ABR traffic, which uses the remaining available bandwidth [9]. Assume that as G-QoS connections come and go, the ABR connections react to the available bandwidth fluctuations by adapting some parameters affecting their traffic generation. More specifically, assume that each ABR source maintains a fixed Markovian fluid model representation, some parameters of which vary, in order to adapt to bandwidth variations. Note that fluctuations due to call-level dynamics (from both classes) happen at a significantly slower time-scale, compared to the burst-scale rate fluctuations of the individual ABR sources, certainly bringing the compound model into the NCD setting. Although the situation cannot be modelled as the superposition of two independent traffic classes (due to the reactive control), the general theory of ref. [6] is still applicable and similar results are obtained. Specifically, the condition (5) for the absence of large-scale dynamics becomes ( ) ( ) r$ + r < C, j (7) j j where r$ ( ) ( ) j is the total peak rate for the G-QoS traffic and r j the total mean rate for ( ) the ABR traffic, both at the call-level state j. (Note that r j is influenced, both by the number of class- sources present at call-level and by the particular parameter setting

16 6 due to the presence of the class- sources at the same call-level). Condition (7) simply states that the cumulative mean rate of the ABR class should not exceed the remaining available bandwidth. When (7) holds, the approximation in (6) is still valid, although now G ( ) j ( x ) must be computed by considering the particular class- sources, and the parameter settings that apply for call-level j. Recall that any reasonable CAC scheme (on the ABR side) and any reactive control should guaranty that, for the present state of affairs, the global G-QoS is respected, i.e. ( G ) j ( B) < ε j, where B is the buffer size and ε the QoS percentile. It is now an immediate consequence of (6) that K K ( ) ( ) ( ) G( B) π G ( B) < ε π = ε. (8) j j j= j= j In words, any ABR CAC scheme, coupled with any parameter-based reactive control that assure the QoS by considering the loading condition at the time of their action are robust in steady state as connections come and go. It is important to notice that the validity of this result is fairly general and does not depend on any particular assumption about the ABR traffic profiles, apart from their Markovian fluid nature. (The effects of reaction are still present, through variation of these models' parameters.) Furthermore, the result is completely independent of any assumptions on the call-dynamics; this is because arbitrary distributions for the sojourn times of any state can be arbitrarily closely approximated by Markovian modelling, at the expense of increasing the state space an issue irrelevant in this setting. Since the stability result is an immediate consequence of (6) the next experiment was set-up to validate it. Since it was not possible to generate coupled traffic streams we simply validated the simple version of (8) with no adaptable parameters. The configuration included a slow traffic component, modelled as a three-state Markovian stream, with sojourn times equal to 5 sec for all three states. The rates were tuned at ( r ) ( =40, r ) ( =45 and r ) 3 =50 Mbit/s, respectively. Since (6) depends only on the steady-state distribution of the slow component, two alternative transition schemes were considered, as shown in fig. 7(a). Due to symmetry, the steady-state probabilities, ( ), are all equal to /3, for all states and transition patterns. π j The fast class consisted of 30 independent and identically distributed ON/OFF streams, featuring exponentially distributed ON and OFF sojourn times with mean values 0.848msec and 4.55msec, respectively, and a peak rate equal to 5 Mbit/s. These parameters yield a mean burst size of 0 cells and a mean rate of 0.Mbit/s per stream. Since $ ( r ) r ( = ) 3 = 50Mb/s and r ( ) = =3Mb/s, equation (5) is satisfied, because the link rate equals 55 Mb/s. Notice that the time-constants of the second class are O( ms ), while those of the first class are in O( s ), verifying the NCD hypothesis. ( The results of the experiment appear in fig. 7(b). The curves G ) ( ( x) to G ) 3 ( x) are obtained by feeding the switch with the ON/OFF traffic plus a CBR component of rate ( r ), j = 3 j, K,. As in previous occasions, the solid lines correspond to data from the experiment, while the dotted lines are derived by fluid-flow analysis. Under the mark G( x), there are three, hardly distinguishable, curves, corresponding to the averaging of the previous curves and to the direct experimental data derived by loading the switch

17 7 with both the ON/OFF traffic and the three-state component, for both transition patterns. The remarkable coincidence verifies all aspects of (6) i.e., both its averaging character and its independence from the particular transition pattern of the slow class. (i) r () 3 =50 Mb/s t 3 = 5 sec r () 3 =50 Mb/s t 3 = 5 sec 3 (ii) 0.5 r () =40 Mb/s t = 5 sec log G(x) 0 - r () =45 Mb/s t = 5 sec () G (x) 3 r () =40 Mb/s t = 5 sec (a) r () =45 Mb/s t = 5 sec - -3 () G (x) G(x) () () () G (x) 3 [ + G (x) +G 3 (x)] -4 () G (x) -5-6 theoretical experimental 6. Conclusion buffer content x (cells) (b) Figure 7: (a) models of the slow-dynamics traffic (b) Buffer occupancy CPDF, validation of (6) Several experiments relating to the statistical multiplexing on ATM systems were conducted on the experimental EXPLOIT testbed. The experimentation provided the means for checking theoretical results, derived by fluid-modelling theories, concentrating at the burst level. The experiments covered: the influence of burst and silence distributions on the multiplexer's performance; class-dependent performance metrics for heterogeneous multiplexing; and the effect of hierarchical time-scale structure in the dynamics of the traffic.

18 8 The outcome of all experiments demonstrated close agreement with the theoretical prediction. This not only validates the particular issues tested but, maybe more importantly, establishes the burst-level (and the associated fluid-modelling techniques) as a suitable level of abstraction for the study of multiplexing issues under bursty traffic. This is even the more so if one takes into account that the switches used in the EXPLOIT testbed belong to the first generation of ATM switches and have small buffers; the new technology employs significantly larger buffers, something that brings the burst-level modelling even more in the forefront. Some of the experiments demonstrated cases featuring a hierarchical time-scale structure and fully verified relevant theoretical results, based on model decomposition. The exploitation of this hierarchy is of particular importance, because it leads to efficient and simple traffic control algorithms and may pave the way to catering for both constraint-bounded traffic and the kind that is amenable to multiplexing gain. Finally, and in a practical respect, it is of importance to highlight the fact that a CBR test-stream experiences a delay-distribution equal to the distribution of the buffer occupancy. This, although being a rather mundane theoretical result, enhances the practice of experimentation, since it permits the measuring of buffer statistics via delay analysis, the latter being typically easier to obtain. References [] F. Bonomi and K. Fendick, "The Rate-Based Flow Control Framework for the Available Bit Rate ATM Service," IEEE Network, Vol. 9, No., pp. 5-39, 995. [] J. Domingo et al. "Switching Block Studies, Network Performance Evaluation and Traffic Engineering for ATM," European Transaction on Telecommunications, Vol. 5, No., pp , Mar.-Apr [3] ITU Recommendation I.37, "Traffic Control and Congestion Control in B- ISDN," Geneva, July 995. [4] G. Konstantoulakis, V. Nellas, "PCA Test Tool User Manual v.0", 994. [5] K.P. Kontovasilis and N.M. Mitrou, "Bursty Traffic Modeling and Efficient Analysis Algorithms via Fluid-Flow Models for ATM IBCN," Annals of Operations Research, Vol. 49, special issue on Methodologies for High Speed Networks, pp , 994. [6] K. Kontovasilis and N. Mitrou, "Markov Modulated Traffic with Near Complete Decomposability Characteristics and Associated Fluid Queueing Models". Advances in Applied Probability, Vol. 7, No. 4, pp , Dec [7] N.M. Mitrou, S. Vamvakos, K. Kontovasilis, "Modelling, Parameter Assessment and Multiplexing Analysis of Bursty Sources with Hyper-exponentially Distributed Bursts," Computer Networks & ISDN Systems, Vol. 7, No. 7, pp. 75-9, 995. [8] N.M. Mitrou, K. Kontovasilis and V. Nellas, "Bursty Traffic Modelling and Multiplexing Performance Analysis in ATM Networks: A Three-moment Approach," in Performance Modelling and Evaluation of ATM Networks, Vol., D. Kouvatsos, Ed. London: Chapman & Hall, pp. 3-54, 995. [9] N. Mitrou, K. Kontovasilis and E.N. Protonotarios, "ATM Traffic Engineering for ABR Service Provisioning", First International 'ATM Traffic Expert' Symposium, Basel, 0 April, 995.

19 [0] V. Nellas, N. Lykouropoulos, G. Konstantoulakis and N. Mitrou, "Experiments with artificial ON/OFF PC-produced traffic, with arbitrary, programmable ON and OFF sojourn time distributions", EXPLOIT Traffic Workshop, Basel, 4-5 Sept [] M. Potts, "An Overview of Experience and Achievements with the Exploit Testbed". This issue. [] RACE Project R06, "Interim Specifications of the Experiments on Resource Management", RACE 06 Deliverable 08, June 993. [3] Wandel&Golterman, "ATM-00, Description and Operation Manual". User's manual [4] S. Wittevrongel and H. Bruneel, "Effect of the on-period distribution on the performance of an ATM multiplexer fed by on/off sources: an analytical study", Proceedings of the Sixth IFIP WG6.3 Conference on Performance of Computer Networks, Istanbul, Turkey, 995. [5] A. Elwalid and D. Mitra: "Effective bandwidth of general Markovian sources and admission control of high-speed networks", IEEE/ACM Trans. on Networking, June 993, pp