Fitting Algorithms for MMPP ATM Traffic Models

Transcription

1 Fitting Agorithms for PP AT Traffic odes A. Nogueira, P. Savador, R. Vaadas University of Aveiro / Institute of Teecommunications, Aveiro, Portuga; e-mai: (nogueira, savador, rv)@av.it.pt ABSTRACT In this paper, we propose and study fitting agorithms for PP() and CPP AT traffic modes, which are specia cases of arkov oduated Poisson Processes (PP). Two fitting agorithms, both based on the ce interarriva times, are considered for the PP() mode: one fits the cumuative distribution and auto-covariance functions and other fits the first three moments and auto-covariance function. The fitting agorithm for the CPP mode is based on the cumuative distribution and autocovariance functions of the arriva rate. The PP() is evauated as a mode for the superposition of IPP sources; the CPP is evauated as a mode for PP(), PP(3), PP(5), IPP, IDP, Pareto and Sef-Simiar traffic. The proposed agorithms can be used in the characterisation of AT traffic streams and in connection admission contro procedures. KEYWRDS PP, CPP, cumuative distribution function, auto-correation, auto-covariance, Ce Loss Ratio, Average Waiting Time.. INTRDUCTIN Broadband networks based on AT technoogy are expected to carry a variety of traffic types, with mutipe characteristics and requirements, in an integrated fashion. The design and contro of these networks is carried out using a set of parameters that describe the main traffic characteristics (e.g. the peak ce rate, the maximum burst size, etc.). Therefore it is important to capture ce arriva fows and describe them through suitabe stochastic modes. An appropriate traffic mode aows a better resource utiisation without performance osses. A traffic mode is a mathematica description of a specific traffic type. In order to buid a mathematica mode from a measured ce arriva fow, some of its statistics must be known. For exampe, the mean, variance and auto-correation function of the ce interarriva time process or the mean, variance and peakedness of the ce counting process. These statistics are measured or cacuated from observed traffic data. The actua set of statistics used in the inference process depends on the impact that those statistics may have in the main performance measures. An effective traffic mode has to reproduce the first and second order statistics of the origina traffic sampe. The distribution function defines the first order statistics whereas the second order statistics can be accounted for by the auto-correation function. The second order statistics pay an important roe in traffic modeing, because traffic auto-correation is an important factor in AT ce osses due to buffer and bandwidth imitations. The arkov oduated Poisson Process with two states, PP(), is a non-renewa mode that has been widey used for modeing AT traffic. In this paper, we present and compare two fitting agorithms for the PP() mode, based on the characterisation of the ce interarriva time process: one approach fits the compementary cumuative distribution function (CDF) and the auto-covariance function (and is adapted from the study reported in []); the other approach fits the first three moments and the auto-covariance function. The Circuant-oduated Poisson Process (CPP) is a particuar case of the PP process, where restrictions in the transition rate matrix assure that the N states of the PP process are equiprobabe. In this paper, we present a modeing methodoogy for the CPP process based on the characterisation of the arriva rate CDF and autocovariance functions (adapted from the study reported in [3] and [4]). This paper is organised as foows. In sections and 3 we present and study the fitting agorithms for the PP() and CPP modes, respectivey. Finay, in section 4, some concusions are drawn.. FITTING ALGRITHS FR THE PP() DEL The defining parameters of the PP() mode are ([]): r r Q ; Λ ; r r p [ r r ] () r + r where Q represents the infinitesima generator, Λ is the matrix of the Poisson arriva rates and p is the

2 initia probabiity vector of the underying arkov process. Assume interva-stationary PP() processes where X i represents the interarriva time between the i th and (i+) th ces. In this case, the distribution of the interarriva time X i is a second order hyperexponentia distribution (H ) [], with compementary CDF: ( Q Λ) x Fc () x pe ( Λ Q) Λe () ux ux qe + ( q) e, < q < and density function: ux ux f () x qu e + ( q) ue, < q < (3) The three parameters of the hyperexponentia distribution, u, u and q, can be reated with the PP() parameters by: + + r + r δ + + r + r + δ u ; u ; r + r u q (4) r + r u u u u ( )( ) where ( + r r ) + 4r r δ. The auto-covariance function, C[k], k, is given by ([]): C k E X E X X E X [] [( { } )( k + { k + } )] k + p( Λ Q) Λ{ ( Λ Q) Λ) ep}(. Λ Q) Aσ where k A σ ( ) r r ( r + r ) ( + r + r ) ( + r + r ) Λe (5) (6) In the foowing section we present two fitting agorithms for the PP() mode. Both procedures estimate the parameters u, u, q and σ.. The interarriva time distribution inference procedure In this approach, u, u, q are estimated by fitting the empirica and theoretica compementary CDFs and σ is estimated by fitting the empirica and theoretica auto-covariance functions. We have used non-inear fitting techniques (the NoninearFit function of ATHEATICA). This approach is different from the one adopted in []. Figure iustrates the appication of this procedure to one of the scenarios under consideration (to be described in section.3) Em pirica distribution function Approximating function histogram bin num ber Empirica correation coefficients Approximating function ag Figure : (Up) Fitting the interarriva time compementary distribution to an hyper-exponentia function; (Down) Fitting the auto-correation to an exponentia function The upper part of Figure represents the fitting of the compementary CDFs. As can be seen, both curves are very cose to each other. The ower part of Figure represents the fitting of the auto-correation functions. In this case, the approximation is not so good and the exponentia function can ony capture the mean behaviour of the empirica auto-correation. The parameters of the PP(),,, r and r, can be cacuated from the estimated parameters u, u, q and σ, through: [ q ( σ )( u u ) + σu + u + ξ ] (7a) uu { q( u u ) u } (7b) u q u u u u r r ( ) ( u )( u ) ( u )( + r u ) u where ξ [ q( σ )( u u ) + σu + u ] 4σu u. (7c) (7d)

3 . The interarriva time moments inference procedure In this approach, u, u, q are estimated by fitting the empirica and theoretica first three moments of the interarriva time and σ is estimated as in previous section. From equation (3), we obtain the Lapace transform of f(x): qu ( q ) f ( s) u + (8) s + u s + u The first three moments are given by the derivatives of (8) cacuated at s: q q m + qγ + ( q) γ (9a) u u m m ( q) q + + γ (9b) qγ ( q) u u ( q) 3 3 6q 6 γ (9c) 3 + 6qγ ( q) u u considering that γ i ui. Introducing the reative second moment r m m and the reative third 3 moment k m 3 6m, the H parameters are given by: m k r k k( r) r ( r) γ ( + r) (a) m k r k k( r) r ( r) γ ( + r) (b) m γ 3 q ( k 3r + 4r γ γ + 3 k k /(( k ( 4 6r) + r ( 3 + 4r) r k + k( 4 6r) + r ( 3 + 4r) ( 4 + 6r + k + k( 4 6r) + r ( 3 + 4r) ) + k + k ( 4 6r) + r ( 3 + 4r))).3 Resuts and discussion + / (c) The evauation of the inference procedures resorts to a superposition of traffic sources. We compare the ce oss ratio (CLR) and the average waiting time (AWT) obtained when feeding a buffer with constant service rate and finite capacity (i) directy with a set of individua traffic sources and (ii) with PP() traffic, where the PP() parameters where inferred from the traffic generated by the set of traffic sources of step (i), using the inference procedures described above. Each individua source is modeed by an Interrupted Poisson Process (IPP), and two types of such sources are considered: type, with mean N and FF durations of 35 ms and 65 ms, respectivey, and with a mean interarriva time in the N state of ms; and type, where the mean N and FF durations are 35 ms and 58 ms, respectivey, and in the N state the mean interarriva time is. ms. The output ink capacity is assumed as.5 b/s. Two sets of traffic sources were considered in this study: the first one consists of the superposition of type sources (homogeneous case) and the second one of the superposition of 8 type and type sources (heterogeneous case). These scenarios where aso considered in []. The resuts obtained for each scenario are represented in Figure and Figure 3. We wi refer to the resuts obtained when driving directy the buffer with the set of traffic sources as the origina case. Wq Ce Loss Ratio,,,,5,,5,,5, CLR (rigina) CLR (cdf method) CLR (moments method) Buffer Size Wq (rigina) Wq (cdf method) Wq (moments method) Buffer size Figure : Resuts for scenario (homogeneous case): (Up) comparison of the CLR vaues; (Down) comparison of the average waiting time vaues

4 Wq Ce Loss Ratio,,,5,,5,,5, CLR (rigina) CLR (cdf method) CLR (moments method) Buffer Size Wq (rigina) Wq (cdf method) Wq (moments method) Buffer size Figure 3: Resuts for scenario (heterogeneous case): (Up) comparison of the CLR vaues; (Down) comparison of the average waiting time vaues From Figure, we can see that for the homogeneous scenario both inference procedures provide good resuts. Concerning the CLR, the CDF method gives resuts that are amost coincident with the origina case. The moments method gives sighty worse resuts. Both curves corresponding to the inference procedures have neary the same sope of the curve obtained with the origina traffic. We note aso that, in this case, the CLR vaues obtained with both inference procedures are greater than the CLR corresponding to the origina traffic yieding upper estimates of the CLR. Regarding the AWT, the resuts obtained using the inference procedures are aso cose to the origina ones. The moments method, for exampe, matches amost perfecty the origina curve. The CDF method, on the other hand, underestimates the AWT, especiay for arger buffer sizes, but this underestimation never exceeds % of the corresponding origina vaue. bserving Figure 3, we see that for the heterogeneous case the modeing based on the inference procedures aways yied greater CLR and AWT vaues. The resuts obtained are gobay worse than the ones corresponding to the homogeneous case. This behavior is party due to a more accurate fitting of the auto-correation function in the homogeneous case, because this function has ess variabiity. 3. FITTING ALGRITH FR THE CPP DEL The CPP is a specia case of a arkov oduated Poisson Process. Like the PP a arkov chain moduates the CPP, but in the CPP the transition rate matrix has to obey the foowing form: a a a an an a a an! Q a a a circ( a) () N N 3 a a a 3! a aa " a N where [ ] a. In summary, if a i,j is the transition rate between state i and state j, then ai, j ai+, j+ i < N, j < N an, j a, j+ () ai, N ai+, Like the PP, the CPP has an associated vector! describing the Poisson arriva rate in each state γ [ γ γ γ N ]. The main advantages of the CPP, over the PP, is the possibiity of constructing ( Q,γ! ) from measured data without the necessity of soving an inverse eigenvaue probem which has no genera soution and the fact that CPP states are equiprobabe. 3. CPP inference procedure In this paper, the CPP mode construction methodoogy, first described in [3] and [4], was adapted and tested with different kinds of traffic streams. The entire methodoogy is based on the characterisation of a arriva rate. The foowing steps are required to construct a CPP mode from the measured data: (i) fitting the empirica autocorreation function with a weighted sum of exponentia functions; (ii) determine the circuant vector a! from the time constants of the exponentia functions, (iii) determine the rate vector! γ from the empirica distribution and auto-correation functions. At the first step, the empirica auto-correation R (τ ) is fitted to the weighted sum of exponentia functions p o R ( τ ) Ψ e (3) τ with Ψ rea non-negative. The cacuation of the Ψ and coefficients is based on the Prony agorithm. We have restricted R (τ ) to be a sum of two p

5 exponentia functions, to improve speed and because most traffic types have auto-correation with exponentia or hyper-exponentia behaviour. However, if required, more exponentia functions can be used in the fitting process. The Prony method does not assure non-negative Ψ vaues. In this case, the foowing procedure is used: (i) an expansion of the set is made, i.e., some new vaues are added using inear interpoation; (ii) a non-negative east squares agorithm, matching R (τ ), is used to cacuate new Ψ vaues, (iii) ony positive Ψ and its associated vaues are used (we denote the number of positive Ψ vaues by L). The second order statistics of a CPP are represented by! * N af! (4) and β! * N γf! (5) where F * represents the conjugate transpose Fourier matrix. Vector β! can be decomposed in a power vector! β!! F * Ψ N γ (6) and a phase vector!!! θ arg{ β} arg{ γf } * (7) In the second step, equation (4) is used to cacuate vector a!. The number of states N is pre-defined; generay states are sufficient. Vector! is a vector with N eements, where L < N are the cacuated in previous step and the remaining N L have zero vaues. At this point, both the position of the eements in the! vector and vector a! are unknowns. The soution of equation (4) is found by combining an index search agorithm (that changes the positions of the eements) and a phase I simpex agorithm. The positions of the Ψ eements in the Ψ! vector are the same as the positions of the eements in the! vector. To construct the vector! γ (third step) we need to determine an empirica vector for the rates associated with each state. This vector wi be denoted by! γ e. The empirica rates are found by inverting the empirica distribution function, under the condition that a states are equiprobabe, i.e., n γ en F, n N,,, N-. To obtain γ!, the function N n γ (8) en γ n has to be minimised. The γ n eements can be obtained by inverting equation (5): N γ γ + Ψ cos(πn / N θ ). (9) n Note that the Ψ vaues and positions have aready been cacuated in previous steps. 3. Resuts and discussion The inference procedure of the CPP parameters described in previous section was tested by comparing the CLR obtained when driving a buffer with finite capacity and constant service rate with (i) data generated (through simuation) by the origina CPP mode and (ii) data generated by the a CPP mode which was inferred (using the procedure of previous section) from data generated in step (i). We aso compared some statistics of the origina and modeed data: mean, standard deviation, maximum, minimum, Hurst parameter and peakedness. We tested the CPP mode (and its inference procedure) as an approximation to different types of traffic. We considered PP(), PP(3), PP(5), IPP, IDP, Pareto and Sef-simiar traffic modes. The comparison was made using the above procedure. The Hurst parameter (H. P.) and peakedness (Peaked.) were considered in order to assess the sef-simiar behaviour of the generated traffic streams. These two parameters were measured using the methods described in [5]. The first step in this test was to appy the inference procedure to traffic streams of 5 ces generated according to the severa modes referred above: CPP with states, PP(), PP(3), PP(5), IPP, IDP, Pareto, and Sef-simiar traffic with H.8 and H.6. Sef-simiar traffic was generated through the methods described in [6]. The second step is the generation of the CPP traffic using the resuts of the inference procedure. Here repicas of ces each are generated. The comparison of the statistics of origina and modeed traffic is made in Tabe. The tabe shows the mean and 95% confidence intervas corresponding to the repicas.

6 ean ax in Std H. P. Peaked. Buffer Len. Trans. Rate CLR CPP ± ±39.6 ± ± ±..7 ±.8 CPP 5 ces bps.63e- 5.E-3 ±.8E-3 PP() ± ±85.4 ± 857. ± ±.6.74 ±. PP() 5 ces bps.e-.44e- ± 3.9E-3 PP(3) PP(5) IPP IDP Pareto S-S H.6 S.SH ± ±8.8 ± 3 ± ± ± ± ±37.7 ± 665 ± ±.9.38 ± ± ±8.6 ± 3 ± ±.8.74 ± ± ±5. ± 35. ± ± ± ± ± ± ± ±..39 ± ± ±.6 69 ± ± ± ± ±6.5 6 ± ± ± ±.4.45 ±. Tabe : Statistics of origina () and modeed () traffic streams. The third step was to drive a buffer with finite capacity and constant service rate with the origina and modeed traffic streams, and estimate the CLR through simuation. The resuts are presented in Tabe. PP(3) PP(5) IPP IDP Pareto S-S H.6 S-S H.8 5 ces bps ces bps ces bps 5 ces bps ces.7 bps ces 8 Kbps ces 78 Kbps.4E-3 4.3E-3 ±.6E-3 3.8E-.36E- ± 7.3E-3 3.3E- 5.73E- ± 9.E-3 6.6E- 5.8E- ±.77E- 6.58E-3.45E- ± 4.5E-3 8.6E- 8.3E- ± 4.6E E-4 3.E-3 ±.3E-3 Tabe : CLR for the origina () and modeed () traffic streams. The appication of the inference procedure to CPP traffic, shows that the procedure can capture a major statistics of the origina traffic. The statistics of the origina traffic are cose to the confidence intervas of the modeed traffic. This is aso true for the CLR. In Figure 4 and Figure 5, we show the CDF and the auto-correation function of the origina and modeed traffic.

7 F(x) Figure 4: CDF for the CPP origina traffic rate (darker) and modeed traffic rate. Auto-correation Figure 5: Auto-correation function for the CPP origina traffic (darker) and modeed traffic. In Figure 6 it can be seen that the CDF of the modeed traffic has softer transitions than the CDF of the origina PP(3) traffic, refecting the existence of ony three states in the arkov chain. However the overa behaviour of the two CDFs is simiar. The same is true for the auto-correation function as represented in Figure 7. F(x),8,6,4, 3 Rate (ces/sec) , 5,8,6,4, rigina odeed rigina odeed Time Lag (sec) rigina odeed Rate (ces/sec) Figure 6: CDF for the origina PP(3) traffic (darker) and modeed traffic rate. It can be concuded that, under the conditions assumed in this study, PP(), PP(3), PP(5) and IPP traffic are we approximated by the CPP mode. Some differences observed in the CLR we beieve can be attributed to the number of generated ces not being sufficienty arge. Auto-correation Figure 7: Auto-correation function for the origina PP(3) traffic (darker) and modeed traffic. The above concusions aso appy to IDP traffic. The CDF curves, Figure 8, shows that the N state of the origina IDP traffic (with a rate of 4 ces/s) is not matched perfecty since the CDF of the modeed traffic is not abe to foow the sharp transition around 4 ces/s of the origina CDF. However, this mismatch around the peak rate does not seem to have a great infuence in the main traffic characteristics. Pareto traffic is not we modeed by the CPP. Pareto traffic has an auto-correation that is theoreticay zero but the CPP inference procedure can not refect this behaviour. The modeed traffic has higher auto-correation (Figure 9) as we higher Hurst parameter. The other statistics are within the confidence intervas. The CLR of the origina traffic is significanty ower than the CLR of the modeed one refecting the ack of correation in the Pareto traffic. F(x) ,,8,6,4, rigina odeed Time Lag (sec) rigina Rate (ces/sec) odeed Figure 8: CDF for the IDP origina traffic rate (darker) and modeed traffic rate. The CPP mode can not match the vaue of the Hurst parameter of the Sef-simiar traffic. It was seen that the H.P. of the CPP is aways smaer than the H.P. of the origina Sef-simiar traffic. The modeing process induces an increase in the variation of the arriva rate, Figure, as we as a higher autocorreation, Figure. This shows that this modeing process may not be appied to Sef-simiar traffic with ow rate variation.

8 Auto-correation Figure 9: Auto-correation function for the origina Pareto traffic (darker) and modeed traffic. F(x) Figure : CDF for the origina traffic rate with H.8 (darker) and modeed traffic rate. Auto-correation ,9,8,7,6,5,4,3,, rigina odeed Time ag (sec) Rate (ces/sec) - rigina odeed rigina odeed Time Lag (sec) Figure : Auto-correation function for the origina sef-simiar traffic with H.8 (darker) and modeed traffic. PP() was evauated as a mode for the superposition of IPP sources; the CPP was evauated as a mode for PP(), PP(3), PP(5), IPP, IDP, Pareto and Sef-Simiar traffic. The proposed agorithms were seen to be appropriate for the characterisation of AT traffic streams and in connection admission contro procedures. REFERENCES [] Kang, Sang H. and Dan K. Sung, Two-state PP odeing of AT Superposed Traffic Streams Based on the Characterisation of Correated Interarriva Times, 995. [] Fischer, Wofgang and Katheen eier-hestern, The arkov-oduated Poisson Process Cookbook, Performance Evauation, nº 8, pp [3] Hao Che and San-qi Li, Fast Agorithms for easurement-based Traffic odeing, IEEE JSAC, vo. 6, no, 5, June 998, pp [4] San-qi. Li and C. Hwang, n the convergence of Traffic easurement and Queuing Anaysis: A Statistica-atch and Queueing (SQA) Too, IEEE/AC Trans. Networking, Apr. 997, pp [5] ine Cagar, K.R. Krishnan and Iraj Saniee, Estimation of Traffic Parameters in High-Speed Data Networks, ITC 6, 999, pp [6] Z.. Yin, New ethods for Simuation of Fractiona Brownian otion, Journa of Computationa Physics, v.7 n., August 996, pp CNCLUSINS In this paper, we proposed and studied fitting agorithms for PP() and CPP AT traffic modes, which are specia cases of arkov oduated Poisson Processes (PP). Two fitting agorithms, both based on the ce interarriva times, were considered for the PP() mode: one fits the cumuative distribution and auto-covariance functions and other fits the first three moments and autocovariance function. The fitting agorithm for the CPP mode is based on the cumuative distribution and auto-covariance functions of the arriva rate. The