On generating self-similar traffic using pseudo-pareto distribution

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Shannon Cummings
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1 On generatng self-smlar traffc usng pseudo-pareto dstruton It has een shown n the lterature that self-smlar or long-range-dependant (LRD) network traffc can e generated y multpleng several sources of Pareto-dstruted and perods. In a contet of a packet-swtched network the perods correspond to packet tran packets transmtted ack to ack, or separated only y a relatvely small preamle (as defned n IEEE standard 802.3, for eample). perods are the perods of slence etween packet trans. Multple sources contrutng to resultng synthetc traffc trace may e thought of as ndvdual flows (connectons). It s reasonale to assume that packet szes wthn a connecton reman constant. Dfferent connectons, however, wll have packets of dfferent szes. To generate a Pareto-dstruted seuence of perods, one can generate a Paretodstruted seuence of packet tran szes. The mnmum tran sze s 1, whch corresponds to a sngle packet transmtted. Pareto dstruton has the followng proalty densty functon: P( ), (1) + 1 Where s a shape parameter (tal nde), and s mnmum value of. When 2, the varance of the dstruton s nfnte. When 1, the mean value s nfnte as well. For self-smlar traffc, should e etween 1 and 2. The lower the value of, the hgher the proalty of an etremely large. Fgure 1 shows the densty functons for varous values of. 1

2 a 1.1 a 1.4 a 1.9 Fgure 1. Proalty densty functons for Pareto dstruton wth a 1.1, 1.4, 1.9 Mean value of a Pareto dstruton eual s E ( ). (2) The formula to generate a Pareto dstruton s X PARETO (3) 1/ U where U s a unformly dstruted value n the range (0, 1] Very often t s desrale to generate a synthetc traffc of a predefned load. Ovously, the resultng load L s just a sum of loads L generated y each ndvdual source. Gven N sources, L N L 1 Thus, t s mportant to e ale to get a good estmaton of the load generated y one source. The load generated y one source s mean sze of a packet tran dvded over mean sze of packet tran and mean sze of nter-tran gap, or puttng t dfferently, t s a mean sze of perod over mean sze of and perods. (4) L (5) + Formula (2) gves the mean value of a true Pareto dstruton. However computers usng euaton (3) generate a pseudo-pareto dstruton. One of prolems comes from the fact that computers cannot generate artrarly large value. However, any true Pareto dstruton of suffcently large length wll have values that eceed the range generated y computers. Thus, what we have s a truncated-value dstruton. 2

3 Let s denote S to e the smallest non-zero value that unform random generator may produce. Then, the generated Pareto-dstruted values wll not eceed : (6) 1/ S Then, the mean value of a Pareto dstruton can e calculated as shown elow: E( ) f ( ) d d d (7) Susttutng (6) nto (7) we get E ( ) 1 S (8) Euaton (8) gves the mean value of a truncated-value Pareto dstruton. Now, f we are gven load L and the packet sze k for a gven source, we can fnd the mnmum value of the perod. Frst, lets fnd the mean value of perod. From euaton (2) we get L L (9) Let s denote M and M to e the mnmum and perods respectvely. We mentoned aove that the mnmum packet tran sze s just one packet,.e., M 1 Then, M M L 1 1 S k 1 S (10) L where s the shape parameter for the perods, and s the shape parameter for the perods. Denotng T and T, we get 3

4 M T T S 1 k T T S L (11) Thus, gven values for k, L,, and, the formula (10) gves us the value for M that would result n lnk load closer to L. However, f we generate traffc usng the aove formulas, we wll notce the mean values for and perods n the generated seres stll slghtly off. The prolem appears to e n the way computers generate Pareto-dstruted values (formula 3). Whle Pareto dstruton assumes contnuous sample space, computers generate dscrete values wth unform proalty. The Pareto-lke dstruton s acheved y havng hgher densty of samples toward lower end of the scale. The Fgure 2 llustrates ths dea. It shows proalty dstruton functon for the pseudo-pareto dstruton. Note that every value has eactly the same proalty of eng chosen. Also note that there are no values etween 12 and 16, or 16 and Fgure 2. Proalty densty functon for the pseudo-pareto dstruton If we uld dstruton functon y aggregatng samples over some wndow sze, we wll get a plot somewhat close to the one shown n Fgure 1. But stll, no matter how large our wndow s, at the tal end the dstance etween two neghorng ponts wll eceed the wndow sze. That means that some wndows wll contan zero samples, even f numer of samples approach nfnty. Of course, that ntroduces an error to the mean sze of and perods. To correct for ths error, we found that the calculated values and should e multpled y coeffcent C 4

5 0.027 C ( ) (12) Thus, formula (11) ecomes M T C T S 1 k T C T S L (13) On a fnal remark, f we choose, and to e the same, the euaton (13) wll reduce to 1 M k (14) L That, however, may lmt the usefulness of the traffc generator. It s very reasonale to assume that n real traffc, proalty of havng etremely large perod s hgher then the proalty of havng etremely large perod. That means that the shape parameter should e larger than. The aove heurstc coeffcent results n generated load eng very close to the specfed load wth all comnatons of, and. 5

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